Lessons In Electric Circuits
A free series of textbooks on the w:hosted by
subjects of electricity and ibiblio
electronics
Copyright (C) 2000-2020, Tony R. Kuphaldt
These books and all related files are published under the terms
and conditions of the Design Science License. These terms and
conditions allow for free copying, distribution, and/or
modification of this document by the general public.
A copy. of the Design Science License is included at the end of
each book volume. For more information about the License,
visit https://www.gnu.org/licenses/dsi.html
As an open and collaboratively developed text, this book is
distributed in the hope that it will be useful, but WITHOUT ANY
WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
See the Design Science License for more details.
Access individual volumes, | through
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Lessons In Electric Circuits
ft Volume !- DC
Book |Volumel| Volume Il - AC Volume Ill -
Volume: - a an
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Volume:
Edition:
Last [ath | ae
revised: | 27, 2007 January 18, 2006
Minor August
revision:| 28, 2015 Feb 17, 2020
Checkout the Socratic Instrumentation Project.
If you are interested in industrial instrumentation, Checkout
the book "Lessons in Industrial Instrumentation" at the
Socratic Instrumentation project. The project provides work
sheets too. You will also find links to public-domain textbooks
on subjects related to industrial instrumentation.
Checkout the Socratic Electronics Project.
We are sometimes asked for homework questions. While the
Socratic Electronics Project does not provide questions keyed
to specific chapters, it does provide questions related to
various electrical/electronic topics in the form of "work
sheets". The basic concept is to encourage active as opposed
to passive learning.
Checkout allaboutcircuits.
Have questions about electronics, math, physics, embedded
systems, programming? Need help with homework? Checkout
these and other forums at allaboutcircuits. Ask your question
at one of the forums. Curious about questions your peers ask?
Want to report errors you find in our text? Report errors at
feedback and suggestions forum. See example of an error
submission to allaboutcircuits.
Checkout Romanian utranslation.
Mihai Olteanu has a Romanian translation of Vol 1, "Lessons in
Electricity".
If you have a web site with a translation of "Lessons in
Electricity" into another language, contact us. We can put
your link here.
Edition numbers reflect major structural changes to a book
volume such as the addition of new chapters, the substantial
expansion of existing chapters, or a change in markup
language (source code formatting). | may also increment the
edition number of a volume due to the accumulation of many
smaller changes. For a volume under active revision, one
edition per year is normal.
"Last revised" dates reflect non-trivial changes only. Minor
changes | make such as typographical error correction and
stylistic changes to the text do not warrant increments to this
date. New topics added to the text, as well as any outside
contributions, are the minimum change level warranting a new
revision date. The "Minor revision" date reflects minor error
corrections: typographical, spelling, minor changes not
involving addition of new content. See changelog for details.
Please submit errors, typos, or suggestions to All About Circuits
> Forums > AllAboutCircuits.com - Feedback and Suggestions
allaboutcircuits-feedback, Feedback and Suggestions forum.
Like to see an example of an error submission to
allaboutcircuits.com? Otherwise, see Contacts section for
address to submit corrections.
Click here for a detailed changelog of all books.
Note to instructors:
My commitment to those using these texts as student
resources in instructional curricula is to never delete subject
matter content as the books evolve through succeeding
revisions and editions. New subjects will be added, and
existing subjects expanded in coverage, but | will never omit
"old" subjects. My experience is that even "obsolete" subjects
in electronics hold important lessons for students, and
sometimes serve to catalyze creativity in new design work.
Unlike publishers, who must consider the page count (printing
costs) of a book, my publication costs are zero. Instructors may
pattern their lesson plans around the subjects contained in this
book series without being forced to change their plans as the
series matures.
Interested in contributing to this project? Click
here.
News flashes (Reverse chronologic order)
January 18, 2010 Volume 6 Experiments: Ch 8 555 Timer
Circuits, new Chapter 8 completed thanks to Bill Marsden.
April 05, 2009 Volume 3 Semiconductors: Ch 4 Bipolar
junction transistors, completed, Ch 7 Thyristors, completed.
November 01, 1007 Thank-you to David Zitzelsberger, who
bears the distinction of being the second contributor to submit
an entire chapter! Go to Combinational Logic Functions in
Volume IV to see his considerable work. This brings the Digital
volume IV nearly to completion.
July 2, 2007, Volume 3, Semiconductors, incomplete chapters
and sections are being completed over the next year or two.
Chapter 1 is proofread and has a new "Attenuator" section.
Chapter 2 and 3 have been completed, but need proofreading.
Please submit errors and corrections to the forum thread at Ch
2. Warren Young has written "Input to output phase shift" for
chapter 8, Operational amplifiers. Read all about it. Expect
Chapter 4, "Bipolar junction transistors" in a few months. See
changelog. for short new text additions to Volume 2, AC. (D
Crunkilton)
June 15 2006; Volume 2, AC has been reformatted to look
more like a printed book. The PDF version has floating
captioned figures. Not much change in the appearance of the
HTML version. No plans to reformat the other volumes due to
the labor involved-- unless there is a lot of interest in printing
them. My best guess is that the most interest will be in viewing
the HTML and PDF's not printing. Some new content in the new
AC motors chapter. March 6;The pdf version of volumes are
now more navigable with hyperlinks- bookmarks. January 1; --
All volumes have a mini table of contents at chapter head, see
changelog. Volume 2 has a new AC motors chapter. (D.
Crunkilton)
June 21, 2005; revised October 30 -- All volumes have at
least minor corrections, see changelog. Volume 3 has a new
Shift Registers Chapter. Spice plots have been replaced by
Spice-nutmeg graphic plots, improving the appearance of
volumes 2, and 3.
July 2004 -- IMPORTANT -- PLEASE READ THIS! It has come
to my attention that | can no longer continue my role as project
coordinator and primary author for this textbook series. My life
has simply become too busy, and | lack the free time necessary
to do a good job administrating this project. See goodnews and
badnews for more details. Fortunately, the open-source nature
of this project has led others to develop it in different
directions, where it will continue to live. The best example of
this to date is AllAboutCircuits.com. Please pay them a visit to
see what neat things are being done with the books.
A huge thank-you to Dennis Crunkilton, who bears the
distinction of being the first contributor to submit an entire
chapter! Go to Karnaugh Mapping in Volume IV to see his fine
work.
At a reader's suggestion, | made a changelog for all the books.
This is a very good idea and | should have done it long ago! In
this changelog, you will find a complete listing of all the
changes made, and when.
Download the entire collection of books
in HTML format
oO oO
liechtml.tar.gz
<HTML> | <--- Click Here!
All volumes! HTML code plus graphic images in JPEG format --
about 36 megabytes in size, in .tar.gz format
Download individual volumes in PDF
format
Click on an individual volume above. A link near the bottom of
the volume table of contents page is provided for downloading
the PDF version, viewable or printable -- a few megabytes each.
Adobe Acrobat viewer can access the bookmarks in the table of
contents and index. Otherwise, the open source Xpdf viewer
works well, sans bookmarks.
Download COMPLETE source code for
the entire collection of books
liecsrc.tar
<--- Click Here!
All volumes! One file (liecsrc.tar) containing *src.tar.gz files
for each volume. Each of these gzipped .tar archives contains
all the makefiles, conversion scripts, SUbML text source, image
libraries, and graphic images (all formats) needed to compile
each volume. About 100 megabytes in its entirety.
Download MINIMAL source code for the
entire collection of books
o o
liectiny. tar
<--- Click Here!
All volumes! One file (liectiny.tar) containing *tiny.tar.gz files
for each volume. The difference between this source code
package and the one shown above is that this package
contains only one format type for each image (EPS for
schematics and illustrations, JPG for photographs), instead of
both formats (EPS and JPG) for all images. This archive is much
smaller (because the omitted EPS photographic image files are
huge!), but requires that you do a lot of image file conversion
to produce either HTML or PostScript/PDF output. About 8
megabytes in its entirety.
Some of the free software used in this
project
me)
GNU/Linux operating system: & (what else?)
: . , |Edited with
Vim text editor: improved. (ip
Xcircuit drafting program for illustrations, tables, charts, and
equations:
“tet oes ME = X Circuit
Gimp graphics
manipulation program (a Photoshop clone):
1 e
2
= . 7"
Ae
The Gir
HW
‘pulation Prod
Miscellaneous UNIX utilities, obtainable from the Free Software
De, al
Foundation:
You can download an Microsoft Windows executable of the sed
utility, necessary for processing source files for the type of
markup language used in this book project, here.
Spice version 2G6, a public-domain program used to simulate
analog circuits. Download a statically-linked executable for
Linux systems here (spice), or the following three files for
execution on MS-DOS systems: spice.exe, 32rtm.exe, and
dpmi32vm.ovl (keep these three files in the same directory).
wihosted by ibiblio
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[ $Id: dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]
4
Back to Master Index
Contributing to this
project
If you feel inclined to contribute to this project, please
contact us via email (See Contacts, below) with your ideas. |
will thoughtfully consider any and all constructive criticism,
suggestions, and content, no matter how small or
"insignificant" in your estimation. Your ideas are important,
as they allow this project to become better than anything
one person could make alone. Bear in mind, though, that
any work contributed to this project falls under the terms
and conditions of the Design Science License and is by
definition "copylefted" for maximum public benefit. Any
content you develop for this project is your intellectual
property, but may be freely copied and distributed by
anyone along with my work.
Listed below are several pages of information pertinent to
contributors. As with everything else on this project, you are
encouraged to submit suggestions for improvement
regarding contributor policy. Being that the whole
phenomenon of "open" books is rather novel, participants in
this project or others like it are pioneers of a sort. Together
we will explore this brave new world of writing and
publishing, figuring out what works and what doesn't as we
go! For questions others have asked try this thread at
www.allaboutcircuits.com.
Contacts: Your primary point of contact is now
(liecibiblio@ gmail.com), the maintainer of this archive. If
you want more information on the recent changes in
administration, read this thread at www.allaboutcircuits.com.
If you really need to contact Tony Kuphaldt, click on his
name above his avtar in the above thread. This will take you
to a page where you can send a "personal message" or
"email". You can alSo read Tony's comments on the index
page of this site, and follow the "good news" and "bad news"
links.
Project worklist
Writing guidelines for authors
Graphic image conventions
Document markup format
Design Science License Be sure to read this legal
document thoroughly before contributing to the project!
A note on software used in the books
One restriction beyond the Design Science License that | feel
compelled to place upon contributors is a prohibition against
the use of non-free software in the authoring of this book
series. The fundamental principle is this: anyone should be
free to "compile" the source code of this book series and
fully explore the circuit simulations shown therein without
having to pay for any software, or be bound by any legal
restrictions regarding copying or distribution. This does not
necessarily mean that all software need be copylefted
(open-source), but that it must be freely available and
executable by anyone.
Examples of unacceptable software use include showing
circuit simulations or general simulations in any of the books
using software such as Pspice, MultiSim, Saber, Matlab,
Mathematica, Maple, where readers could not explore the
same simulations without having to pay for the use of that
software. Also, if anyone modifies the book in such a way
that compilation of the source files cannot be done without
the use of non-free software (i.e. all source files translated
into Quark format, and released as such), this is
unacceptable as well. The use of free, but closed-source,
software within the text such as Constantin Zeldovich's
Winscope program is acceptable, because there are no
restrictions other than that its use being non-commercial
(commercial use requires a fee be paid to Dr. Zeldovich).
| will not prohibit the use of proprietary software such as
Photoshop, Illustrator, Visio, or AutoCAD for the creation of
illustrations to accompany contributed text, or with using
non-free text editing software to type the text, because no
one who reads these books or "compiles" the source files
into a readable format would have any need to use the same
software. Any software whose operation is discussed in the
text as an aid to understanding circuit analysis, though,
should be freely accessible to readers. Otherwise, some
readers will be excluded from the full educational benefit of
the books, and perhaps from contributing to the project, by
their inability to purchase the necessary software.
Having said this, though, | would prefer that all contributors
use the same application software that | do (most notably,
Xcircuit for illustrations), so that there is consistency in the
appearance of all the books, and so all developers will be
able to modify the source files thus created without having
to purchase expensive software.
This restriction regarding non-free software is not legally
binding. It is merely a standard that | will vigilantly maintain
with regard to accepting contributions to the "official"
version of the book hosted at www.ibiblio.org. If anyone
wants to convert the book to Quark format, and/or substitute
Pspice simulations in place of the existing public-domain
Spice software simulations, they are legally free to do so.
The Design Science License merely states that all source
files for the books, before and after modification by
contributors, be freely accessible to all.
Back to Master Index
PROJECT WORKLIST
A friend of mine has a sign hanging in his workshop that
reads: ° Projects are born pregnant." Like many projects,
this book series keeps growing and evolving, reproducing
itself in the form of new volumes and new chapters. Will it
ever be complete? Probably no, but it should always be
improving!
The following is a "to-do" list of work items for the book. For
each volume, work items are listed in order of my own
personal priority from first to last. Do not feel limited though,
merely by what /think should be done first. I'll take any help
| can get! If you think of a work item that isn't in this list, tell
me and I'll include it with the rest.
All volumes
e Make SubML markup language XML-compliant, or else
go to a different markup language entirely.
e Add section links to top of each chapter page, to
improve navigation and content display. mini-TOC now
at top of each chapter of html! and pdf (DC).
e Add section links to the main index page of each
volume, to improve navigation and content display.
Edit to improve readability of text, especially for those
with limited English proficiency.
Convert all plain-text SPICE plot analyses to true graphic
format using the nutmeg postprocessor utility. Mostly
complete (DC).
e Write instructions for compiling book from downloaded
"tar" archive files. README added to main directory
(DC).
e Volume 3, Semiconductors has many missing chapters
and sections. Need help here
Something else I've wanted to do for each volume is to make
a series of practice problems (complete with answers) for
readers to test and hone their skills on. As an electronics
instructor, I've already done this for my college curriculum,
but unfortunately it had to be done on school time and with
school computer equipment, which means | cannot "open
source" it like | can the contents of this book series. What I'd
rather not have is a slew of multiple-choice or numerical
answer problems like so many textbooks, but rather
problems engaging higher levels of thinking (synthesis and
evaluation), complete with detailed answers explaining
problem-solving strategies and different ways of
approaching a problem.
Practice problems might be better located in a separate
volume (volume VII ?) rather than at the end of every
chapter, as some of the volumes are getting pretty big
already. The DC volume already exceeds 500 pages when
printed on 8-1/2 x 11 paper, so I'd rather not add bulk if |
don't have to.
Volume I - DC
e Write "Electric Motors" chapter.
e Discuss strain gauge "Rosettes" and Anderson Loops in
the "Electrical Instrumentation Signals" chapter (#9).
Discuss RTDs and Thermistors in the "Electrical
Instrumentation Signals" chapter (#9).
Expand coverage of Magnetism (chapter 14) to include
magnetic circuit calculations.
Include a discussion of Murray and Varley loop testing in
the "DC Metering Circuits" chapter.
Edit section on circuit grounding in Safety chapter -- tree
touching power line wire may not be best illustration of
why we ground power systems.
Volume Il - AC
Write "AC Motors" chapter (#13). Completed (DC).
Add section(s) discussing modulation to chapter 7
(Mixed-Frequency AC Signals), including AM and FM
sidebands.
Make "Filters" chapter (#8) more mathematically
rigorous.
Upgrade SPICE plots in "Filters" chapter (#8) using
Nutmeg graphical image output instead of plain-text
output. Completed (DC).
Add "Scott-T" transformer discussion to Transformers
chapter (#9). Completed (DC).
Add a section or two discussing "Smith charts" to
chapter 14: "Transmission Lines"
Discuss balanced versus unbalanced transmission lines
in chapter 14, and the operation of "balun" transformers.
Re-take screenshots of Winscope in time-domain and
frequency-domain modes. Designate the new screenshot
files 22*.ong, according to the naming convention for
screenshot image files.
Volume Ill - Semiconductors
e Complete chapter 6: "Insulated-Gate Field-Effect
Transistors."
e Complete chapter 3: "Diodes and Rectifiers." Completed
2007 (DC)"
e Complete "Practical Analog Semiconductor Circuits"
chapter. See section headings within this chapter for an
idea of the content I'm planning on. The completion of
the first section ("Power supply circuits") should be top
priority in this chapter.
Write "Active Filters" chapter (#10).
Write "DC Motor Drives" chapter (#11).
Write "Inverters and AC Motor Drives" chapter (#12).
Complete chapter 2: "Solid-State Device Theory." What
I'm looking for is a chapter that explains the quantum
mechanisms of semiconductor devices in as much detail
possible without involving calculus. Impossible?
Perhaps, but it's worth a try! | cringe every time | read
an introductory text on semiconductors that attempts to
describe electron and hole interaction in terms of
classical physics. . .(TK) Completed Fall 2007. However,
we need to keep up with new developments in this field
(DC)
Complete chapter 4: "Bipolar Junction Transistors."
Complete chapter 5: "Junction Field-Effect Transistors."
Complete chapter 7: "Thyristors." Discuss 4-quadrant
firing of TRIACs.
e Complete chapter 8: "Operational Amplifiers." Add
section on chopper-stabilization of amplifiers.
e Expand coverage of electron tubes in chapter 13.
Volume IV - Digital
e Complete chapter 7: "Boolean Algebra."
e Write "Combinational Logic Functions" chapter (#9).
Completed by David Zizelsberger, 2007
Complete chapter 11: "Counters."
Write "Shift Registers" chapter (#12). Completed 2005.
Expand coverage of microprocessor architecture and
function in chapter 16.
e Expand coverage of digital memory to include more
modern technology in chapter 15, especially optical and
magnetic media.
Volume V - Reference
e Write a chapter on basic algebra techniques, especially
equation-solving and "story problem" solving.
e Write a chapter on oscilloscope usage.
Volume VI - Experiments
e Write new experiments for any and all chapters.
This is perhaps the easiest way for someone to contribute to
the book: write a short electric/electronic circuit experiment,
complete with parts list, diagrams and illustrations, and
instructions. A lot less work than writing a whole chapter or
chapter section!
4
4
WRITING GUIDELINES FOR
AUTHORS
1. The intended audience includes self-taught
experimenters, advanced high school students, two-year
community/technical college students, and beginning
four-year undergraduate students. Assume no prior
knowledge on the part of the reader, except a basic
understanding of algebra, and whatever else has been
taught in the book series prior to your section or
chapter.
2. There is no limit to how complex the subject matter
becomes in this book series, only in how complete the
coverage is and how fast the complexity increases.
Whatever you contribute to a book, make sure there are
no "gaps" in the subject matter from basic electrical
theory (volume I, chapter 1) all the way to whatever it is
you're writing about. Never assume that the reader will
be able to follow all significant cognitive "leaps" made in
your writing. This is probably the most important thing
I've learned as an educator! It is better that you
thoroughly explain every little step at the expense of a
lengthier chapter than to rush through explanations and
leave some readers unable to follow along.
3.1 recommend structuring your prose in such a way that
the reader is "led through" the lesson as though they
were being taught by an instructor in a laboratory
session. Present hypothetical situations and practical
problems to provoke thought. Pose rhetorical questions.
Make the reader feel as though they are right there with
you, building circuits and observing the results (make
frequent use of first-person plural tense: "we," "our,"
etc.).
. Identify and reference major connecting ideas
throughout the book series. Examples include:
o Kirchhoff's and Ohm's Laws
o Scientific method and circuit troubleshooting
strategies
o Signal feedback
o Fundamental calculus principles (derivative and
integral)
. Avoid colloquial language and any other references not
likely to be understood by people of other nationalities
and cultures.
. Although this is not intended to be a math book, many
abstract mathematical principles become much clearer
when applied to circuits. Logarithms (exponential
functions in RC and L/R circuits), complex numbers (AC
voltage, current, and impedance), and calculus
principles (derivative in capacitor and inductor
calculations) are examples of mathematical concepts at
the far end of the expected mathematical proficiency
level of the intended audience. When there is
opportunity to apply and clarify these concepts via
circuit design and analysis, please do so! Eventually, |
would like to take the mathematical complexity as high
as differential equations, and do so in the context of
analog op-amp circuits (analog computer circuitry). Are
there any "old-timers" out there with practical
experience programming differential equations into
analog computers, who would like to contribute their
expertise to the project?
7. When introducing a new term, italicize it and leave an
index reference for it above the introductory paragraph.
Set off the term in quotation marks for its second use, if
there is a need to reinforce the term's novelty. Format
the term normally after that.
8. Please be so kind as to spell-check and grammar-check
all submissions prior to emailing them to me!
4
4
GRAPHIC IMAGE
CONVENTIONS
1. In keeping all source files available for copy to the user,
all images created by Xcircuit (*.eps) will be maintained
in the distribution, and all photographs (*.jpg) will as
well.
2. Keep all Xcircuit library files (*.lps) in the distribution,
for the benefit of all Xcircuit users.
3. When using Xcircuit to draw equations, here are some
general style rules:
o The final .eps image should not exceed 480 pixels in
width. At 100 dpi print resolution, this makes for a
4.8 inch wide picture. After conversion to PNG
format, the image should not exceed 600 pixels in
width. Included here are two files, sample.eps and
sample.png as examples of the maximum width I'd
like all illustrations to have.
o Use Helvetica font for descriptions, worded
quantities (i.e. "Seventeen"), notes, etc.
o Use Times New Roman font for numbers, component
labels (i.e. Rigags Criterrs Lchoker Qi), and equations.
o Use Symbol (Greek) font for special characters.
Some really special characters (like the "R"
reluctance symbol and the "angle" symbol for polar
notation can be found in the font map of the Symbol
set. Access this map in Xcircuit by pressing the
"backslash" key when in the text mode.
o Try to type all single-line equation expressions as a
single, uninterrupted string of text. If an equation is
written as multiple strings of text pieced together,
sometimes conversion from .eps to another graphic
format will reveal this "splicing."
o Use Courier font only for Boolean variables, where
the monospacing works well for referencing the
locations of complementation bars.
4. When processing images taken with a digital camera,
follow these steps:
o Take image in highest-quality JPEG format available.
o Using Gimp, convert camera image into TIFF format
for lossless manipulation.
o When done manipulating photo, use Gimp to scale
image to 640x480, then "Save As" an EPS image
with Width and Height image size parameters both
set to "100".
o Re-scale the image to 480x360, then "Save As" a
JPEG image. Use quality setting of "0.75".
5. When using Nutmeg to process SPICE output for true
graphic images, follow these steps:
o Run spice with the -r option, to produce a "rawfile"
for Nutmeg to process. (spice -r old.raw < input.cir)
o If using SPICE2g6, you must use the sconvert utility
to convert the old rawfile format into the new
format, like this:
sconvert o old.raw b new. raw
o Run Nutmeg (nutmeg new. raw). You will have to
manually enter the data points to be plotted. When
plotting AC values, be sure to use the "m" modifier
so that the polar magnitude gets plotted. For
example, to plot the voltage at node 3, type: vm(3)
rather than v(3), or else Nutmeg (and SPICE3F5!)
will only plot the "real," rectangular value.
o If uSing an X-windows based graphic environment
(i.e. UNIX/Linux), you may capture the screen image
using the /mport utility:
import junk.png
o Use Gimp to cut away the "Quit" and "Hardcopy"
buttons, then save as the same format (PNG), under
the desired name. Save also as an EPS file with a
width of 100 mm.
6. File names: Each graphic file has a numerical, five-digit
name, and it exists in two of three different file formats.
Encapsulated PostScript (.eps) is for generating
PostScript and PDF output using LaTeX, while PNG (.png)
or JPEG (.jpg) is for generating HTML output. The choice
between using PNG or JPEG depends on the type of
image. PNG is preferred for images created by Xcircuit
and computer screenshots, while JPEG is preferred for
photographic images.
XXXXX.epS
XXXXX. jpg
O0Oxxx = Volume I (DC) -- Schematic diagrams (.eps source)
10xxx = Volume I (DC) -- Tables and Equations (.eps
source)
20xxx = Volume I (DC) -- Computer screenshots (.png
source)
40xxx = Volume I (DC) -- Artwork (.jpg source)
50xxx = Volume I (DC) -- Photographs (.jpg source)
O2xxx = Volume II (AC) -- Schematic diagrams (.eps
source)
12xxx = Volume II (AC) -- Tables and Equations (.eps
)
22xXxx = Volume
source)
42xxx = Volume
52xxx = Volume
03xxx = Volume
source)
13xxx = Volume
source)
23xxx = Volume
source)
43xxx = Volume
53xxx = Volume
04xxx = Volume
source)
14xxx = Volume
source)
24xxx = Volume
source)
44xxx = Volume
54xxx = Volume
Q01xxx = Volume
source)
11xxx = Volume
(.eps source)
21xxx = Volume
(.png source)
41xxx = Volume
51xxx = Volume
05xxx = Volume
(.eps source)
15xxx = Volume
(.eps source)
25xxx = Volume
(.png source)
45xxx = Volume
55xxx = Volume
source)
II (AC)
II
II
(AC)
(AC)
III (Semi)
III (Semi)
III (Semi)
III
LET
(Semi)
(Semi)
IV (Digital)
IV (Digital)
IV (Digital)
IV (Digital)
IV (Digital)
V (Reference) --
V (Reference) --
V (Reference) --
V (Reference) --
V (Reference) --
VI (Experiments)
VI (Experiments)
VI (Experiments)
VI
VI
(Experiments)
(Experiments)
-- Computer screenshots (.png
-- Artwork (.jpg source)
-- Photographs (.jpg source)
-- Schematic diagrams (.eps
-- Tables and Equations (.eps
-- Computer screenshots (.png
-- Artwork (.jpg source)
-- Photographs (.jpg source)
-- Schematic diagrams (.eps
-- Tables and Equations (.eps
-- Computer screenshots (.png
-- Artwork (.jpg source)
-- Photographs (.jpg source)
Schematic diagrams (.eps
Tables and Equations
Computer screenshots
Artwork (.jpg source)
Photographs (.jpg source)
Schematic diagrams
Tables and Equations
Computer screenshots
Artwork (.jpg source)
Photographs (.jpg
To answer the question, "why do the Volume | (DC) files
begin with 00 and Volume V (Reference) files begin with
01?", when | first began writing this book, | only
intended to have two volumes, and "Reference" was the
second volume. By the time | realized that all | had to
write on circuits wasn't going to fit well within a single
volume, | had already created hundreds of files for the
"Reference" volume, beginning with the prefix "01". So, |
made the second volume (AC) files begin with "02" and
SO on.
When submitting graphic image files for inclusion into
the book(s), name the files according to your own
convention (i.e. "imageO1.eps," "image02.eps," etc.). Do not
try to follow my numbering scheme, as you would have
to know what the last file number is in order that your
filename isn't the same as another graphic file already in
use. Just send them to me with your own filenames and
I'll re-name them to fit in with all the other files.
4
4
DOCUMENT MARKUP
FORMAT
Submissions from contributors
When submitting content for inclusion into the "official"
distribution at www.ibiblio.org, the preferred formats are
plain text or hand-coded HTML. Please, please do not send
me HTML files created by web page software such as
FrontPage or Netscape Composer! Also, do not send me
content in any word processor format (i.e. Word,
Wordperfect). If you use a word processing program to write,
please export your work in plain text (.txt) format. The
reason for this is because | must perform some rudimentary
conversions of your text into the markup language used for
this book project, and this is easier to do if the text you send
me is in a more primitive form.
If you wish to make LARGE contributions to the project
(multiple chapters, or translations of the English text into
other languages), | would recommend that you learn to write
your documents(s) using the SUODML markup language, so
that | do not have to re-type large portions of your work. You
may learn more about the SUbML markup language in the
last section of this page. [Click Here! ]
If you are not familiar with what a markup language is, refer
to the second-to-the-last section of this page before reading
anything else. [Click Here! ]
History of markup languages used tin
“Lessons In Electric Circuits" book
project
There is a history of markup languages and formats used in
the creation and presentation of this book series that
readers may find interesting (or at least amusing!). Here, |
will describe how the project began, where it has gone,
where it is now, and hopefully where it is going with regard
to markup.
At first, the entire book was written in plain-ASCIlI text
format. That's right: plain vanilla text, with not a single
graphic image to be found, except for "ASCII art"
illustrations and graphs. Believe it or not, there's a
surprising amount of illustration that may be done using
nothing but monospaced font and the characters found ona
keyboard. Take for instance this "ASCII art" circuit schematic:
R3
Te ORG re a aya a eet ae aes LNIEND NE SO Sees = t
| | 1.5k |
aac /
Battery - R2 \ 2.2k R4 \ 10k
ae / /
\ \
| R1 | |
atghe at LNIN ee Rb ree ee ene as oe Ieee +
1k
The rationale behind ASCII formatting was universal
readability, and small file size. Anyone, using practically any
computer in the world, can view and edit plain ASCII text
files! Also, | was hosting the book on my own personal web
page, with very limited hard drive space, and file size was an
important issue. However, the limitations of "ASCII art" soon
became apparent, and | was forced to go with something
better or else be severely limited in what | could present in
the books.
Later, in 1999, | tried converting the plain text files into
Microsoft Word format, so that at least the paragraphs would
not have to be rendered in Courier (ugly!) font. The
illustrations were still rendered in ASCll-art, but the book
text appeared in Times New Roman font, which was much
easier to read.
It was then that | learned the limitations of word processors
with regard to large documents. | was hoping to use the
capabilities of Microsoft Word to provide page numbers for
the book, but was disappointed at the results. | seemed to
have very little freedom in how the page numbers appeared
on the paper, and | noticed how much variance there was
between the text as it appeared on the computer screen,
and the text as it appeared on paper after printing (margins,
paragraph breaks, etc.). Additionally, | could find no way to
get Word to generate an index, or a table of contents, both
of which | knew would be important for a book to have.
Worse yet, formatting with Word limited the electronic
readership of the book to those who had Microsoft Word on
their computers. Word is an expensive program, and the
“Wordpad" mini-processor that comes with Microsoft
Windows doesn't always read Word files properly. All in all,
the experience with Microsoft Word was negative in general,
and | did not foresee better results using any other brand of
word processor.
Then, in the May of 2000, | read about Yorktown High
School's Open Book Project in an issue of Linux Journal
magazine. Managed by Jeffrey Elkner, the Open Book Project
is a site intended to host "open" textbooks for free,
educational use. | immediately contacted Jeff and requested
permission for my book to be hosted on their server instead
of my own webpage. He agreed, and began to offer advice
on how to improve the book's appearance. One of his
students at Yorktown HS, Jason Starck, became involved with
the task of translating the plain-ASCII text into HTML format
for better appearance. At this point, there were still no real
graphic images (still "ASCII art" diagrams), but the book's
appearance and ease of navigation were vastly improved.
Over the 2000 summer break (July-September), | worked
feverishly on the task of creating real graphic images for the
book using Xcircuit, an X-Windows based drafting program
intended for drawing electronic schematic diagrams. By Fall
quarter of 2000, the book had a whole new appearance.
In October of 2000, the Open Book Project moved to the
servers of www.ibiblio.org, away from Yorktown High School's
servers. Accessibility and visibility increased dramatically
with this relocation, and with those improvements it became
more important to make the book's appearance as
professional as possible. One major problem with HTML
formatting was its poor translation to printed paper copy. My
students needed a paper version of the book, and printed
HTML lacked all the necessary elements for paper
navigation: page numbers, table of contents, and an index.
From past experience | knew that going to a word processor
format such as Microsoft Word was not going to help me
here. What | needed to do was use a markup language
designed to produce printed copy, as opposed to HTML
(Hypertext Markup Language) which is intended only for
electronic presentation.
The Open Book Project was already collaboratively
developing a computer programming textbook by Professor
Allen Downey called "How to Think Like a Computer
Scientist," using a language called LaTeX as the official
source markup standard. LaTeX makes wonderful printed
copy, but is not directly viewable over the internet and thus
requires translation to HTML for online viewing. In discussing
some legal issues with Richard Stallman over email, | was
directed toward a markup language called Texinfo that was
supposed to address both needs: one source language that
translated easily to TeX for printed copy and HTML for online
viewing (as well as to a special hyperlinked info format
intended as a "man" page substitute for UNIX systems).
Being that Texinfo was the official markup language for
Stallman's Free Software Foundation documentation, |
thought it fitting that it be used to create an open-source
textbook, and | committed the book series to that style of
markup.
In email conversation with Jeff Elkner, a new markup
language called DocBook was brought up. Like HTML,
DocBook is an instance of SGML, with a feature set
specifically designed for rendering technical literature. It
promised to be the Holy Grail of markup for textbooks,
generating professional-quality print and web-viewable
output from a single source markup format, with just about
every feature imaginable. Unfortunately, neither Jeff nor |
knew how to use DocBook yet, so he remained committed to
LaTeX as the official markup language of Downey's "How to
Think..." book while | remained with Texinfo for the
“Lessons...” book series. Another "open book" author,
David Sweet, encouraged me to consider DocBook as the
markup language of choice for my text, but after reading
Norman Walsh and Leonard Muellner's "DocBook, The
Definitive Guide", | was put off by the language's
complexity.
As the year 2000 rolled over into 2001, | realized that
Texinfo was not as great a solution to the markup language
problem as | originally thought. It suffered from two major
disadvantages: an inability to render superscripts and
subscripts, as well as Greek characters. In electronics and
mathematical work, these three features are almost essential
to proper text formatting. Up to this point | had tolerated
Texinfo's limitations in this area because it did such a fine
job of creating both printed output and HTML output from a
single set of source files. | considered doing what Jeff Elkner
was doing with Allen Downey's programming book
(switching to LaTeX as the source markup language), but
decided against it because they were having to write their
own conversion software to translate into HTML the way they
wanted it.
By the summer break of 2001, | Knew | had to abandon
Texinfo for something else. Having learned more about
DocBook in the mean time, | became convinced it was the
ultimate markup language for what | was doing, but despite
significant effort | could not get the parsing software to work
as it should on my home computer. Now I'm no Linus
Torvalds, but I'm not exactly a slouch when it comes to
computers, either. Even if | did manage to get DocBook fully
operational on my home computer, | reasoned, chances were
that many others would not be able to get it to work on their
computers, thus effectively barring some people from being
able to use the book to its full potential. Also, if | were to
switch to DocBook markup, | would have to make sure that
all the proper parsing software was set up on ibiblio's server,
so that | could continue my policy of uploading just the
source files over the internet and have the ibiblio computer
“compile” them into HTML and PostScript. The alternative --
to compile all the source files on my home machine and
upload the finished files to ibiblio's server -- would magnify
the size of my uploads by several times.
At this point, | had familiarized myself with several markup
languages in my search for the "perfect" solution: HTML,
TeX, LaTeX, Texinfo, groff, Qwertz, and DocBook. There were
many similarities in structure between these markup
languages, although syntax varied greatly between them. It
became apparent that the structures were similar enough to
allow for search-and-replace translation from one format to
another, so long as only the basic features of the individual
languages were used. This is analogous to discovering
several different sooken languages where only the words
differed, but the grammar was approximately the same.
Given this fortuitous situation, it becomes technically
possible to translate from one markup language to another
using simple search-and-replace routines, just as it would be
possible to translate flawlessly between the hypothetical
spoken languages using nothing but a multilingual
dictionary.
So | thought to myself, "why not make my own markup
language loosely based on DocBook, structured in sucha
way that translation to any of the other markup languages
requires only search-and-replace substitutions?" In effect, |
would identify whatever structures were common to
DocBook, LaTeX, and HTML, and design SGML/XML-style tags
to represent them. The result would be a markup language
limited to those features common to the intersection of the
different languages’ structures, but very easily translated to
any of those common markup languages for final output. If |
designed this language as close as | could to the structure of
DocBook, it would be just as easy to convert the files to
DocBook at some later date with the same search-and-
replace approach. In honor of its intended purpose, | decided
to call my language SUbML, meaning Substitutionary
Markup Language.
It was then that | discovered a remarkable little program
called sed, which stands for stream editor. Its singular
purpose is to execute bulk search-and-replace operations on
any ASCII file, according to scripts written using UNIX
regular expressions. | developed the SUbML language and all
the necessary sed scripts to translate a SUbML file into Tex,
LaTeX, and HTML over the 2001 summer break, as | was
taking a course on comparative religion at a local
community college. SUBML became the official markup
language for my class papers that quarter, and | used the
experience to "debug" the language before applying it to
the "Lessons..." book series.
Since then, SUbML has remained the official markup
language of the "Lessons In Electric Circuits" book series.
Being that the sed executable file and associated conversion
scripts are quite small, and sed is available in versions for
many different computer operating systems, the SUbML
language Is very portable. It supports all the normal
chapter/section/subsection structuring you would expect
from a textbook markup language, plus full Greek alphabet
support and sub/superscripting. It does not, however,
support either tables or mathematical equations, so | use
graphic illustrations generated with Xcircuit for these
features.
| eventually plan to move to DocBook, but I'm waiting fora
couple of things to take place. First, DocBook must become
easier to set up and use on a home computer. Every once in
a while I'll try to parse a simple "Hello, world" DocBook file,
but I still can't get the @*# *$%! thing to work. Secondly,
I'd like to see the DocBook standard (especially the XML
version of it) reach a point of greater stability. At present,
there are so many changes planned in the vocabulary of
DocBook (new tags, plus tags destined for obsolescence)
that | fear writers will be forced to constantly update their
source files to keep up with the latest version of DocBook.
So, what exactly is a markup
language?
Let's start at the beginning: The ASCII (American Standard
Code for Information Interchange) standard is a set of binary
codes, 7 bits for each text character, that describe every
letter in the English alphabet, both lower-case and capital,
plus numbers, punctuation marks, and other miscellaneous
symbols. Every text character that you see displayed ona
computer screen is, at some level in the computer system,
represented by a 7-bit binary number according to the ASCII
standard. The capital letter "A", for example, is the binary
number 1000001. The number "6" as a single character in
the ASCII standard is represented by the binary number
0110110. The "equals" sign (=) is the binary number
0111101. The exclamation point (!) is the binary number
0100001.
Just as Morse Code provides a digital means of transmitting
text, the ASCII code standard provides a much fuller means
of digitally transmitting, storing, and displaying text data. A
file comprised of strings of these 7-bit codes (+ 1 bit to
"pad" each character up to eight full bits, or one byte per
character) will appear as text characters when viewed by
any word processor, text editor, or text viewer software,
because all these different computer programs have been
designed to recognize the ASCII code set. Imagine a world
where everyone understood the same language. This is how
computers are with regard to ASCII.
However, ASCII is as limited as it is universal. If ASCII were
all we had to encode text documents in digital form, the
documents you would see on computers would be very dull.
All characters would appear in the same, boring font,
without any form of emphasis such as /ta/ics, boldface, or
underlining. There could be no SUPEs“"PLING OF oi rinting, ANC
there could certainly be no Greek characters such as "pi" (tt)
or "beta" (8).
When you use a word processing program such as Microsoft
Word to format a text document, the file generated by that
program is a mix of ASCII codes in addition to a lot of binary
codes that do not correspond to the ASCII standard, the
latter used to delineate all the special formatting functions
such as italics, boldface, underlining, page margins, font
type, font sizes, etc. If you were to try to view a word
processor file using a text editor, or some other computer
program that only understands ASCII codes, all the non-
ASCII codes will appear as "gibberish." In fact, the majority
of the document is comprised of these special codes due to
all the detail that is necessary to describe how the text is to
appear on the page.
Different word processor manufacturers invented their own
"standards" for these formatting codes, and the result is that
a document composed using one word processor may not be
viewable using a different word processor. In later years,
word processor programs became more adept at translating
between formats (Microsoft Word versus WordPerfect versus
AmiPro...), but the translations were often far from perfect,
much like translations between different human languages.
Because all the word processor file formats would appear as
gibberish when viewed with a text editor (or with another
word processor that couldn't understand all the formatting
codes), the person trying to read or modify the document
would be left helpless without the proper software. They
could not, for instance, "manually" re-write the codes in the
document file so that their word processor could understand
it. This is one major limitation of word-processor document
formatting.
Far more significant than this, however, is the fact that word
processor file formats tend to be very concrete rather than
abstract; specific rather than general. In computer
programming terms, they would be classified as very "low-
level" languages. This makes them difficult to translate to
other formats, even by a computer. Imagine the comparison
between translating a "high-level" verbal command ("Go to
the store and purchase a loaf of bread!") from English to
Japanese, versus translating a very detailed ("low-level")
document from English to Japanese describing every detail
involved with the task of buying bread ("Go to the store,
open the front door, walk down the bread aisle, choose a
loaf, walk to the cash register, .. ."), especially if this
document is replete with idiomatic expressions and
colloquial terms. Obviously, the more abstract ("high-level")
command would be far easier to accurately translate than
the concrete ("low-level") set of instructions. Computer
programmers are very familiar with this problem. It is far
easier to translate a computer program between high-level
languages (example: from Fortran to Pascal) than between
low-level languages (example: from Intel 80386 assembly
language to Motorola 68020 assembly language).
The computer programming solution to this problem is to
write software in a high-level language, where all the
"codes" resemble a human language such as English, then
have another piece of software called a compiler or an
interpreter automatically translate these high-level codes
down to the very verbose, specific, low-level codes that the
computer will need to run the program. The high-level code
that the human programmer types is exclusively composed
of ASCII characters: the same characters you see ona
standard keyboard. As a result, the written code fora
computer program looks every bit as dull as a plain-ASCIl
text document, but this simplicity of formatting means that
any programmer, anywhere in the world, using any kind of
computer, will be able to read the code and modify it if they
can obtain a copy of it, and do so with far greater ease than
if the code were low-level microprocessor codes (assembly
language).
Another benefit of high-level computer programming is
portability. |\deally, a high-level program need only be
written once, then it may be compiled (translated) to as
many different low-level microprocessor languages (Intel
x86, Motorola 68xxx, SPARC, whatever), for as many
different operating systems (Microsoft Windows, Unix, BeOS,
whatever), as needed. The concept of "write once, run
many" is the Holy Grail of computer programming, and is
attainable only by writing software in high-level, as opposed
to low-level, languages.
In summary, a markup language is a standardized
set of high-level instructions, written using ASCII
character sequences within a plain-text document,
describing how the text is supposed to appear in
final form. Here is a simple example, showing plain (un-
marked) text first, then HTML markup code for formatting
the text to use different font styles, then the final output:
Plain text, with no markup:
This is a some text that I wish to format.
I would like to use italics, boldface,
and underlined fonts in this short paragraph,
as well as typeset a math statement: 3%2 = 9.
HTML "source code" markup for the above
paragraph, viewed as plain text:
<p>
This is a some text that I wish to format.
I would like to use <i>italics</i>, <b>boldface</b>,
and <u>underlined</u> fonts in this short paragraph,
as well as typeset a math statement: 3<sup>2</sup> = 9.
</p>
Source code, as interpreted and presented by your
web browser:
This is a some text that | wish to format. | would like to use
italics, boldface, and underlined fonts in this short
paragraph, as well as typeset a math statement: 32 = 9.
When viewed as plain text, the HTML source code for this
brief paragraph appears as sets of matching "tags" using
"less-than" (<) and "greater-than" (>) characters, plus
letters, to represent font style commands. A text editor
would present this document showing all the HTML tags, as
seen in the middle rendition of the paragraph. You web
browser, however, interprets those special character
sequences as commands to obey, and renders the enclosed
text accordingly.
HTML is not the only markup language in existence. Another
markup language, intended for creating professional paper
copy (print), is called TeX. Here is how TeX would be used to
format the same sample paragraph:
TeX "source code" markup for the above paragraph,
viewed as plain text:
This is a some text that I wish to format.
I would like to use {\it italics}, {\bf boldface},
and \underbar{underlined} fonts in this short paragraph,
as well as typeset a math statement: $3%2 = 9$.
To translate this TeX source code into something printable,
you would have to process the source file using a computer
program called TeX (freely available, by the way) which
would output another file cast in a "DeVice Independent"
(.dvi) format, then use a program called "dvips" (also free) to
convert the .dvi file into Adobe PostScript (.ps) format for
printing to a PostScript printer, or with a PostScript
interpreter program such as GhostScript (also free). Believe
me, this whole process is actually easier than it sounds, and
the quality of the final print is superb!
The markup language | use for the "Lessons In Electric
Circuits" book series is called SUbML (SUBstitutionary
Markup Language), an invention of my own. SUbML would be
used to mark up the sample paragraph like this:
SubML "source code" markup for the above
paragraph, viewed as plain text:
<para>
This is a some text that I wish to format.
I would like to use <italic>italics</italic>,
<bold>boldface</bold>,
and <underline>underlined</underline> fonts in this short
paragraph,
as well as typeset a math statement:
3<superscript>2</superscript> = 9.
</para>
Documents written in a markup language generally include
as little mechanical detail (margins, font sizes, font types) as
possible, and when they do it is in the form of ASCII
character codes that may be seen by anyone using any kind
of text editor or word processor program, so that nothing is
ever "hidden" from view. Like high-level computer
languages, document markup languages also require that
there be special software available to "compile" or
"translate" the markup codes into some final format suitable
for presentation, such as PostScript or PDF. Ideally,
documents written using a markup language are completely
portable: that is, any single document may be automatically
converted to any number of electronic formats for
presentation, without any further intervention from the
author, because the document uses general terms rather
than computer- or printer-specific terms to specify structure
and appearance.
Writing documents using a markup language requires more
technical knowledge on the part of the author, though.
Instead of just clicking on a little icon in a word-processor
environment to select italicized text, for instance, the author
must know what code(s) to insert into that portion of the
document to command the use of an italicized font. Then,
the author must "compile" their source document using
software designed to translate the markup codes into a
presentation format. Computer programmers find this
development cycle (write, compile, review, debug) a natural
process. Others may not.
Another very important advantage of composing a
document in a markup language instead of using a word
processor, from the perspective of "open source" projects, is
that nothing is hidden from anyone wishing to modify or
duplicate the document's structure. For instance, | have
seen many fantastic-looking documents composed using
Microsoft Word, and wondered to myself, "How did they do
that?" Also, | have been given Word documents in electronic
form that | wished to modify, but could not without
destroying the original markup because | was not as
proficient with Word's features as the person who made it.
When you read a document composed using a word
processor, you can see the results, but you cannot see what
functions and methods were used by the original author to
obtain those results.
| remember an older word processor program named
"WordStar" equipped with a "reveal codes" feature that
could show you some of the special formatting codes within
a document used to make it look the way it did. This was a
step in the right direction, but still not as powerful a concept
as a true markup language, where all formatting codes are
available for viewing, copying, and/or modification via a
simple text editor.
The "openness" of a markup language makes it possible for a
person to learn how to write their own documents in that
language just by viewing what others have written: an
impossibility with any word processor document. For
example, most of my knowledge of HTML has come from
viewing the markup codes of web pages written by other
people, rather than by reading tutorials on the subject.
Markup languages naturally foster learning and sharing,
values held in high esteem in the "open source" culture.
Because markup languages differ little from formal computer
languages, spelling and context of the markup codes is
critical. This makes it possible to write a document that has
"bugs" in it: one that does not appear the way the author
intended it to, due to some type of syntactical or error with
the markup tags. Because the author does not see the
results of the code as they type it (the code must be
compiled before the results may be viewed), errors are not
immediately evident. This can be frustrating.
Markup languages, however, prove their worth when any
large document projects are involved. Documents written in
a word processor format become more and more difficult to
manage (revising, expanding, publishing) as the size of the
document increases. Documents written in a markup
language, however, become easier to manage as they
increase in size. In other words, a word processor is probably
the easiest way to write and publish a business letter, but
using a markup language is probably the easiest way to
write and publish a book.
The SubML Markup language
Rather than present a tutorial on SUbML here, | will provide
links for you to download all the necessary sed scripts, plus
a tutorial on SUbML written in that language. To use any of
these files, you will have to have sed installed and working
on your computer system. A Microsoft Windows-compatible
executable version of sed may be downloaded here. All
Linux and other UNIX systems should come equipped with
sed as a Standard utility. If installing sed on a Microsoft
system, make sure you have the "sed.exe" executable file
installed in a directory on your hard drive where your
operating system knows to find it (C:\Windows is a good
place).
Tutorial on using SUbML -- uses all features of the language
(tutorial. sml)
SubML-to-HTML conversion script (smi2html. sed),
SubML-to-LaTexX conversion script (sml2latx.sed)
SubML-to-text conversion script (sml2txt. sed)
TAR archive file containing all of the above, and more
(cmar0301. tar)
When you have the tutorial file, sed, and the sm12html.sed
conversion script downloaded on your home computer, try
converting the tutorial file into HTML with this command
(typed in the "command line" environment, with a final
"Enter" keystroke at the end of each command you type):
sed -f sml2html.sed tutorial.sml > tutorial.html
You should be able to view the resulting tutorial.html file
using Internet Explorer, Netscape Navigator, or any other
web browser software. It should look like this.
To generate LaTeX code from SubML source code, use sed
like this:
sed -f sml2latx.sed tutorial.sml > tutorial. latex
To generate LaTeX output, of course, you will need to have a
LaTeX/TeX compiler installed on your computer, along with
all the associated LaTeX/TeX macro and font files. Packaged
installations are freely available over the internet from a
variety of sources. Once this is all installed on your
computer, you may translate the tutorial.sml file into .dvi
format by first converting it into LaTeX format as shown
above, then running this command:
Latex tutorial. latex
The resulting file, tutorial.dvi, may be viewed with any DVI
file viewer (such as xdv/ on UNIX systems), or converted into
PostScript format using the free utility dvips like this:
dvips -o tutorial.ps tutorial.dvi
If Adobe PDF is more to your liking, you may convert the .dvi
file to PostScript using a special option of dvips like this:
dvips -Ppdf -o tutorial.ps tutorial.dvi
... then, convert the resulting PostScript file into PDF using
another free utility, ps2pdf.
ps2pdf tutorial.ps tutorial.pdf
If successful, you should end up with a file named
tutorial.pdf, viewable with Adobe's Acrobat viewer, or any
free PDF viewer software such as Ghostview or xpdf.
For the "Lessons..." book series, | used a set of Makefiles to
manage all these command-line utilities, and automate the
packaging of the output files into a final product that people
can download and use. Anyone is free, of course, to
download the source files for the book series and peruse the
Makefiles for themselves to see how this works.
4
The SubML markup language
Copyright © 2001-2006, Tony R. Kuphaldt
Introduction
SubML stands for Substitutionary Markup Language. Similar in
structure to an SGML-based language, SubML is intended for simple
text formatting with very few frills, but providing the capability of
standard font emphasis modes, itemized lists, and image inclusion.
SubML is designed so that it may be translated into practically any
markup language with nothing more than some search-and-replace
commands (hence the term substitutionary), executed in the sed
stream editor. Rather than rely on complex translational algorithms
(i.e. a Perl or Python script), the philosophy here is to design ease of
conversion into the structure of the original markup so that any fool
can write a sed script to convert to any new markup. So far, the
following conversions are provided in a set of sed scripts supplied with
this tutorial:
¢ SubML to TeX
¢ SUbML to L4T_X
e SubML to HTML
e SubML to plain text (ASCII)
More conversion routines are planned. As far as | can see, none of them
should present any unordinary difficulties in conversion. | simply
haven't got around to writing and testing all the scripts yet. These
include:
e SUbML to nroff/troff/groff
SubML to Texinfo
SubML to DocBook (SGML and/or XML)
SubML to Lout
SubML to Qwertz
Also, it should be fairly easy to write an XML DTD for SubML, making it
directly readable by XML-compatible browsers and other software.
Platform compatibility is limited only to the availability of a sed binary
to perform the conversion. And since sed is such a widely used and
robust utility (free, too, thanks to the Free Software Foundation!), this
should not be a problem. I've successfully “compiled” SubML
documents on both Linux and Microsoft Windows 95 with equal ease.
Characters usually interpreted as escape characters in other markup
languages like \, & $,%, |, ~, ~, and _ are handled without special
tagging as well (100% of the time, too -- this makes SUbML worth
$1,000,000 & that's not all!). The only characters SUbML requires you
to specially code (not type verbatim in your source document) are the
< and > symbols, simply because SubML itself uses them as escape
characters to mark the beginning and end of tags.
Levels of sections under each chapter
This is text contained in the first true section of this tutorial.
This is the first subsection (titlebar)
This is text contained in the first subsection of this tutorial.
This is the second subsection (titlebar)
This is text contained in the second subsection of this tutorial.
This is the first subsubsection (titlebar)
This is text contained in the first subsubsection of this tutorial, which is
within the second subsection.
Gallery of inline text formatting tricks
In this section, we will explore the various inline (embedded within a
sentence) formatting commands provided by SubML.
Note that this may not be the fanciest array of formatting commands,
but it should suffice for most common formatting requirements.
If the standard SubML philosophy is followed, additional formatting
Capabilities may be included at a later date. The only real restriction is
that whatever formatting capability is added must be translatable to
the desired output type (T—EX, HTML, DocBook, etc.) using nothing more
than simple search-and-replace algorithms.
Sub- and super-scripting
This is a test of the SUDgcripting aNd supers“"PtINS Capabilities of SUDML.
This is useful to create simple mathematical (-2°3 = -0.125) and
chemical (H50, 9,U2?°) expressions.
While the following displays in html, it does not display properly in ps
or pdf due to tex/latex errors when using the normal <subscript>,
<superscript>, as above. Instead, we use <subscript2>,
<superscript2>.
10!0910(Vi)
10!0°910(Vi/Vo)
Un-comment line here to create error.
Note the <math> </math> around the whole subscript and
superscript line in the tutorial.sml source above.(You need to be
looking at tutorial.sml) Only use this if you have tex/Latex errors, no ps
or pdf. Complex mixtures of both superscripts and subscripts are a
reason.
Boolean overline negation
Boolean negation (not) is supported in LaTeX by the \overline{ }
command, available in the math environment. HTML provides no such
support for overline. However, it is customary in some texts to indicate
negation with a single quote (') post-fixed to the negated variable.
Thus, we support Boolean negation in SML with the <ob> and </ob>
tags (overbar) enclosing the negated variable.The sed processed Latex
output will show (dvi, ps, pdf) overline negated variables, the html has
the post-fixed single quote form of negation. Equations with any
negated variables must be surrounded by <math> and </math> tags
to activate the "math" environment for latex.
Any extensive use of Boolean equations should be xcircuit images so
that real overlines will be available in html as well as LaTex. The
methods here are meant to support simple in-sentence Boolean
expressions, not free-standing equations.
<math>Y = (<ob>A</ob> + <ob>B</ob>)</math> This markup
gives the result below:
Y =(A' +B') This result.
<math>Y = <ob>(<ob>A</ob> + <ob>B</ob>)</ob>
</math> This markup gives:
Y =(A' +B')' This result with long overline is due to outer tags.
The span of the overline is analogous to the span of a pair of bold tags.
While the parenthesis are not necessary in the LaTeX rendition, they
are mandatory in the "single quoted" html version to indicate the
extent of the negation.
Some other examples follow:
Y =(A’ +B')' =((AB)' )'
Y =(A' B'C'ED')' Incorrect in LaTeX, we wanted broken bar BC
like AB.
Ye(A* B CVED*) This is incorrect in LaTteX, OK on html. We
wanted broken bar between ABC.
Y =(A' B' C'ED')' Like this by putting spaces between ABC. See
tutorial.sml
Y=((A(BC')')')' This is better as an xcircuit image; html is
difficult to follow.
Emphasis fonts
Italicized, boldface, and underlined type are also available in SUDML.
Special dashes
The regular dash, such as that used for hyphenation, looks-like-this. A
dash specifically used for subtraction is typeset using a special SUbDML
tag, so that 5-3 (math dash) looks distinct from 5-3 (ordinary dash).
Some people don't care too much about this, so use this tag at your
discretion.
Sometimes it is useful to show a pair of dashes -- not the “em-dash”
used in setting off a section of text like this -- but a real pa/r of dashes.
In this case, another special SUbML tag has been created to do this --
and you just read over it! | use it to denote series-connected electronic
components in symbolic form. For example, a pair of resistors (R; and
R>) are connected in parallel with each other, but together they're in
series with R3. Symbolically, | represent such a configuration like this:
(R3//R2)--R3.
Block formatting
An important feature I've found in document processing is the ability
to typeset a literal segment of text. That is, a section of print ina
monospaced font with all normal paragraph formatting features of the
target markup language turned off.
One common usage of this feature is for the typesetting of computer
programming code. An example follows:
File listing: hello.c
#include <stdio.h>
int main(void)
printf("\nHello, world! \n");
return (0);
The dots are inserted manually within the SUBML document to “set off”
the literal block of text from the rest of the document. Also, the leading
dots (at very left of each line) help overcome a problem I'm having
with T—_X formatting where leading spaces get discarded and
everything ends up smashed against the left margin.
Without the dots, it looks like this:
#include <stdio.h>
int main(void)
printf("\nHello, world! \n");
return (0);
}
The "set off" leading dot may be replaced by the <sp> tag to avoid the
dot in your literal block.
#include <stdio.h>
int main(void)
printf("\nHello, world! \n");
return (0);
Another kind of block formatting is the inclusion of offset quotations.
Note the following example:
"Vague and insignificant forms of speech, and abuse of language,
have so long passed for mysteries of science; and hard or
misapplied words with little or no meaning have, by prescription,
such a right to be mistaken for deep learning or height of
speculation, that it will not be easy to persuade either those who
speak or those who hear them, that they are but the covers of
ignorance and hindrance of true knowledge." - John Locke
Italics may also be added to “set off” a quotation from the rest of the
text, especially in HTML. Combining the italic and bold tag sets inside
of the quotation tag set accomplishes this goal nicely:
"Vague and insignificant forms of speech, and abuse of language,
have so long passed for mysteries of science; and hard or
misapplied words with little or no meaning have, by prescription,
such a right to be mistaken for deep learning or height of
speculation, that it will not be easy to persuade either those who
speak or those who hear them, that they are but the covers of
ignorance and hindrance of true knowledge." - John Locke
While perhaps not a true block-formatting feature, itemized lists can
be created using SubML. Take the following example:
e This is the first item
e This is the second item
e This is the third item
e Isn't this fun?
In the spirit of simplicity, | haven't created the option of enumerated
lists, indented lists, or anything fancy like that within the language of
SubML.
Including graphic images in a document
Graphic image inclusion is perhaps the best feature of SUbML. Note the
following example:
Have a nice day!
You must be sure to specify an HTML-compatible image in the markup
code. This means an image file specified with a filename ending in
.png, .jpg, .bmp, or .gif (three-character extensions only: .jpg, not
.jpeg!). For TeX or LAT_X output, there must be an Encapsulated
Postscript image file .eps in the same directory, but not specified in the
markup code.
For example, the markup code necessary to place the "happy face"
image shown above is as follows:
<image>test.png</image>
Two versions of the image exist: test.png for inclusion into the HTML
output, and test.eps for inclusion into the T-X or L‘T;-X output, but only
the HTML-compatible file need be specified in the SUbML source code.
Have a nice day!
This Is a fine caption.
Below is the markup code necessary to place the "happy face" image
with a caption shown in figure above. A "Figure x.x" string precedes
the caption in LATEX. It also generates LAT;-X code for a //lable test.eps,
which is used to reference the figure. The caption is included in the
html without the "Figure x.x" designation.
<image>test.png<caption>This is a fine caption.</caption></image>
Note that in the previous paragraph, we reference "figure 1.1" or
"figure above" in tutorial.ps and tutorial.html, respectively. The markup
below, between the ref tags, is for referencing the above image as a
figure. The image name, test.png, is a symbolic reference, replaced by
1.1, 1.2, etc., during "latex tutorial.latex" processing. Put the image
name between the tags.
See figure<ref>test.png</ref> for a "happy face".
If you read about Latex figures, labels, and references, you will find
that the label is completely arbitrary. The only requirement is that the
//ref command must call out the label associated with the figure. In our
case the sml2latx.sed file contains substitutions which fill in the image
number, eg: test.png, 02041.png, for the label. Thus, we do not have
to manually fill that in for each of our images, which we may or may
not reference. If we do wish to reference a figure we reference the
image number. It may be necessary to run "latex tutorial.latex" twice
to resolve the references.
As an option for html, a word may follow the image name as below. Eg.,
"test.png above" will put "above" into the tutorial.html. We have no
way to generate numbered figures in the html. So, figure above, figure
below, may be usefull. View tutorial.html vs tutorial.ps for "figure 1.1"
vs "figure above", respectively. Here we reference figure again, but
only in tex/latex, no html as in the above markup. The markup below
shows the optional html word.
See figure<ref>test.png above</ref> for a "happy face".
In the case of html, we do not have the referencing facilities provided
by L4T_X. The best we can do is refer to the figure above or below as
shown in the above markup.
Unrelated, take a look at tutorial.html to see how we have indented
the above markup code without a leading dot. Compare to previous
unindented markups.
See caution in next section: only one reference per line (pair of <ref>
tags). Else, split line with (return).
Labeling a figure
Do not confuse the "Labeling" with the caption on a figure. In most all
cases you can skip this section and let the sed processing
automatically generate the label which the "figure" requires so that it
may be referenced. The automatic label is the same as the image file
name (eg 02221.png). The previous section covers this. The only
reason to read this section is in the rare event that a second instance
of a figure is being used. In which case, it needs a new, unique, not
automatically generated label, not the (automatic) label for the first
instance of the image. You may also skip this, if there is no caption for
the figure. We will give the second instance of the image a unique
label so that it will not be confused with the first instance when we
reference it. See Figure below
Have a nice day!
Caption for the second instance of our image.
Note that our new figure is captioned as Figure above. The caption is
different than the caption for the previous Figure 2nd-above. We are
able to assign a label to it:
<image>test.png<caption>This is a fine caption.</caption>
<Label>newtest.png</label></image>
Note that the above markup must be on one line. It is too wide for our
page. So, we wrapped it. It may wrap in the text editor. But there
cannot be a (return) except at the end of the line. The sed script
processes a line at a time for each command. We process all the tags in
the line with one command for image, caption, label, and ref tags.
Once it has a label, we can distinguish it from the other figure by
referencing it the same way we reference other figures (just a different
label):
If we compare the above image caption for newtest.png to the
previous caption for for test.png, we find that both specify the same
image, test.png. The latter has a different label "newtest.png" This is
just a label. There is no image by that name.
See figure<ref>newtest.png above</ref> for a 2nd “happy face".
Caution, a limitation of the sed script for caption processing is that
only one figure reference ( eg.: <ref>newtest.png</ref>) may be
processed properly per line. Typically, there is only one line, all the
words up to the end-of-line between <para> tags. If we need more
than one <ref></ref> in a paragraph, the paragraph may be split into
two or more lines between the two paragraph tags. See tutorial.sml for
an example of this in the paragraph "Note that our new figure is
captioned..."
Scaling an image
Once in along while, an image which is of satisfactory size in the html
version of a document is too small in the LaTeX produced pdf
document. The solution is to make the image the "right size" for the
html document, then scale it to a suitable size in the LaTeX file. This is
done by a sed (string substitution program) command in sml2latx.sed.
When the sml source is processed, a scale factor is added to the .latex
file, but not the .html file.
The scale factor must be added to the .sml as a modification between
the <image> tag and the file name of the image. This markup
produces Figure below).
<image>[ scale=0.5] test.png<caption>This is scaled down in LaTex.</
caption><Label>smalltest.png</label></image>
The image command must be on a single line, a CR only at the end,
none in the middle. Though, we wrapped it above for appearance. And,
don't put two on one line- split into two lines. This scale parameter,
[scale=0.5], only works if the <caption> tags are used, due to sed
script limitations. The same is true of the <label> tags. The caption
tags generate a figure number, even if there is nothing between the
tag. There must be a unique label between the <label> tags, else
LaTeX give an error. There must be no space between [scale=0.5] and
test.png. LaTeX doesn't want a space in front of the image file name. It
must be like this [scale=0.5]test.png.
Have a nice day!
This 1s scaled down in LaTex.
Special characters
In addition to special logos like TeX, SUbML provides for certain often-
used characters of the Greek alphabet.
The ratio of a circle's circumference to its diameter is symbolized by
the Greek letter “pi,” which SUbML represents like this: 1. The area of a
circle is given as A=nr?. Not many people realize that the standard
symbol tt is actually the /owercase version of the Greek letter. The
capital version looks like this: , and it does not represent the same
thing in mathematics.
But there are other useful Greek characters for us to use in SUDML as
well. When SubML is converted to plain ASCII text, some of the Greek
characters like and p will be represented by the closest-resembling
Roman (English alphabet) character available. If there is no Roman
character close enough, the Greek character's name will be spelled in
parentheses. T_X, on the other hand, is very Greek-literate and
requires no “fudging” to obtain perfect representation. HTML output
from SubML conversion renders these characters using Unicode. In
order for a web browser to properly display them, it must be set up
with Unicode character support. For your viewing pleasure, we have:
Alpha (lower-case): a
Beta (lower case): B
Gamma (lower case): y...... Gamma (capital): F
Delta (lower case): 5...... Delta (capital): A
Epsilon (lower case): €
Varepsilon (lower case): €
Zeta (lower case): ¢
Eta (lower case): n
Theta (lower case): 0...... Theta (capital): ©
Vartheta (lower case): 9
lota (lower case): t
Kappa (lower case): K
Lambda (lower case): A...... Lambda (capital): A
Mu (lower case): U
Nu (lower case): v
Xi (lower case): €...... Xi (capital): =
Pi (lower case): T...... Pi (capital):
Rho (lower case): o
Varrho (lower case): @
Sigma (lower case): 0...... Sigma (capital): 2
Varsigma (lower case): ¢
Tau (lower case): T
Upsilon (lower case): v...... Upsilon (capital) Y
Phi (lower case): @...... Phi (capital): ®
Varphi (lower case): o
Chi (lower case): x
Psi (lower case): W...... Psi (capital): V
Omega (lower case): W...... Omega (capital): Q
non-breaking space 11112223 3 34 4 4 4
Tau (lower case): T
bigtriangledown: V
oplus, exclusive or sign: ®
almostequal: =
approxequal, approximately equal: =
neq, not equal: #
plusminus, plus or minus: +
cdot, centered dot, times dot: -
leq, less than or equal: s
geq, greater than or equal: =
times, times sign: x
registered, registration sign: ®
Another special symbol available in SUbML is the Z symbol (<angle>),
used in mathematical statements to designate an angle. This is useful
for expressing complex numbers in polar form. Take for example this
impedance: 500 Q Z -34.61°. By the way, the way | typeset the
"degree" symbol is with a superscript letter "o".
Other mathematical symbols included in SUDML's vocabulary are the
integration symbol (J), partial derivative symbol (0), and the infinity
symbol («). Here are some examples of these symbols in use:
V=fQdt+C
ox/ot
co is bigger than BIG!
Note that you cannot show upper and lower integration limits for a
definite integral using the "{" markup tag. It is useful for crude in-line
formatting of an integral equation only. If you want to show lower and
upper integration limits in a SUbML document, you must use a graphic
image -- sorry!
For special characters used in other languages (Spanish, French,
German, etc.), the following are available in the SUbML vocabulary:
¢ "a" with grave (back prime): a...... A
¢ "a" with acute (forward prime): 4...... A
"a" with circumflex (caret):a...... A
"a" with umlaut/dieresis/tremat: 4...... A
"a" with tilde: 2...... A
"a" with "ring" above: a...... A
"c" with cedilla:¢...... C
¢ "e" with grave (back prime): @...... E
¢ "e" with acute (forward prime): é...... E
e "e" with circumflex (caret): é6...... E
¢ "e" with umlaut/dieresis/tremat: 6...... E
¢ "i" with grave (back prime):i...... |
¢ "i" with acute (forward prime): f...... i
¢ "i" with circumflex (caret): f...... i
- "i" with umlaut/dieresis/tremat:7...... |
>
e "n'" with tilde: A...... N
¢ "o" with grave (back prime): 0...... O
¢ "o" with acute (forward prime): 6...... O
¢ "o" with circumflex (caret): 6...... O
¢ "o" with umlaut/dieresis/tremat: 6...... O
e "o" with tilde: 6...... O
"u" with grave (back prime):U...... U
¢ "u" with acute (forward prime): U......U
e "yu" with circumflex (caret): G...... U ;
e "u" with umlaut/dieresis/tremat: U...... U
Inverted question mark ¢
Inverted exclamation mark j
So, now you may impress all your Espanol-speaking amigos with the
following phrases in your documents:
"sDdnde esta el cuarto de bafo?"
"iMas cerveza, por favor!"
"sPuede indicarme dénde esta en el mapa?"
"Por favor, digale tu amigo que voy a llegar cinco minutos tarde."
"Aqui tiene mi casa."
And when your friend asks you this...
"sQué procesador de textos usted utiliza?"
... you may respond with pride:
"No utilizo un procesador de textos.jEn lugar, utilizo un lenguaje
de marcas!"
Tex/Latex only, HTML only
Tags <tex>, </tex>, <htmlo>, </htmlo> are provided to include text
from .sml selectively only in .latex, .tex or only in .Atml. The <tex>
</tex> tags mark text that is only included in the .latex and .tex
outputs of "sed -f sml2latx.sed" and "sed -f sml2tex.sed". Text that is
only to be included in the .html is marked of by the <htmlo>,
</htmlo> tags.
This following markup is to only show text in tutorial.latex and
tutorial.tex. Following the markup, see text in tutorial.latex,
tutorial.tex, but not in tutorial.html
<tex>This only shows in tutorial.latex and tutorial. tex</tex>
This following markup is to only show text in tutorial.html. Following
the markup we see the text in tutorial.html but not tutorial.latex,
tutorial.tex.
<htmlo>This only shows in tutorial.html</htmlo>
This only shows in tutorial.html
Given both a portrait and landscape version on a same-size image, a
practical application of the <tex>, <htmlo> tags is to selectively
direct those images to tutorial.latex or tutorial.html. We do not actually
do this in tutorial.sml, but show the markup. For example, we wish to
send the landscape version of a big image to the html version of our
book so that readers do no have to rotate their monitors. This
landscape is too big for our .latex, .tex, .ps, .pdf 6-inch wide book
pages. We cannot reduce the size of the landscape, which would be
unreadable. So, we rotate our big landscape to a portrait. It started out
4-inches tall and is now 4-inches wide. It fits side ways nicely ona
book page. We have not reduced the size, just rotated it. A book reader
can easily rotate the book to view the large image.
<htmlo><image>landscape. png</image></htmlo>
<tex><image>portrait.png</image></tex>
Hyperlinks and targets
link at end of this section.
sample target located here, jump here from a link (Click) near the
bottom of this section
The <url>, </url> tags provide clickable links to URLs in both the html
and pdf versions of a document. The pdf is derived from LaTex. Internal
links are provided by <hyperlink>, </hyperlink> tags, which link to
targets defined by the <hypertarget>, </hypertarget> tags. The
syntax for these tags takes the following form:
<url>url_lLink[ text] </url>
<hyperlink>Link[ text] </hyperlink>
<hypertarget>Link[ optional text] </hypertarget>
The "link" for <hyperlink> must match the "link" at the
<hypertarget> to actually jump there on clicking. The links for
<hypertarget> in the case of multiple targets needs to be unique- no
two targets the same. The "link" for<hyperlink> and <hypertarget>
may not contain any underscores, eg., invisible link. Though, it works
in html, the pdf links will be dead. And, no errors are generated. The
<url> and <hyperlink> text will appear colored in both html and pdf
when viewed. The <hypertarget> text is not colored, and is optional.
The following markup provides an external link to a URL in both html
and pdf documents:
Go to
<url>http:www.ibiblio.org/obp/electricCircuits/index.htm[ Lessons in
Electric Circuits] </url>
to learn about electricity.
Go to Lessons in Electric Circuits to learn about electricity.
Why are there no quotes around the URL above? While the quotes are
needed in html code, they are not used in L“T_X. Therefore, we do not
include them here. They are added by the sml2html.sed script to the
html document.
Click this link to jump to invisible target at end of section. At the top of
this section click on "link at" to also jump to the end of the section.
The following markup provides the link below it to the top of this
section:
<hyperlink>LINK[ Click] </hyperlink> to go to target at top of section.
Click to go to target at top of section.
Here is the markup for an "invisible" target at the end of this section:
<hyperlink>invisibleTarget[ ] </hyperlink>
Bibliography and citations
The <thebibliography>,</thebibliography> tags mark a section of
text to be treated as a list of bibliographic references. Contained
therein are individual bibliographic entries delimited by <bibitem>
</bibitem> tags. Theses entries may be referenced from the body of
the main text by <cite></cite> tags. The syntax of these tags is as
follows:
<thebibliography>
<bibiten[ ref] text</bibitem>
<bibiten[ ref2] text2</bibitem>
</thebibliography>
The purpose of this paragraph is to reference the bibliography below.
This paragraph is broken into several lines terminated by a return.
[footnotes] You should skip to the bibliography and look at the first
entry, here.[latex] The second entry in the bibliography is here.[1d]
Note that the fourth bibitem contains a url to link to home of this
project.[4]
The bracketed reference, [ref], in the bibitem needs to be matched by
the corresponding citation reference <cite>ref</cite> in the body of
the text. See above and below. In LaTeX, this is usually an easy to
remember mnemonic. This is replaced by bracketed a number, eg. [2],
in the processed LaTeX version of the document. However, the html
version of the document will not have numbers unless the reference is
a number, eg. <cite>4</cite>. The bibliography in html is a
numbered list. However, these numbers do not necessarily correspond
to the sml bibitem reference. Use numbers instead of mnemonics in
the bibitem reference for numbers in the html.[4].
A sample bibliography with four items follows:
Bibliography
1. [latex]Helmut Kopka and Patrick W. Daly, A Guide to LaTex:
Document Preparation for Beginners and Advanced Users
(Addison-Wesley, Reading, MA, 1999), 3rd. ed.
2. [footnotes]The html sed processing only handles one citation per
line. Though, LaTeX can handle more.
3. [1d]B. C. Freasier, C. E. Woodward, and R. J. Bearman, “Heat
capacity extrema on isotherms in one-dimension: Two particles
interacting with the truncated Lennard-Jones potential in the
canonical ensemble,” J. Chem. Phys. 105, 3686--3690 (1996).
4. [4] Kuphaldt, Tony R., Lessons in Electric Circuits in the open book
project at ibiblio.org
Note that the last entry above contains a url. The whole bibterm must
be on one line, only one return, at the end.
What SubML won't do
SubML is designed to be a simple markup language, and as such it
lacks certain advanced features found in other, more capable
languages like TeX or DocBook. One of these missing features is tables.
However, | have found that it often works well to create a table using a
graphics editor and then insert it into the document as an image. One
advantage to doing tables this way is consistency in appearance
between different outputs (T-X, HTML, etc.).
Another thing SUbML makes no provision for is easy, verbatim display
of its own markup code. In order to show verbatim SubML code, you
must mark all < and > symbols with the appropriate <It> and <gt>
tags. The following paragraph shows the markup required for this
paragraph. For a really wild experience, view the source code of this
file to see how | mark up that paragraph:
<para>
Another thing SubML makes no provision for is easy, verbatim display
of its own markup code. In order to show verbatim SubML code, you
must mark all <lt> and <gt> symbols with the appropriate
<lt>lt<gt> and <lt>gt<gt> tags. The
following paragraph shows the markup required for this paragraph.
For a really wild experience, view the source code of this file to
see how I mark up <italic>that</italic> paragraph:
</para>
| could carry the recursion one step further, but that would be cruel
and unusual punishment for both of us.
How to do the conversion
First, you need to have sed installed and operational on your computer.
Next, be sure that all conversion scripts (smi2tex.sed, sml2html.sed, etc.)
have been installed in the same directory as the SUbML document that
you wish to convert. If you wish to convert your SUBML document to
TeX, groff, or some other markup language requiring further
processing, you must of course have the necessary software installed
on your computer to process the markup format(s) of choice.
For instance, if you converted your SUbML document into a T-X
document using the sml2tex.sed script provided with this tutorial, but
didn't have Donald Knuth's TX processing system installed on your
computer, all the sed script will do is produce a T-X source file: a new
document marked up with TeX commands and tags in place of SUDML
tags. In other words, these scripts simply convert SUbML source code
into source code for other markup languages. With the exceptions of
HTML and plain ASCII text, none of the output formats generated by
these sed scripts will be ready-to-use.
If you wish to convert your source document (entitled foo.sml) to HTML,
here is what you would have to type at the command prompt:
sed -f sml2html.sed foo.sml > foo.htm
The -f option tells sed to look to file sml2html.sed for instructions
rather than take direct search-and-replace commands from the
command prompt when processing the input file foo.sml. The output
file is named foo.htm.
The redirection command ( > ) is necessary, otherwise sed will simply
send the converted text to standard output (the computer's command-
line screen) and all of it will flash before your very eyes instead of
being saved in a file. Of course, you can name the target file anything
you wish, so long as the extension is appropriate to the type of
converted document that it is (i.e. .htm or .html for HTML output, so
that a browser will recognize the filename).
The use of standard input and standard output in a sed script allows for
great flexibility in the use of SUbML. For instance, | have a book I'm
writing (Lessons In Electric Circuits), in which I'm using Makefiles to
direct compilation from SubML to LAT_X and HTML. By using
stdin/stdout redirection within the Makefile commands, I'm able to
prepend and append files containing special LATEX and HTML code to
the basic text (written in SUObML format) to achieve markup capabilities
beyond the basic scope of SUbML. For instance, | may want to generate
a coverpage for my book using a series of special L4T-X commands.
SubML doesn't specify detailed layout tags, and so! write the
necessary LAT-X code in a file that gets prepended to the sed-
converted output of the main text body. Same for the generation of an
index: a special file containing the necessary LAT-X commands gets
appended to the very end, after sed has converted the main body of
the text. Same for navigation buttons at the beginning and end of
each HTML file generated from SubML.
How mini TOC works
A mini Table of Contents (TOC) is automatically inserted after the
chapter title for (1) html, (2) LATEX which provices dvi, ps, and pdf.
There is no mini TOC support for other formats: txt, tex, or groff. This
requires different packages for (1) html, (2) L4T_X. Thus, the method of
generation of the mini TOC is different for the two case. In both cases
the automatic generation is initiated by the sed command file
substitution for the </chaptertitle> tag. Other features in headers or
makefiles cause the required software to generate and insert the mini
TOC after the chapter title.
In the case of html, the sml2htm.sed file contains the </chpatertitle>
tag substitution: <!--minitoc-> which flags the html for inclusion of
the mini TOC. We use a a perl script, htmltoc, modified for our
requirements to htmltoc2 for placing a mini TOC at the <!--minitoc->
tag. The original script placed the mini TOC before the chapter title.
So, we modified it to place the mini TOC at our tag, which is after the
title. The Makefile has a line calling minitoc with appropriate
parameters:
./htmltoc2 -inline -noorg -toclabel " " -tocmap toc.map \
-minitoc "<\! \-\-\minitoc\-\->" AC_1.html
See the minitoc documentation for details. We added the -minitoc
parameter to the htmlitoc perl script for our htmitoc2 so that it looks for
the quoted tag which follows it. In our case we want the mini TOC at
the <!--minitoc-> tag, so that tag with escaping backslashes follows.
The makefile for each book has a make target for each of the book
chapters. The chapters for which we want a mini TOC require the
above htmltoc2 command in the make targets. We include it in
chapter targets, 1, 2, etc., but not the appendix targets, Al, A2,
_A3. Thus, all chapters but the appendices have a mini TOC after the
chapter title. Eg., see AC/Makefile targets: AC_14.html, AC _Al.latex for
chapter vs appendix.
In the case of the L“T_X translation, .latex, the </chaptertitle> in .sml
is replaced by /minitoc. See sml2latx.sed. This /minitoc tells L4T-X
where to place the mini TOC.
Also, the header, hi.latex, contains \usepackage{minitoc} and
\dominitoc to load the minitoc package and "do" the minitable of
contents respectively. The table will be placed where the /minitoc
command is encountered in the chapter text.
Nothing unusual is required of the makefile to generate the mini TOC.
However, if we do not want the mini TOC in the appendices, a sed
script in each of the latex appendix targets, removes the /minitoc
command from the .latex. Normal target processing, puts a chapter
mini TOC in for all chapters but appendices. Eg., see AC/Makefile
targets: lines.latex, about.latex for chapter vs appendix.
Table of contents - TOC
The LaTeX table of contents is due to commands in the hi.latex header
file. The command \setcounter{tocdepth}{1} limits the depth of the
TOC entries to one level below chapter. Thus, we get chapter and
section entries. The file hi_appendix, inserted between the chapters
and appendices by Makefile, sets the counter to the chapter level with
\settocdepth{chapter}. This leaves a single TOC entry for each
appendix. The package tocvsec2 is required to reset the counter. See
\usepackage{../bin/tocvsec2} in hi.latex
The hyperref package (hi.latex) generates a list of bookmarks along
the left side of the acrobat viewer. The depth of this bookmark TOC
only extends to the chapter level if there is a "real" TOC. It is possible
to generate expandable bookmarks to more levels, if the TOC is
suppressed by removing \tableofcontents, \setcounter{tocdepth }
{1},\settocdepth{chapter}. At this time we opt for the printed TOC
over the expanded bookmark version.
Copyright (C) 2000-2020, Tony R.
Kuphaldt
See the Design Science License (Appendix 3)
for details regarding copying and distribution
Revised April 05, 2009
Master Index
Chapter 1: AMPLIFIERS AND ACTIVE DEVICES
Chapter 2: SOLID-STATE DEVICE THEORY
Chapter 3: DIODES AND RECTIFIERS
Chapter 4: BIPOLAR JUNCTION TRANSISTORS
Chapter 5: JUNCTION FIELD-EFFECT TRANSISTORS
***| NCOMPLETE***
Chapter 6: INSULATED-GATE FIELD-EFFECT
TRANSISTORS ***INCOMPLETE***
Chapter 7: THYRISTORS
Chapter 8: OPERATIONAL AMPLIFIERS
Chapter 9: PRACTICAL ANALOG SEMICONDUCTOR
CIRCUITS ***INCOMPLETE***
Chapter 10: ACTIVE FILTERS ***PENDING***
Chapter 11: DC MOTOR DRIVES ***PENDING***
Chapter 12: INVERTERS AND AC MOTOR DRIVES
***P EN DING***
Chapter 13: ELECTRON TUBES
Appendix 1: ABOUT THIS BOOK
Appendix 2: CONTRIBUTOR LIST
Appendix 3: DESIGN SCIENCE LICENSE
Download printable versions of this
volume
Adobe PDF format:
SEMI.pdf
Approximately 2 megabytes
Adobe PDF
{
Adobe PostScript (compressed) format:
SEMI.ps.gz
Approximately 3 megabytes
PostScript
1
"How do! view and/or print PostScript documents," you ask?
Easy! Just download some free software at:
www.cs.wisc.edu/~ ghost.
There you'll find GSview and Ghostscript, two progams
necessary to display and print Postscript files (they'll even
display and print compressed PostScript files!). These
programs also display and format Adobe PDF files as a bonus.
Versions for Windows, OS/2, and Linux available.
Download source files for this volume
0 O
SEMIsrc.tar.gz
<SubML> Approximately 8 megabytes
a o
SEMitiny. tar.gz
<SubMl> | Approximately 1 megabyte
To "compile" these source files into a viewable format, you
will need the following pieces of software (all available freely
over the internet):
e Make, a project management utility originally intended
as a programming tool, but useful for managing just
about any kind of computer project composed of many
files. /f you cannot obtain a copy of Make for your
computer system, you can get by with a little skill and a
few batch files (also known as shell scripts). The master
"Makefile" in this directory is readable with a text editor
or word processor, and contains all the instructions
carried out by the other utilities.
e Sed (stands for Stream EDitor), a common UNIX utility
for performing search-and-replace commands on text
files. Required to convert SUbML source code into HTML,
TeX, LaTeX, and other formats. This is all you need for
generating HTML output!
LaTeX2e, a document formatting system designed as an
extension to TeX, Donald Knuth's outstanding text
processing system. You can also get by with just plain
TeX, but your printed output won't look quite as nice and
it will lack table-of-contents and index entries.
If you opt for the smaller of the two files (SEMItiny.tar.gz),
you'll also need a set of graphic manipulation utilities
released as a package called ImageMagick. Specifically, the
utility you'll need is named Mogrify. The larger of the two
source archive files contains all graphic images in two
formats, Encapsulated PostScript (*.eps) and JPEG (*.jpg).
This makes for a large file. The smaller source archive file
only contains Encapsulated PostScript for schematic
diagrams and JPEG images for photographs. This makes for a
much smaller file, but it requires that you do some image
conversion on your end. If you have access to other image
manipulation software capable of converting hundreds of
files with a batch command, you won't have to use
ImageMagick.
Back to Master Index
—/ | 4]
Lessons In Electric Circuits
-- Volume lll
Chapter 1
AMPLIFIERS AND ACTIVE
DEVICES
From electric to electronic
Active versus passive devices
Amplifiers
Amplifier gain
Decibels
Absolute dB scales
Attenuators
o Decibels
o T-section attenuator
o Pl-section attenuator
L-section attenuator
Bridged T attenuator
Cascaded sections
RF attenuators
Contributors
(2)
°
12)
°
From electric to electronic
This third volume of the book series Lessons /n Electric
Circuits makes a departure from the former two in that the
transition between e/ectric circuits and e/ectronic circuits is
formally crossed. Electric circuits are connections of
conductive wires and other devices whereby the uniform
flow of electrons occurs. Electronic circuits add a new
dimension to electric circuits in that some means of contro!
is exerted over the flow of electrons by another electrical
signal, either a voltage or a current.
In and of itself, the control of electron flow is nothing new to
the student of electric circuits. Switches control the flow of
electrons, as do potentiometers, especially when connected
as variable resistors (rheostats). Neither the switch nor the
potentiometer should be new to your experience by this
point in your study. The threshold marking the transition
from electric to electronic, then, is defined by how the flow
of electrons is controlled rather than whether or not any
form of control exists in a circuit. Switches and rheostats
control the flow of electrons according to the positioning of a
mechanical device, which is actuated by some physical force
external to the circuit. In electronics, however, we are
dealing with special devices able to control the flow of
electrons according to another flow of electrons, or by the
application of a static voltage. In other words, in an
electronic circuit, electricity is able to control electricity.
The historic precursor to the modern electronics era was
invented by Thomas Edison in 1880 while developing the
electric incandescent lamp. Edison found that a small
current passed from the heated lamp filament to a metal
plate mounted inside the vacuum envelop. (Figure below
(a)) Today this is known as the “Edison effect”. Note that the
battery is only necessary to heat the filament. Electrons
would still flow if a non-electrical heat source was used.
(a) Edison effect, (b) Fleming valve or vacuum diode, (c)
DeForest audion triode vacuum tube amplifier.
By 1904 Marconi Wireless Company adviser John Flemming
found that an externally applied current (plate battery) only
passed in one direction from filament to plate (Figure above
(b)), but not the reverse direction (not shown). This
invention was the vacuum diode, used to convert alternating
currents to DC. The addition of a third electrode by Lee
DeForest (Figure above (c)) allowed a small signal to control
the larger electron flow from filament to plate.
Historically, the era of electronics began with the invention
of the Audion tube, a device controlling the flow of an
electron stream through a vacuum by the application of a
small voltage between two metal structures within the tube.
A more detailed summary of so-called e/ectron tube or
vacuum tube technology is available in the last chapter of
this volume for those who are interested.
Electronics technology experienced a revolution in 1948
with the invention of the transistor. This tiny device
achieved approximately the same effect as the Audion tube,
but in a vastly smaller amount of space and with less
material. Transistors control the flow of electrons through
solid semiconductor substances rather than through a
vacuum, and so transistor technology is often referred to as
solid-state electronics.
Active versus passive devices
An active device is any type of circuit component with the
ability to electrically control electron flow (electricity
controlling electricity). In order for a circuit to be properly
called e/ectronic, it must contain at least one active device.
Components incapable of controlling current by means of
another electrical signal are called passive devices.
Resistors, capacitors, inductors, transformers, and even
diodes are all considered passive devices. Active devices
include, but are not limited to, vacuum tubes, transistors,
silicon-controlled rectifiers (SCRs), and TRIACs. A case might
be made for the saturable reactor to be defined as an active
device, since it is able to control an AC current with a DC
current, but I've never heard it referred to as such. The
operation of each of these active devices will be explored in
later chapters of this volume.
All active devices control the flow of electrons through them.
Some active devices allow a voltage to control this current
while other active devices allow another current to do the
job. Devices utilizing a static voltage as the controlling
signal are, not surprisingly, called vo/tage-controlled
devices. Devices working on the principle of one current
controlling another current are known as current-controlled
devices. For the record, vacuum tubes are voltage-controlled
devices while transistors are made as either voltage-
controlled or current controlled types. The first type of
transistor successfully demonstrated was a current-
controlled device.
Amplifiers
The practical benefit of active devices is their amplifying
ability. Whether the device in question be voltage-controlled
or current-controlled, the amount of power required of the
controlling signal is typically far less than the amount of
power available in the controlled current. In other words, an
active device doesn't just allow electricity to control
electricity; it allows a sma// amount of electricity to control a
large amount of electricity.
Because of this disparity between controlling and controlled
powers, active devices may be employed to govern a large
amount of power (controlled) by the application of a small
amount of power (controlling). This behavior is known as
amplification.
It is a fundamental rule of physics that energy can neither
be created nor destroyed. Stated formally, this rule is known
as the Law of Conservation of Energy, and no exceptions to
it have been discovered to date. If this Law is true -- and an
overwhelming mass of experimental data suggests that it is -
- then it is impossible to build a device capable of taking a
small amount of energy and magically transforming it into a
large amount of energy. All machines, electric and electronic
circuits included, have an upper efficiency limit of 100
percent. At best, power out equals power in as in Figure
below.
Poss > Perfect machine > an
P
output
Efficiency = = 1= 100%
input
The power output of a machine can approach, but never
exceed, the power input for 100% efficiency as an upper
limit.
Usually, machines fail even to meet this limit, losing some of
their input energy in the form of heat which is radiated into
surrounding space and therefore not part of the output
energy stream. (Figure below)
P. Realistic machine Pp
input output
LL» Piost (USUally waste heat)
P
output
Efficiency = < 1=less than 100%
input
A realistic machine most often loses some of its input
energy as heat in transforming it into the output energy
stream.
Many people have attempted, without success, to design
and build machines that output more power than they take
in. Not only would such a perpetual motion machine prove
that the Law of Conservation of Energy was not a Law after
all, but it would usher in a technological revolution such as
the world has never seen, for it could power itself in a
circular loop and generate excess power for “free”. (Figure
below)
Perpetual-motion
Pinput <> cae e P, mutput
output
P
Efficiency = > 1 = more than 100%
input
ape ae
Perpetual-motion
oe = —> = —
Poutput
Hypothetical “perpetual motion machine” powers itself?
Despite much effort and many unscrupulous claims of “free
energy” or over-unity machines, not one has ever passed the
simple test of powering itself with its own energy output and
generating energy to spare.
There does exist, however, a class of machines known as
amplifiers, which are able to take in small-power signals and
output signals of much greater power. The key to
understanding how amplifiers can exist without violating the
Law of Conservation of Energy lies in the behavior of active
devices.
Because active devices have the ability to contro/a large
amount of electrical power with a small amount of electrical
power, they may be arranged in circuit so as to duplicate the
form of the input signal power from a larger amount of
power supplied by an external power source. The result is a
device that appears to magically magnify the power of a
small electrical signal (usually an AC voltage waveform) into
an identically-shaped waveform of larger magnitude. The
Law of Conservation of Energy is not violated because the
additional power is supplied by an external source, usually a
DC battery or equivalent. The amplifier neither creates nor
destroys energy, but merely reshapes it into the waveform
desired as shown in Figure below.
External
power source
Pat —>»> Amplifier —> Bis
Tw eo ae,
While an amplifier can scale a small input signal to large
output, its energy source is an external power supply.
In other words, the current-controlling behavior of active
devices is employed to shape DC power from the external
power source into the same waveform as the input signal,
producing an output signal of like shape but different
(greater) power magnitude. The transistor or other active
device within an amplifier merely forms a larger copy of the
input signal waveform out of the “raw” DC power provided
by a battery or other power source.
Amplifiers, like all machines, are limited in efficiency toa
maximum of 100 percent. Usually, electronic amplifiers are
far less efficient than that, dissipating considerable amounts
of energy in the form of waste heat. Because the efficiency
of an amplifier is always 100 percent or less, one can never
be made to function as a “perpetual motion” device.
The requirement of an external source of power is common
to all types of amplifiers, electrical and non-electrical. A
common example of a non-electrical amplification system
would be power steering in an automobile, amplifying the
power of the driver's arms in turning the steering wheel to
move the front wheels of the car. The source of power
necessary for the amplification comes from the engine. The
active device controlling the driver's “input signal” is a
hydraulic valve shuttling fluid power from a pump attached
to the engine to a hydraulic piston assisting wheel motion. If
the engine stops running, the amplification system fails to
amplify the driver's arm power and the car becomes very
difficult to turn.
Amplifier gain
Because amplifiers have the ability to increase the
magnitude of an input signal, it is useful to be able to rate
an amplifier's amplifying ability in terms of an output/input
ratio. The technical term for an amplifier's output/input
magnitude ratio is gain. As a ratio of equal units (power out /
power in, voltage out / voltage in, or current out / current in),
gain is naturally a unitless measurement. Mathematically,
gain is symbolized by the capital letter “A”.
For example, if an amplifier takes in an AC voltage signal
measuring 2 volts RMS and outputs an AC voltage of 30
volts RMS, it has an AC voltage gain of 30 divided by 2, or
15:
Vv
output
Ay =
Vinput
30 V
Ay =
2V
Ay = 15
Correspondingly, if we know the gain of an amplifier and the
magnitude of the input signal, we can calculate the
magnitude of the output. For example, if an amplifier with
an AC current gain of 3.5 is given an AC input signal of 28
mA RMS, the output will be 3.5 times 28 mA, or 98 mA:
Toutput = (Ap Tinput)
I = (3.5)(28 mA)
output
I =98 mA
output
In the last two examples | specifically identified the gains
and signal magnitudes in terms of “AC.” This was
intentional, and illustrates an important concept: electronic
amplifiers often respond differently to AC and DC input
signals, and may amplify them to different extents. Another
way of saying this is that amplifiers often amplify changes or
variations in input signal magnitude (AC) at a different ratio
than steady input signal magnitudes (DC). The specific
reasons for this are too complex to explain at this time, but
the fact of the matter is worth mentioning. If gain
calculations are to be carried out, it must first be understood
what type of signals and gains are being dealt with, AC or
DC.
Electrical amplifier gains may be expressed in terms of
voltage, current, and/or power, in both AC and DC. A
summary of gain definitions is as follows. The triangle-
shaped “delta” symbol (A) represents change in
mathematics, so “AVoutout / AVinput” Means “change in
output voltage divided by change in input voltage,” or more
simply, “AC output voltage divided by AC input voltage”:
DC gains AC gains
Voutput AV
Vinput AV
output
Voltage | Ay=
input
Current Ay = Mout Al output
I Al
input input
(AV outyan MAT suipur)
Ap=
(AV inp MAT aren)
Ap=(Ay)(A))
A= "change in..."
If multiple amplifiers are staged, their respective gains form
an overall gain equal to the product (multiplication) of the
individual gains. (Figure below) If a 1 V signal were applied
to the input of the gain of 3 amplifier in Figure below a 3 V
signal out of the first amplifier would be further amplified by
a gain of 5 at the second stage yielding 15 V at the final
output.
Input signal —————> Amplifier
gain=3
Overall gain = (3(5)=15
> Output signal
The gain of a chain of cascaded amplifiers is the product of
the individual gains.
Decibels
In its simplest form, an amplifier's ga/n is a ratio of output
over input. Like all ratios, this form of gain is unitless.
However, there is an actual unit intended to represent gain,
and it is called the be/.
As a unit, the bel was actually devised as a convenient way
to represent power /oss in telephone system wiring rather
than gain in amplifiers. The unit's name is derived from
Alexander Graham Bell, the famous Scottish inventor whose
work was instrumental in developing telephone systems.
Originally, the bel represented the amount of signal power
loss due to resistance over a standard length of electrical
cable. Now, it is defined in terms of the common (base 10)
logarithm of a power ratio (output power divided by input
power):
P
output
Ap, ratio) ~
input
P
output
A
Appel) = log
input
Because the bel is a logarithmic unit, it is nonlinear. To give
you an idea of how this works, consider the following table
of figures, comparing power losses and gains in bels versus
simple ratios:
Table: Gain / loss in bels
Loss/gain as Loss/gain Loss/gain as Loss/gain
a ratio in bels a ratio in bels
Poutput log P output Pourput log P output
Prnput Pinput Pinput Pinput
ee ee
ee
te | om |
]
It was later decided that the bel was too large of a unit to be
used directly, and so it became customary to apply the
metric prefix deci (meaning 1/10) to it, making it decibels, or
dB. Now, the expression “dB” is so common that many
people do not realize it is a combination of “deci-” and “-
bel,” or that there even is such a unit as the “bel.” To put
this into perspective, here is another table contrasting
power gain/loss ratios against decibels:
Table: Gain / loss in decibels
Loss/gain as Loss/gain Loss/gain as Loss/gain
a ratio in decibels a ratio in decibels
Pourput Poutput Poutput Poutput
10 log 10 log
Prnput Pinput P input Pinput
Te [ma [oe [oe
As a logarithmic unit, this mode of power gain expression
covers a wide range of ratios with a minimal span in figures.
It is reasonable to ask, “why did anyone feel the need to
invent a /ogarithmic unit for electrical signal power loss ina
telephone system?” The answer is related to the dynamics of
human hearing, the perceptive intensity of which is
logarithmic in nature.
Human hearing is highly nonlinear: in order to double the
perceived intensity of a sound, the actual sound power must
be multiplied by a factor of ten. Relating telephone signal
power loss in terms of the logarithmic “bel” scale makes
perfect sense in this context: a power loss of 1 bel translates
to a perceived sound loss of 50 percent, or 1/2. A power gain
of 1 bel translates to a doubling in the perceived intensity of
the sound.
An almost perfect analogy to the bel scale is the Richter
scale used to describe earthquake intensity: a 6.0 Richter
earthquake is 10 times more powerful than a 5.0 Richter
earthquake; a 7.0 Richter earthquake 100 times more
powerful than a 5.0 Richter earthquake; a 4.0 Richter
earthquake is 1/10 as powerful as a 5.0 Richter earthquake,
and so on. The measurement scale for chemical pH is
likewise logarithmic, a difference of 1 on the scale is
equivalent to a tenfold difference in hydrogen ion
concentration of a chemical solution. An advantage of using
a logarithmic measurement scale is the tremendous range of
expression afforded by a relatively small soan of numerical
values, and it is this advantage which secures the use of
Richter numbers for earthquakes and pH for hydrogen ion
activity.
Another reason for the adoption of the bel as a unit for gain
is for simple expression of system gains and losses. Consider
the last system example (Figure above) where two amplifiers
were connected tandem to amplify a signal. The respective
gain for each amplifier was expressed as a ratio, and the
overall gain for the system was the product (multiplication)
of those two ratios:
Overall gain = (3)(5) = 15
If these figures represented power gains, we could directly
apply the unit of bels to the task of representing the gain of
each amplifier, and of the system altogether. (Figure below)
Appel) = log Apyratin)
Apert) = log 3 Aner = log 5
ones Amplifier
Input signal ———5 =3 > gain =5 —. Output signal
waite 1" 477 B gain = 0.699 B
Overall gain = (3)(5) = 15
Overall gain. = log 15 = 1.176 B
Power gain in bels is additive: 0.477 B + 0.699 B = 1.176 B.
Close inspection of these gain figures in the unit of “bel”
yields a discovery: they're additive. Ratio gain figures are
multiplicative for staged amplifiers, but gains expressed in
bels add rather than multiply to equal the overall system
gain. The first amplifier with its power gain of 0.477 B adds
to the second amplifier's power gain of 0.699 B to makea
system with an overall power gain of 1.176 B.
Recalculating for decibels rather than bels, we notice the
Same phenomenon. (Figure below)
Apyap) =10 log Agyeatio)
Apjan) = 10 log 3 Apyan) = 10 log 5
canes
Am she
Input signal ——— ain = —S gai in —> Output signal
oP 477 dB gain = "6 oe
Overall gain = (3)(5) = 15
Overall gain;ge, = 10 log 15 = 11.76 dB
Gain of amplifier stages in decibels is additive: 4.77 dB +
6.99 dB = 11.76 AB.
To those already familiar with the arithmetic properties of
logarithms, this is no surprise. It is an elementary rule of
algebra that the antilogarithm of the sum of two numbers'
logarithm values equals the product of the two original
numbers. In other words, if we take two numbers and
determine the logarithm of each, then add those two
logarithm figures together, then determine the
“antilogarithm” of that sum (elevate the base number of the
logarithm -- in this case, 10 -- to the power of that sum), the
result will be the same as if we had simply multiplied the
two original numbers together. This algebraic rule forms the
heart of a device called a s/ide rule, an analog computer
which could, among other things, determine the products
and quotients of numbers by addition (adding together
physical lengths marked on sliding wood, metal, or plastic
scales). Given a table of logarithm figures, the same
mathematical trick could be used to perform otherwise
complex multiplications and divisions by only having to do
additions and subtractions, respectively. With the advent of
high-speed, handheld, digital calculator devices, this
elegant calculation technique virtually disappeared from
popular use. However, it is still important to understand
when working with measurement scales that are logarithmic
in nature, such as the bel (decibel) and Richter scales.
When converting a power gain from units of bels or decibels
to a unitless ratio, the mathematical inverse function of
common logarithms is used: powers of 10, or the antilog.
If:
Apvpel) = log Apwatio)
Then:
f — Apel)
Apvratio) = 10
Converting decibels into unitless ratios for power gain is
much the same, only a division factor of 10 is included in
the exponent term:
If:
Apap) = 10 log Ap ratio)
Then:
Apap)
=-190'°
A
P(ratio)
Example: Power into an amplifier is 1 Watt, the power out is
10 Watts. Find the power gain in cB.
Apap) = 10 logi9(Po / P}) = 10 logyp (10 /1) = 10 logy,
(10) = 10 (1) = 10 dB
Example: Find the power gain ratio Apratio) = (Po / Pi) for a
20 dB Power gain.
Apvap) =20=10 logio Ap(ratio)
20/10 = logig Apvratio)
1029/10 = 1Q!0910 (Ap(ratioy)
100 = Apiratio) = (Po / Pi)
Because the bel is fundamentally a unit of power gain or
loss in a system, voltage or current gains and losses don't
convert to bels or dB in quite the same way. When using bels
or decibels to express a gain other than power, be it voltage
or current, we must perform the calculation in terms of how
much power gain there would be for that amount of voltage
or current gain. For a constant load impedance, a voltage or
current gain of 2 equates to a power gain of 4 (22); a voltage
or current gain of 3 equates to a power gain of 9 (32). If we
multiply either voltage or current by a given factor, then the
power gain incurred by that multiplication will be the square
of that factor. This relates back to the forms of Joule's Law
where power was calculated from either voltage or current,
and resistance:
E-
P= —
R
P=IR
Power is proportional to the square
of either voltage or current
Thus, when translating a voltage or current gain ratio into a
respective gain in terms of the bel unit, we must include this
exponent in the equation(s):
Ap, Bel) > log Apiratio)
Avipet) = log Aviratioy “~~ Exponent required
a
Apel) = log A
I(ratio)
The same exponent requirement holds true when expressing
voltage or current gains in terms of decibels:
Apia) = 10 log A piratio)
Ayiapy = 10 log Avcraticy, “~~ Exponent required
5
Ayap) = 10 log Atvratioy
However, thanks to another interesting property of
logarithms, we can simplify these equations to eliminate the
exponent by including the “2” as a multiplying factor for the
logarithm function. In other words, instead of taking the
logarithm of the square of the voltage or current gain, we
just multiply the voltage or current gain's logarithm figure
by 2 and the final result in bels or decibels will be the same:
For bels:
Avipel) = log Aviratio) Apel) = log Ay ratio)
...isthesameas... .../sthesameas...
Avipel) = 2 log Av iratio) Awpel) = 2 log A iratio)
For decibels:
Avis) = 10 log Aviratio) Avap) = 10 log Aicratio).
...isthesameas... ...isthesameas...
Aviap) = 20 log Ay; ratio) Aap) = 20 log Aicratio)
The process of converting voltage or current gains from bels
or decibels into unitless ratios is much the same as it is for
power gains:
j —_ 9 / / a | /
Avipel) oe log Aviratio) Anpel) — log Ajvatio)
Then:
Aven Ay Teel)
>) >
Aviratioy = 10 A ivratio) =10
Here are the equations used for converting voltage or
current gains in decibels into unitless ratios:
If:
/ eats f a |
Avia) = 20 log Ay ratio) Aap) = 20 log Aicratio)
Then:
Avian) Awa)
' = 20 ; _ 20
Av (ratio) = 10 Ay ratio) ~ 10
While the bel is a unit naturally scaled for power, another
logarithmic unit has been invented to directly express
voltage or current gains/losses, and it is based on the
natural logarithm rather than the common logarithm as bels
and decibels are. Called the neper, its unit symbol is “N,;
though, lower-case “n” may be encountered.
= Voutput = output
Aviratio) = A ratio) =
Vinput input
p P
A =InA A =InA
Vineper) Viratio) I(neper) I(ratio)
For better or for worse, neither the neper nor its attenuated
cousin, the decineper, is popularly used as a unit in
American engineering applications.
Example: The voltage into a 600 Q audio line amplifier is
10 mV, the voltage across a 600 Q load is 1 V. Find the
power gain in dB.
Aigpy = 20 logig(Vo / V;) = 20 logyg (1 /0.01) = 20 logy
(100) = 20 (2) = 40 dB
Example: Find the voltage gain ratio Aywratio) = (Vo / Vi) for
a 20 dB gain amplifier having a 50 Q input and out
impedance.
Ayas) = 20 10919 Av(ratio)
20 = 20 10910 Av(ratio)
20/20 = 10910 Ap(ratio)
1029/20 = 1Q!0910 (Aviratioy)
10 = Aviratioy = (Vo / Vi)
REVIEW:
Gains and losses may be expressed in terms of a unitless
ratio, or in the unit of bels (B) or decibels (dB). A decibel
is literally a deci-bel: one-tenth of a bel.
The bel is fundamentally a unit for expressing power
gain or loss. To convert a power ratio to either bels or
decibels, use one of these equations:
Appel) = log Aprnio) Apia) =10 log Apanio)
When using the unit of the bel or decibel to express a
voltage or current ratio, it must be cast in terms of an
equivalent power ratio. Practically, this means the use of
different equations, with a multiplication factor of 2 for
the logarithm value corresponding to an exponent of 2
for the voltage or current gain ratio:
—_> -m”
Avnet) es log Avintioy Avian) = <0 log Avimiia)
=3 i -7n of
Aypel) == log Atintioy Ayan y= =U log Ajirnio)
To convert a decibel gain into a unitless ratio gain, use
one of these equations:
Nien
mp 20
Avirsiv) =10
re
Ajvratio) = LO a
Apyratioy = LO a
A gain (amplification) is expressed as a positive bel or
decibel figure. A loss (attenuation) is expressed as a
negative bel or decibel figure. Unity gain (no gain or
loss; ratio = 1) is expressed as zero bels or zero decibels.
When calculating overall gain for an amplifier system
composed of multiple amplifier stages, individual gain
ratios are multiplied to find the overall gain ratio. Bel or
decibel figures for each amplifier stage, on the other
hand, are added together to determine overall gain.
Absolute dB scales
It is also possible to use the decibel as a unit of absolute
power, in addition to using it as an expression of power gain
or loss. Acommon example of this is the use of decibels as a
measurement of sound pressure intensity. In cases like
these, the measurement is made in reference to some
standardized power level defined as 0 dB. For measurements
of sound pressure, O dB is loosely defined as the lower
threshold of human hearing, objectively quantified as 1
picowatt of sound power per square meter of area.
A sound measuring 40 dB on the decibel sound scale would
be 10% times greater than the threshold of hearing. A 100 dB
sound would be 10!° (ten billion) times greater than the
threshold of hearing.
Because the human ear is not equally sensitive to all
frequencies of sound, variations of the decibel sound-power
scale have been developed to represent physiologically
equivalent sound intensities at different frequencies. Some
sound intensity instruments were equipped with filter
networks to give disproportionate indications across the
frequency scale, the intent of which to better represent the
effects of sound on the human body. Three filtered scales
became commonly known as the “A,” “B,” and “C” weighted
scales. Decibel sound intensity indications measured
through these respective filtering networks were given in
units of dBA, dBB, and dBC. Today, the “A-weighted scale” is
most commonly used for expressing the equivalent
physiological impact on the human body, and is especially
useful for rating dangerously loud noise sources.
Another standard-referenced system of power measurement
in the unit of decibels has been established for use in
telecommunications systems. This is called the dBm scale.
(Figure below) The reference point, 0 dBm, is defined as 1
milliwatt of electrical power dissipated by a 600 © load.
According to this scale, 10 dBm is equal to 10 times the
reference power, or 10 milliwatts; 20 dBm is equal to 100
times the reference power, or 100 milliwatts. Some AC
voltmeters come equipped with a dBm range or scale
(sometimes labeled “DB”) intended for use in measuring AC
signal power across a 600 Q load. 0 dBm on this scale is, of
course, elevated above zero because it represents
something greater than 0 (actually, it represents 0.7746
volts across a 600 Q load, voltage being equal to the square
root of power times resistance; the square root of 0.001
multiplied by 600). When viewed on the face of an analog
meter movement, this dBm scale appears compressed on
the left side and expanded on the right in a manner not
unlike a resistance scale, owing to its logarithmic nature.
Radio frequency power measurements for low level signals
encountered in radio receivers use dBm measurements
referenced to a 50 QO load. Signal generators for the
evaluation of radio receivers may output an adjustable dBm
rated signal. The signal level is selected by a device called
an attenuator, described in the next section.
Table: Absolute power levels in dBm (decibel milliwatt)
Power in Power in Power in Power in Power in
watts milliwatts dBm milliwatts dBm
Absolute power levels in dBm (decibels referenced to 1
milliwatt).
cc
An adaptation of the dBm scale for audio signal strength is
used in studio recording and broadcast engineering for
standardizing volume levels, and is called the VU scale. VU
meters are frequently seen on electronic recording
instruments to indicate whether or not the recorded signal
exceeds the maximum signal level limit of the device, where
significant distortion will occur. This “volume indicator”
scale is calibrated in according to the dBm scale, but does
not directly indicate dBm for any signal other than steady
sine-wave tones. The proper unit of measurement for a VU
meter is volume units.
When relatively large signals are dealt with, and an absolute
dB scale would be useful for representing signal level,
specialized decibel scales are sometimes used with
reference points greater than the 1 mW used in dBm. Such is
the case for the dBW scale, with a reference point of 0 dBW
established at 1 Watt. Another absolute measure of power
called the dBk scale references 0 dBk at 1 KW, or 1000
Watts.
REVIEW:
e The unit of the bel or decibel may also be used to
represent an absolute measurement of power rather
than just a relative gain or loss. For sound power
measurements, 0 dB is defined as a standardized
reference point of power equal to 1 picowatt per square
meter. Another dB scale suited for sound intensity
measurements is normalized to the same physiological
effects as a 1000 Hz tone, and is called the dBA scale. In
this system, 0 dBA is defined as any frequency sound
having the same physiological equivalence as a 1
picowatt-per-square-meter tone at 1000 Hz.
An electrical dB scale with an absolute reference point
has been made for use in telecommunications systems.
Called the dBm scale, its reference point of 0 dBm is
defined as 1 milliwatt of AC signal power dissipated by a
600 O load.
A VU meter reads audio signal level according to the
dBm for sine-wave signals. Because its response to
signals other than steady sine waves is not the same as
true dBm, its unit of measurement is vo/ume units.
dB scales with greater absolute reference points than
the dBm scale have been invented for high-power
signals. The dBW scale has its reference point of O dBW
defined as 1 Watt of power. The dBk scale sets 1 kW
(1000 Watts) as the zero-point reference.
Attenuators
Attenuators are passive devices. It is convenient to discuss
them along with decibels. Attenuators weaken or attenuate
the high level output of a signal generator, for example, to
provide a lower level signal for something like the antenna
input of a sensitive radio receiver. (Figure below) The
attenuator could be built into the signal generator, or bea
stand-alone device. It could provide a fixed or adjustable
amount of attenuation. An attenuator section can also
provide isolation between a source and a troublesome load.
od
Z, | Z
Attenuator
Q 1% © Ex
et od
Constant impedance attenuator is matched to source
impedance Z, and load impedance Zo. For radio frequency
eguioment Z is 50 Q.
In the case of a stand-alone attenuator, it must be placed in
series between the signal source and the load by breaking
open the signal path as shown in Figure above. In addition,
it must match both the source impedance Z, and the load
impedance Zo, while providing a specified amount of
attenuation. In this section we will only consider the special,
and most common, case where the source and load
impedances are equal. Not considered in this section,
unequal source and load impedances may be matched by an
attenuator section. However, the formulation is more
complex.
T attenuator II attenuator
T section and [1 section attenuators are common forms.
Common configurations are the T and fl networks shown in
Figure above Multiple attenuator sections may be cascaded
when even weaker signals are needed as in Figure below.
Decibels
Voltage ratios, as used in the design of attenuators are often
expressed in terms of decibels. The voltage ratio (K below)
must be derived from the attenuation in decibels. Power
ratios expressed as decibels are additive. For example, a 10
dB attenuator followed by a 6 dB attenuator provides 16dB
of attenuation overall.
10 dB + 6db = 16 dB
Changing sound levels are perceptible roughly proportional
to the logarithm of the power ratio (P; / Po).
sound level = logj9(P; / Po)
A change of 1 dB in sound level is barely perceptible to a
listener, while 2 db is readily perceptible. An attenuation of
3 dB corresponds to cutting power in half, while a gain of 3
db corresponds to a doubling of the power level. A gain of -3
dB is the same as an attenuation of +3 dB, corresponding to
half the original power level.
The power change in decibels in terms of power ratio is:
dB = 10 logio(P, / Po)
Assuming that the load R, at P; is the same as the load
resistor Ro at Po (R; = Ro), the decibels may be derived from
the voltage ratio (V; / Vo) or current ratio (1) / Io):
Po =Volo=Vo*/R=107R
P=Vjl=Ve/R=17R
dB = 10 logio(P)/ Po) = 10 logio(V;2 / Vo2) = 20
logio(V/Vo)
dB = 10 logyo(P; / Po) = 10 logi(|;2 / Io2) = 20
logio(I/lo)
The two most often used forms of the decibel equation are:
dB = 10 logj9(P;/ Po) or dB = 20 logi9(V| / Vo)
We will use the latter form, since we need the voltage ratio.
Once again, the voltage ratio form of equation is only
applicable where the two corresponding resistors are equal.
That is, the source and load resistance need to be equal.
Example: Power into an attenuator is 10 Watts, the power
out is 1 Watt. Find the attenuation in dB.
dB = 10 1ogi9(P;/ Po) = 10 logyg (10 /1) = 10 log; (10)
= 10(1) =10dB
Example: Find the voltage attenuation ratio (K= (V,/ Vo))
for a 10 dB attenuator.
dB = 10= 20 logig(V,/ Vo)
10/20 = lodig(V| / Vo)
1910/20 _ 1Q9!0910(V / Vo)
3.16 = (V,;/ Vo) = Aprratio)
Example: Power into an attenuator is 100 milliwatts, the
power out is 1 milliwatt. Find the attenuation in dB.
dB = 10 logj9(P; / Po) = 10 logyg (100 /1) = 10 logyg
(100) = 10 (2) = 20 dB
Example: Find the voltage attenuation ratio (K= (V,/ Vo))
for a 20 dB attenuator.
dB = 20= 20 logi9(V,/ Vo )
1020/20 = 19 logio(Vi / Vo)
10 = (V,/Vo) =K
T-section attenuator
The T and fl attenuators must be connected to a Z source
and Z load impedance. The Z-(arrows) pointing away from
the attenuator in the figure below indicate this. The Z-
(arrows) pointing toward the attenuator indicates that the
impedance seen looking into the attenuator with a load Z on
the opposite end is Z, Z=50 Q for our case. This impedance
is a constant (50 Q) with respect to attenuation- impedance
does not change when attenuation is changed.
The table in Figure below lists resistor values for the T and Nn
attenuators to match a 50 QO source/ load, as is the usual
requirement in radio frequency work.
Telephone utility and other audio work often requires
matching to 600 Q. Multiply all R values by the ratio
(600/50) to correct for 600 Q matching. Multiplying by 75/50
would convert table values to match a 75 Q source and load.
GB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
K = Vi = 10 dB20
Vo
K- |
Ri =Z())
2K
R,=Z (= )
K*- |
Formulas for T-section attenuator resistors, given K, the
voltage attenuation ratio, and Z, = Zp = 50.
The amount of attenuation is customarily specified in dB
(decibels). Though, we need the voltage (or current) ratio K
to find the resistor values from equations. See the dB/20
term in the power of 10 term for computing the voltage ratio
K from dB, above.
The T (and below f1) configurations are most commonly
used as they provide bidirectional matching. That is, the
attenuator input and output may be swapped end for end
and still match the source and load impedances while
supplying the same attenuation.
Disconnecting the source and looking in to the right at Vj,
we need to see a Series parallel combination of Ry, R2, Rj,
and Z looking like an equivalent resistance of Z;y, the same
as the source/load impedance Z: (a load of Z is connected to
the output.)
Zin = Ry + (Ro |[(Rz + Z))
For example, substitute the 10 dB values from the 50 O
attenuator table for Ry and R2 as shown in Figure below.
Zy = 25.97 + (35.14 ||(25.97 + 50))
Zw = 25.97 + (35.14 || 75.97 )
Zw = 25.97 + 24.03 = 50
This shows us that we see 50 Q looking right into the
example attenuator (Figure below) with a 50 Q load.
Replacing the source generator, disconnecting load Z at Vo,
and looking in to the left, should give us the same equation
as above for the impedance at Vg, due to symmetry.
Moreover, the three resistors must be values which supply
the required attenuation from input to output. This is
accomplished by the equations for Ry and Rg above as
applied to the T-attenuator below.
y R,=26.0 -R,
T attenuator
10 dB attenuators for matching input/output to Z= 50 Q.
10 OB T-section attenuator for insertion between a 50Q
source and load.
Pl-section attenuator
The table in Figure below lists resistor values for the N
attenuator matching a 50 QO source/ load at some common
attenuation levels. The resistors corresponding to other
attenuation levels may be calculated from the equations.
dB = attenuation in decibels
Z = source/load impedance (resistive) Resistors for M-section
K>1
Vv,
K = —=10%”
Vo
R,=2(5e")
R, =Z (Ke ) IT attenuator
Formulas for [l-section attenuator resistors, given K, the
voltage attenuation ratio, and Z,; = Zp = 50.
The above apply to the n-attenuator below.
What resistor values would be required for both the NM
attenuators for 10 dB of attenuation matching a 50 Q source
and load?
Z
II attenuator
10 dB Il-section attenuator example for matching a 50 Q
source and load.
The 10 dB corresponds to a voltage attenuation ratio of
K=3.16 in the next to last line of the above table. Transfer
the resistor values in that line to the resistors on the
schematic diagram in Figure above.
L-section attenuator
The table in Figure below lists resistor values for the L
attenuators to match a 50 QO source/ load. The table in Figure
below lists resistor values for an alternate form. Note that
the resistor values are not the same.
dB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
(K-1) L attenuator
L-section attenuator table for 50 Q source and load
impedance.
The above apply to the L attenuator below.
dB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
R; /Vo
K = hs = 10%" o AW 0 “1
oO : <2
4
R; = Z( K-1 ) 5
K . 0
Ry=Z (Ky) is
Alternate form L-section attenuator table for 50 Q source
and load impedance.
Bridged T attenuator
The table in Figure below lists resistor values for the bridged
T attenuators to match a 50 QO source and load. The bridged-
T attenuator is not often used. Why not?
dB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
V, dB20
K= —=10 ~~
Vo
2
6 (K-1)
R,; = Z(K-1)
Bridged T attenuator
Formulas and abbreviated table for bridged-T attenuator
section, Z = 50 Q.
Cascaded sections
Attenuator sections can be cascaded as in Figure below for
more attenuation than may be available from a single
section. For example two 10 db attenuators may be
cascaded to provide 20 dB of attenuation, the dB values
being additive. The voltage attenuation ratio K or V,/Vo for
a 10 dB attenuator section is 3.16. The voltage attenuation
ratio for the two cascaded sections is the product of the two
Ks or 3.16x3.16=10 for the two cascaded sections.
section 1 section 2
Cascaded attenuator sections: dB attenuation is additive.
Variable attenuation can be provided in discrete steps by a
switched attenuator. The example Figure below, shown in
the 0 dB position, is capable of 0 through 7 dB of
attenuation by additive switching of none, one or more
sections.
SS Seen ll eee eee, is ee ee
! S1 ! §2 §3
oo oT “o-o- peers T~o-0
4 0B 2dB 1dB
Switched attenuator: attenuation is variable in discrete
steps.
The typical multi section attenuator has more sections than
the above figure shows. The addition of a 3 or 8 dB section
above enables the unit to cover to 10 dB and beyond. Lower
signal levels are achieved by the addition of 10 dB and 20
dB sections, or a binary multiple 16 dB section.
RF attenuators
For radio frequency (RF) work (<1000 Mhz), the individual
sections must be mounted in shielded compartments to
thwart capacitive coupling if lower signal levels are to be
achieved at the highest frequencies. The individual sections
of the switched attenuators in the previous section are
mounted in shielded sections. Additional measures may be
taken to extend the frequency range to beyond 1000 Mhz.
This involves construction from special shaped lead-less
resistive elements.
metalic conductor
resistive disc
resistive rod
Coaxial T-attenuator for radio frequency work
Coaxial T-attenuator for radio frequency work.
A coaxial T-section attenuator consisting of resistive rods
and a resistive disk is shown in Figure above. This
construction is usable to a few gigahertz. The coaxial N
version would have one resistive rod between two resistive
disks in the coaxial line as in Figure below.
metalic conductor
resistive rod
resistive disc
Coaxial [l-attenuator for radio frequency work
Coaxial [l-attenuator for radio frequency work.
RF connectors, not shown, are attached to the ends of the
above T and lM attenuators. The connectors allow individual
attenuators to be cascaded, in addition to connecting
between a source and load. For example, a 10 dB attenuator
may be placed between a troublesome signal source and an
expensive spectrum analyzer input. Even though we may
not need the attenuation, the expensive test equipment is
protected from the source by attenuating any overvoltage.
Summary: Attenuators
e An attenuator reduces an input signal to a lower level.
e The amount of attenuation is specified in decibels (dB).
Decibel values are additive for cascaded attenuator
sections.
e dB from power ratio: © dB = 10 10g j0(P,/ Po)
¢ dB from voltage ratio: dB = 20 logj9(V,/ Vo)
e Tand /7section attenuators are the most common circuit
configurations.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Colin Barnard (November 2003): Correction regarding
Alexander Graham Bell's country of origin (Scotland, not the
United States).
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—|/|+4]l\—
—| | +4/l—
Lessons In Electric Circuits
-- Volume Ill
Chapter 2
SOLID-STATE DEVICE
THEORY
Introduction
Quantum physics
Valence and Crystal structure
Band theory of solids
Electrons and “holes”
The P-N junction
Junction diodes
Bipolar junction transistors
Junction field-effect transistors
Insulated-gate field-effect transistors (MOSFET)
Thyristors
Semiconductor manufacturing techniques
Superconducting devices
Quantum devices
Semiconductor devices in SPICE
Contributors
Bibliography
Introduction
This chapter will cover the physics behind the operation of
semiconductor devices and show how these principles are
applied in several different types of semiconductor devices.
Subsequent chapters will deal primarily with the practical
aspects of these devices in circuits and omit theory as much
as possible.
Quantum physics
“| think it is safe to say that no one understands
quantum mechanics.”
Physicist Richard P. Feynman
To say that the invention of semiconductor devices was a
revolution would not be an exaggeration. Not only was this
an impressive technological accomplishment, but it paved
the way for developments that would indelibly alter modern
society. Semiconductor devices made possible miniaturized
electronics, including computers, certain types of medical
diagnostic and treatment equipment, and popular
telecommunication devices, to name a few applications of
this technology.
But behind this revolution in technology stands an even
greater revolution in general science: the field of quantum
physics. Without this leap in understanding the natural
world, the development of semiconductor devices (and more
advanced electronic devices still under development) would
never have been possible. Quantum physics is an incredibly
complicated realm of science. This chapter is but a brief
overview. When scientists of Feynman's caliber say that “no
one understands [it],” you can be sure it is a complex
subject. Without a basic understanding of quantum physics,
or at least an understanding of the scientific discoveries that
led to its formulation, though, it is impossible to understand
how and why semiconductor electronic devices function.
Most introductory electronics textbooks I've read try to
explain semiconductors in terms of “classical” physics,
resulting in more confusion than comprehension.
Many of us have seen diagrams of atoms that look something
like Figure below.
@ © =electron
= proton
(N) = neutron
Rutherford atom: negative electrons orbit a small positive
nucleus.
Tiny particles of matter called protons and neutrons make up
the center of the atom; e/ectrons orbit like planets around a
star. The nucleus carries a positive electrical charge, owing to
the presence of protons (the neutrons have no electrical
charge whatsoever), while the atom's balancing negative
charge resides in the orbiting electrons. The negative
electrons are attracted to the positive protons just as planets
are gravitationally attracted by the Sun, yet the orbits are
stable because of the electrons’ motion. We owe this popular
model of the atom to the work of Ernest Rutherford, who
around the year 1911 experimentally determined that atoms'
positive charges were concentrated in a tiny, dense core
rather than being spread evenly about the diameter as was
proposed by an earlier researcher, J.J. Thompson.
Rutherford's scattering experiment involved bombarding a
thin gold foil with positively charged alpha particles as in
Figure below. Young graduate students H. Geiger and E.
Marsden experienced unexpected results. A few Alpha
particles were deflected at large angles. A few Alpha particles
were back-scattering, recoiling at nearly 180°. Most of the
particles passed through the gold foil undeflected, indicating
that the foil was mostly empty space. The fact that a few
alpha particles experienced large deflections indicated the
presence of a minuscule positively charged nucleus.
Rutherford scattering: a beam of alpha particles is scattered
by a thin gold foil.
Although Rutherford's atomic model accounted for
experimental data better than Thompson's, it still wasn't
perfect. Further attempts at defining atomic structure were
undertaken, and these efforts helped pave the way for the
bizarre discoveries of quantum physics. Today our
understanding of the atom is quite a bit more complex.
Nevertheless, despite the revolution of quantum physics and
its contribution to our understanding of atomic structure,
Rutherford's solar-system picture of the atom embedded
itself in the popular consciousness to such a degree that it
persists in some areas of study even when inappropriate.
Consider this short description of electrons in an atom, taken
from a popular electronics textbook:
Orbiting negative electrons are therefore attracted
toward the positive nucleus, which leads us to the
question of why the electrons do not fly into the atom's
nucleus. The answer ts that the orbiting electrons remain
in their stable orbit because of two equal but opposite
forces. The centrifugal outward force exerted on the
electrons because of the orbit counteracts the attractive
inward force (centripetal) trying to pull the electrons
toward the nucleus because of the unlike charges.
In keeping with the Rutherford model, this author casts the
electrons as solid chunks of matter engaged in circular orbits,
their inward attraction to the oppositely charged nucleus
balanced by their motion. The reference to “centrifugal force
is technically incorrect (even for orbiting planets), but is
easily forgiven because of its popular acceptance: in reality,
there is no such thing as a force pushing any orbiting body
away from its center of orbit. It seems that way because a
body's inertia tends to keep it traveling in a straight line, and
since an orbit is a constant deviation (acceleration) from
straight-line travel, there is constant inertial opposition to
whatever force is attracting the body toward the orbit center
(centripetal), be it gravity, electrostatic attraction, or even
the tension of a mechanical link.
”
The real problem with this explanation, however, is the idea
of electrons traveling in circular orbits in the first place. Itisa
verifiable fact that accelerating electric charges emit
electromagnetic radiation, and this fact was known even in
Rutherford's time. Since orbiting motion is a form of
acceleration (the orbiting object in constant acceleration
away from normal, straight-line motion), electrons in an
orbiting state should be throwing off radiation like mud from
a spinning tire. Electrons accelerated around circular paths in
particle accelerators called synchrotrons are known to do
this, and the result is called synchrotron radiation. \f
electrons were losing energy in this way, their orbits would
eventually decay, resulting in collisions with the positively
charged nucleus. Nevertheless, this doesn't ordinarily
happen within atoms. Indeed, electron “orbits” are
remarkably stable over a wide range of conditions.
Furthermore, experiments with “excited” atoms
demonstrated that electromagnetic energy emitted by an
atom only occurs at certain, definite frequencies. Atoms that
are “excited” by outside influences such as light are known
to absorb that energy and return it as electromagnetic waves
of specific frequencies, like a tuning fork that rings at a fixed
pitch no matter how it is struck. When the light emitted by
an excited atom is divided into its constituent frequencies
(colors) by a prism, distinct lines of color appear in the
spectrum, the pattern of spectral lines being unique to that
element. This phenomenon is commonly used to identify
atomic elements, and even measure the proportions of each
element in a compound or chemical mixture. According to
Rutherford's solar-system atomic model (regarding electrons
as chunks of matter free to orbit at any radius) and the laws
of classical physics, excited atoms should return energy over
a virtually limitless range of frequencies rather than a select
few. In other words, if Rutherford's model were correct, there
would be no “tuning fork” effect, and the light spectrum
emitted by any atom would appear as a continuous band of
colors rather than as a few distinct lines.
4102A
4340
4861
di
y aoe lamp
slit
6563
Balmer series
Bohr hydrogen atom (with orbits drawn to scale) only allows
electrons to inhabit discrete orbitals. Electrons falling from
n=3,4,5, or 6 to n=2 accounts for Balmer series of spectral
lines.
A pioneering researcher by the name of Niels Bohr attempted
to improve upon Rutherford's model after studying in
Rutherford's laboratory for several months in 1912. Trying to
harmonize the findings of other physicists (most notably,
Max Planck and Albert Einstein), Bohr suggested that each
electron had a certain, specific amount of energy, and that
their orbits were quantized such that each may occupy
certain places around the nucleus, as marbles fixed in
circular tracks around the nucleus rather than the free-
ranging satellites each were formerly imagined to be. (Figure
above) In deference to the laws of electromagnetics and
accelerating charges, Bohr alluded to these “orbits” as
stationary states to escape the implication that they were in
motion.
Although Bohr's ambitious attempt at re-framing the
structure of the atom in terms that agreed closer to
experimental results was a milestone in physics, it was not
complete. His mathematical analysis produced better
predictions of experimental events than analyses belonging
to previous models, but there were still some unanswered
questions about why electrons should behave in such
strange ways. The assertion that electrons existed in
stationary, quantized states around the nucleus accounted
for experimental data better than Rutherford's model, but he
had no idea what would force electrons to manifest those
particular states. The answer to that question had to come
from another physicist, Louis de Broglie, about a decade
later.
De Broglie proposed that electrons, as photons (particles of
light) manifested both particle-like and wave-like properties.
Building on this proposal, he suggested that an analysis of
orbiting electrons from a wave perspective rather than a
particle perspective might make more sense of their
quantized nature. Indeed, another breakthrough in
understanding was reached.
node node
|
antinode antinode
String vibrating at resonant frequency between two fixed
points forms standing wave.
The atom according to de Broglie consisted of electrons
existing as standing waves, a phenomenon well known to
physicists in a variety of forms. As the plucked string of a
musical instrument (Figure above) vibrating at a resonant
frequency, with “nodes” and “antinodes” at stable positions
along its length. De Broglie envisioned electrons around
atoms standing as waves bent around a circle as in Figure
below.
ge 2
iw My.
wr <
s %
= 8
ode ° =i =
nucleus -. 3
® ror
®
ve
e 2g e
OK “Re ae
(a) “© (b) "%
“Orbiting” electron as standing wave around the nucleus, (a)
two cycles per orbit, (b) three cycles per orbit.
Electrons only could exist in certain, definite “orbits” around
the nucleus because those were the only distances where the
wave ends would match. In any other radius, the wave
should destructively interfere with itself and thus cease to
exist.
De Broglie's hypothesis gave both mathematical support and
a convenient physical analogy to account for the quantized
states of electrons within an atom, but his atomic model was
still incomplete. Within a few years, though, physicists
Werner Heisenberg and Erwin Schrodinger, working
independently of each other, built upon de Broglie's concept
of a matter-wave duality to create more mathematically
rigorous models of subatomic particles.
This theoretical advance from de Broglie's primitive standing
wave model to Heisenberg's matrix and Schrodinger's
differential equation models was given the name quantum
mechanics, and it introduced a rather shocking characteristic
to the world of subatomic particles: the trait of probability, or
uncertainty. According to the new quantum theory, it was
impossible to determine the exact position and exact
momentum of a particle at the same time. The popular
explanation of this “uncertainty principle” was that it was a
measurement error (i.e. by attempting to precisely measure
the position of an electron, you interfere with its momentum
and thus cannot know what it was before the position
measurement was taken, and vice versa). The startling
implication of quantum mechanics is that particles do not
actually have precise positions and momenta, but rather
balance the two quantities in a such way that their combined
uncertainties never diminish below a certain minimum value.
This form of “uncertainty” relationship exists in areas other
than quantum mechanics. As discussed in the “Mixed-
Frequency AC Signals” chapter in volume II of this book
series, there is a mutually exclusive relationship between the
certainty of a waveform's time-domain data and its
frequency-domain data. In simple terms, the more precisely
we know its constituent frequency(ies), the less precisely we
know its amplitude in time, and vice versa. To quote myself:
A waveform of infinite duration (infinite number of
cycles) can be analyzed with absolute precision, but the
less cycles available to the computer for analysis, the
less precise the analysis. .. The fewer times that a wave
cycles, the less certain its frequency is. Taking this
concept to its logical extreme, a short pulse -- a
waveform that doesn't even complete a cycle -- actually
has no frequency, but rather acts as an infinite range of
frequencies. This principle is common to all wave-based
phenomena, not just AC voltages and currents.
In order to precisely determine the amplitude of a varying
signal, we must sample it over a very narrow span of time.
However, doing this limits our view of the wave's frequency.
Conversely, to determine a wave's frequency with great
precision, we must sample it over many cycles, which means
we lose view of its amplitude at any given moment. Thus, we
cannot simultaneously know the instantaneous amplitude
and the overall frequency of any wave with unlimited
precision. Stranger yet, this uncertainty is much more than
observer imprecision; it resides in the very nature of the
wave. It is not as though it would be possible, given the
proper technology, to obtain precise measurements of both
instantaneous amplitude and frequency at once. Quite
literally, a wave cannot have both a precise, instantaneous
amplitude, and a precise frequency at the same time.
The minimum uncertainty of a particle's position and
momentum expressed by Heisenberg and Schrodinger has
nothing to do with limitation in measurement; rather it is an
intrinsic property of the particle's matter-wave dual nature.
Electrons, therefore, do not really exist in their “orbits” as
precisely defined bits of matter, or even as precisely defined
waveshapes, but rather as “clouds” -- the technical term is
wavefunction -- of probability distribution, as if each electron
were “spread” or “smeared” over a range of positions and
momenta.
This radical view of electrons as imprecise clouds at first
seems to contradict the original principle of quantized
electron states: that electrons exist in discrete, defined
“orbits” around atomic nuclei. It was, after all, this discovery
that led to the formation of quantum theory to explain it.
How odd it seems that a theory developed to explain the
discrete behavior of electrons ends up declaring that
electrons exist as “clouds” rather than as discrete pieces of
matter. However, the quantized behavior of electrons does
not depend on electrons having definite position and
momentum values, but rather on other properties called
quantum numbers. |n essence, quantum mechanics
dispenses with commonly held notions of absolute position
and absolute momentum, and replaces them with absolute
notions of a sort having no analogue in common experience.
Even though electrons are known to exist in ethereal, “cloud-
like” forms of distributed probability rather than as discrete
chunks of matter, those “clouds” have other characteristics
that are discrete. Any electron in an atom can be described
by four numerical measures (the previously mentioned
quantum numbers), called the Principal, Angular
Momentum, Magnetic, and Spin numbers. The following is
a synopsis of each of these numbers' meanings:
Principal Quantum Number: Symbolized by the letter n,
this number describes the she//that an electron resides in.
An electron “shell” is a region of space around an atom's
nucleus that electrons are allowed to exist in, corresponding
to the stable “standing wave” patterns of de Broglie and
Bohr. Electrons may “leap” from shell to shell, but cannot
exist between the shell regions.
The principal quantum number must be a positive integer (a
whole number, greater than or equal to 1). In other words,
principal quantum number for an electron cannot be 1/2 or
-3. These integer values were not arrived at arbitrarily, but
rather through experimental evidence of light spectra: the
differing frequencies (colors) of light emitted by excited
hydrogen atoms follow a sequence mathematically
dependent on specific, integer values as illustrated in Figure
previous.
Each shell has the capacity to hold multiple electrons. An
analogy for electron shells is the concentric rows of seats of
an amphitheater. Just as a person seated in an amphitheater
must choose a row to sit in (one cannot sit between rows),
electrons must “choose” a particular shell to “sit” in. As in
amphitheater rows, the outermost shells hold more electrons
than the inner shells. Also, electrons tend to seek the lowest
available shell, as people in an amphitheater seek the closest
seat to the center stage. The higher the shell number, the
greater the energy of the electrons in it.
The maximum number of electrons that any shell may hold is
described by the equation 2n2, where “n” is the principal
quantum number. Thus, the first shell (n=1) can hold 2
electrons; the second shell (n=2) 8 electrons, and the third
Shell (n=3) 18 electrons. (Figure below)
o K L M N O P Q
3 4
|
5,2
1 =
2 g 18 32
observed fill= 2 8 18 32 18 18 2
Principal quantum number n and maximum number of
electrons per shell both predicted by 2(n?), and observed.
Orbitals not to scale.
Electron shells in an atom were formerly designated by letter
rather than by number. The first shell (n=1) was labeled K,
the second shell (n=2) L, the third shell (n=3) M, the fourth
Shell (n=4) N, the fifth shell (n=5) O, the sixth shell (n=6) P,
and the seventh shell (n=7) Q.
Angular Momentum Quantum Number: A shell, is
composed of subshells. One might be inclined to think of
subshells as simple subdivisions of shells, as lanes dividing a
road. The subshells are much stranger. Subshells are regions
of space where electron “clouds” are allowed to exist, and
different subshells actually have different shapes. The first
subshell is shaped like a sphere, (Figure below(s) ) which
makes sense when visualized as a cloud of electrons
surrounding the atomic nucleus in three dimensions. The
second subshell, however, resembles a dumbbell, comprised
of two “lobes” joined together at a single point near the
atom's center. (Figure below(p) ) The third subshell typically
resembles a set of four “lobes” clustered around the atom's
nucleus. These subshell shapes are reminiscent of graphical
depictions of radio antenna signal strength, with bulbous
lobe-shaped regions extending from the antenna in various
directions. (Figure below(d) )
y
X
= = = —_—
/ J W/
Z 1 of 1 1 of 3 1 of 5 1 of 5
p, shown d,2_y2 shown d,z shown
Py, Pz similar dy. dy». d,, similar
(s) (P) (d,2.y2) (dz)
Orbitals: (s) Three fold symmetry. (p) Shown: p,, one of three
possible orientations (py, Py, Pz ), about their respective
axes. (d) Shown: d,7-/7 similar to dyy, Ayy Ayz. Shown: d/.
Possible d-orbital orientations: five.
Valid angular momentum quantum numbers are positive
integers like principal quantum numbers, but also include
zero. These quantum numbers for electrons are symbolized
by the letter I. The number of subshells in a shell is equal to
the shell's principal quantum number. Thus, the first shell
(n=1) has one subshell, numbered 0; the second shell (n=2)
has two subshells, numbered O and 1; the third shell (n=3)
has three subshells, numbered O, 1, and 2.
An older convention for subshell description used letters
rather than numbers. In this notation, the first subshell (l=0)
was designated s, the second subshell (l=1) designated p,
the third subshell (I=2) designated d, and the fourth subshell
(I=3) designated f. The letters come from the words sharp,
principal (not to be confused with the principal quantum
number, n), diffuse, and fundamental. You will still see this
notational convention in many periodic tables, used to
designate the electron configuration of the atoms' outermost,
or valence, shells. (Figure below)
ll
.) Vth
Yrey Nl) } ii il
n= 1 2 3 4 5 n= i :
spectroscopic Is> 2s°2p° 3s73p°'3d" 4s74p°4d” 5s
notation
(a) (b)
1
(a) Bohr representation of Silver atom, (b) Subshell
representation of Ag with division of shells into subshells
(angular quantum number !). This diagram implies nothing
about the actual position of electrons, but represents energy
levels.
Magnetic Quantum Number: The magnetic quantum
number for an electron classifies which orientation its
subshell shape is pointed. The “lobes” for subshells point in
multiple directions. These different orientations are called
orbitals. For the first subshell (s; 1=0), which resembles a
sphere pointing in no “direction”, so there is only one orbital.
For the second (p; |=1) subshell in each shell, which
resembles dumbbells point in three possible directions. Think
of three dumbbells intersecting at the origin, each oriented
along a different axis in a three-axis coordinate space.
Valid numerical values for this quantum number consist of
integers ranging from -| to |, and are symbolized as m, in
atomic physics and I, in nuclear physics. To calculate the
number of orbitals in any given subshell, double the subshell
number and add 1, (2:1 + 1). For example, the first subshell
(I=0) in any shell contains a single orbital, numbered 0; the
second subshell (l=1) in any shell contains three orbitals,
numbered -1, 0, and 1; the third subshell (l=2) contains five
orbitals, numbered -2, -1, 0, 1, and 2; and so on.
Like principal quantum numbers, the magnetic quantum
number arose directly from experimental evidence: The
Zeeman effect, the division of spectral lines by exposing an
ionized gas to a magnetic field, hence the name “magnetic”
quantum number.
Spin Quantum Number: Like the magnetic quantum
number, this property of atomic electrons was discovered
through experimentation. Close observation of spectral lines
revealed that each line was actually a pair of very closely-
spaced lines, and this so-called fine structure was
hypothesized to result from each electron “spinning” on an
axis as if a planet. Electrons with different “spins” would give
off slightly different frequencies of light when excited. The
name “spin” was assigned to this quantum number. The
concept of a spinning electron is now obsolete, being better
suited to the (incorrect) view of electrons as discrete chunks
of matter rather than as “clouds”; but, the name remains.
Spin quantum numbers are symbolized as mg, in atomic
physics and s, in nuclear physics. For each orbital in each
subshell in each shell, there may be two electrons, one with a
spin of +1/2 and the other with a spin of -1/2.
The physicist Wolfgang Pauli developed a principle
explaining the ordering of electrons in an atom according to
these quantum numbers. His principle, called the Pauli
exclusion principle, states that no two electrons in the same
atom may occupy the exact same quantum states. That is,
each electron in an atom has a unique set of quantum
numbers. This limits the number of electrons that may
occupy any given orbital, subshell, and shell.
Shown here is the electron arrangement for a hydrogen
atom:
subshell orbital — spin
m m
reer (/) (m) — (ms)
(n = 1) 0 0 'l, —— One electron
Hydrogen
Atomic number (Z) = 1
(one proton in nucleus)
Spectroscopic notation: 1s!
With one proton in the nucleus, it takes one electron to
electrostatically balance the atom (the proton's positive
electric charge exactly balanced by the electron's negative
electric charge). This one electron resides in the lowest shell
(n=1), the first subshell (I=0), in the only orbital (spatial
orientation) of that subshell (m,=0), with a spin value of 1/2.
A common method of describing this organization is by
listing the electrons according to their shells and subshells in
a convention called spectroscopic notation. |n this notation,
the shell number is shown as an integer, the subshell as a
letter (s,p,d,f), and the total number of electrons in the
subshell (all orbitals, all soins) as a superscript. Thus,
hydrogen, with its lone electron residing in the base level, is
described as 1s!.
Proceeding to the next atom (in order of atomic number), we
have the element helium:
subshell orbital = spin
(/) (m) — (ms)
-'/, —~— electron
Kshell _ © 0
(n= 1) 0 0 > —+— electron
Helium
Atomic number (Z) = 2
(two protons in nucleus)
Spectroscopic notation: 1s*
A helium atom has two protons in the nucleus, and this
necessitates two electrons to balance the double-positive
electric charge. Since two electrons -- one with spin=1/2 and
the other with spin=-1/2 -- fit into one orbital, the electron
configuration of helium requires no additional subshells or
Shells to hold the second electron.
However, an atom requiring three or more electrons wil/
require additional subshells to hold all electrons, since only
two electrons will fit into the lowest shell (n=1). Consider the
next atom in the sequence of increasing atomic numbers,
lithium:
subshell orbital = spin
(/) (m) — (Ms)
pe 0 0 “> ~— electron
K shell 0 0 -'/, —~— electron
(n= 1) 0 0 ‘5 —— electron
Lithium
Atomic number (Z) = 3
Spectroscopic notation: 1s°2s'
An atom of lithium uses a fraction of the L shell's (n=2)
capacity. This shell actually has a total capacity of eight
electrons (maximum shell capacity = 2n? electrons). If we
examine the organization of the atom with a completely filled
L shell, we will see how all combinations of subshells,
orbitals, and spins are occupied by electrons:
subshell orbital = spin
(/) (m) — (m,)
1 1 “f
1 1 "1
1 0 “Ip p subshell
|=1
L shell 1 0 ye 6 ia) ae
1 -1 ne
0 0 'j, | Subshell
1 (| = 0)
0 0 IP 2 electrons
K shell 0 0 “fy s subshell
=0
(n=1) 0 0 ye 2 a
Neon
Atomic number (Z) = 10
Spectroscopic notation: 1s°2s*2p°
Often, when the spectroscopic notation is given for an atom,
any shells that are completely filled are omitted, and the
unfilled, or the highest-level filled shell, is denoted. For
example, the element neon (shown in the previous
illustration), which has two completely filled shells, may be
spectroscopically described simply as 2p® rather than
1s¢2s22p®. Lithium, with its K shell completely filled and a
solitary electron in the L shell, may be described simply as
2s! rather than 1s22s!.
The omission of completely filled, lower-level shells is not just
a notational convenience. It also illustrates a basic principle
of chemistry: that the chemical behavior of an element is
primarily determined by its unfilled shells. Both hydrogen
and lithium have a single electron in their outermost shells
(1s! and 2s!, respectively), giving the two elements some
similar properties. Both are highly reactive, and reactive in
much the same way (bonding to similar elements in similar
modes). It matters little that lithium has a completely filled K
Shell underneath its almost-vacant L shell: the unfilled L shell
is the shell that determines its chemical behavior.
Elements having completely filled outer shells are classified
as noble, and are distinguished by almost complete non-
reactivity with other elements. These elements used to be
classified as /nert, when it was thought that these were
completely unreactive, but are now known to form
compounds with other elements under specific conditions.
Since elements with identical electron configurations in their
outermost shell(s) exhibit similar chemical properties, Dmitri
Mendeleev organized the different elements in a table
accordingly. Such a table is known as a periodic table of the
elements, and modern tables follow this general form in
Figure below.
1 1A 18 ‘VIIA
H 1 He 2
Hysiagan Periodic Table of the Elements Hatum
: toate
la 57|Ce 58/Pr 59 |Nd 60 |Pm 61 [Sm 62/Eu 63) Gd 4 | Tb 65 | Dy 66 | Ho 67 |Er 68) Tm 69 | Yb 70 jlu 71
dantmanite | Lanthanum) Cartum = |preectrran|NeodymiumPromathium) Samarium | Europium |Gackinium) Terbeum yaprostum) Holmium Ertaum Trutu Tertium | Lutedum
sor 138.9055 140.115 | 14090765 14424 (145) 180.35 151.965 157 25 158.92594 162.50 164.93032 167.26 168.90421 173.04 174.967
5d‘éa” 4t'sd'ea 4fta" 4'se 4tts” 4a” area 4t'Sa'éa” | aise” at" ea at" és ats? 4t°Gs” 4t' ‘Gs? 4t'td'a"
te 89] Th 90/Pa g1]U @2 | Np 93 | Pu 94] am 95|Ccm 9 | Bk oF jct SB) Es 99 | Fm 100 | Md 101 | No 102 | Lr 103
Actinte Actinium Thorum | Pretectran| Uranium |Neptuntum | Futonium| Amaricium Qaium |Berkdium (Calfomium Einstartum| Formium |wencwevun| Nobelum | Lrwrencum
serves (227) 232.0081 | 23100583 | 233.0259 (237) (2 (243) (247) (247) (251) (252) (257) 258) (259) (260
6d' 7a” 6a’ SPéd'7s” | Sftd'7s" sr'sd'?s? | Sttecf7s” | Sredtrs St'6d'7s” | Sféd’7s* | St'*eo*7s” | St! eatrs” | sored? 7s” | Sr Gd 7s” af7s’ éd'73”
Periodic table of chemical elements.
Dmitri Mendeleev, a Russian chemist, was the first to develop
a periodic table of the elements. Although Mendeleev
organized his table according to atomic mass rather than
atomic number, and produced a table that was not quite as
useful as modern periodic tables, his development stands as
an excellent example of scientific proof. Seeing the patterns
of periodicity (similar chemical properties according to
atomic mass), Mendeleev hypothesized that all elements
should fit into this ordered scheme. When he discovered
“empty” spots in the table, he followed the logic of the
existing order and hypothesized the existence of heretofore
undiscovered elements. The subsequent discovery of those
elements granted scientific legitimacy to Mendeleev's
hypothesis, furthering future discoveries, and leading to the
form of the periodic table we use today.
This is how science should work: hypotheses followed to their
logical conclusions, and accepted, modified, or rejected as
determined by the agreement of experimental data to those
conclusions. Any fool may formulate a hypothesis after-the-
fact to explain existing experimental data, and many do.
What sets a scientific hypothesis apart from post hoc
speculation is the prediction of future experimental data yet
uncollected, and the possibility of disproof as a result of that
data. To boldly follow a hypothesis to its logical conclusion(s)
and dare to predict the results of future experiments is nota
dogmatic leap of faith, but rather a public test of that
hypothesis, open to challenge from anyone able to produce
contradictory data. In other words, scientific hypotheses are
always “risky” due to the claim to predict the results of
experiments not yet conducted, and are therefore
susceptible to disproof if the experiments do not turn out as
predicted. Thus, if a hypothesis successfully predicts the
results of repeated experiments, its falsehood is disproven.
Quantum mechanics, first as a hypothesis and later as a
theory, has proven to be extremely successful in predicting
experimental results, hence the high degree of scientific
confidence placed in it. Many scientists have reason to
believe that it is an incomplete theory, though, as its
predictions hold true more at micro physical scales than at
macroscopic dimensions, but nevertheless it is a
tremendously useful theory in explaining and predicting the
interactions of particles and atoms.
As you have already seen in this chapter, quantum physics is
essential in describing and predicting many different
phenomena. In the next section, we will see its significance
in the electrical conductivity of solid substances, including
semiconductors. Simply put, nothing in chemistry or solid-
state physics makes sense within the popular theoretical
framework of electrons existing as discrete chunks of matter,
whirling around atomic nuclei like miniature satellites. It is
when electrons are viewed as “wavefunctions” existing in
definite, discrete states that the regular and periodic
behavior of matter can be explained.
e REVIEW:
e Electrons in atoms exist in “clouds” of distributed
probability, not as discrete chunks of matter orbiting the
nucleus like tiny satellites, as common illustrations of
atoms show.
e Individual electrons around an atomic nucleus seek
unique “states,” described by four quantum numbers:
the Principal Quantum Number, known as the shel//; the
Angular Momentum Quantum Number, known as the
subshell, the Magnetic Quantum Number, describing the
orbital (subshell orientation); and the Spin Quantum
Number, or simply spin. These states are quantized,
meaning that no “in-between” conditions exist for an
electron other than those states that fit into the quantum
numbering scheme.
The Principal Quantum Number (n) describes the basic
level or shell that an electron resides in. The larger this
number, the greater radius the electron cloud has from
the atom's nucleus, and the greater that electron's
energy. Principal quantum numbers are whole numbers
(positive integers).
The Angular Momentum Quantum Number (/) describes
the shape of the electron cloud within a particular shell
or level, and is often known as the “subshell.” There are
as many subshells (electron cloud shapes) in any given
Shell as that shell's principal quantum number. Angular
momentum quantum numbers are positive integers
beginning at zero and ending at one less than the
principal quantum number (n-1).
The Magnetic Quantum Number (m,) describes which
orientation a subshell (electron cloud shape) has.
Subshells may assume as many different orientations as
2-times the subshell number (/) plus 1, (2I1+1) (E.g. for
l=1, ml= -1, 0, 1) and each unique orientation is called
an orbital. These numbers are integers ranging from the
negative value of the subshell number (/) through 0 to
the positive value of the subshell number.
The Spin Quantum Number (m,) describes another
property of an electron, and may be a value of +1/2 or
-1/2.
Paull's Exclusion Principle says that no two electrons in
an atom may share the exact same set of quantum
numbers. Therefore, no more than two electrons may
occupy each orbital (spin=1/2 and spin=-1/2), 2/+1
orbitals in every subshell, and n subshells in every shell,
and no more.
Spectroscopic notation is a convention for denoting the
electron configuration of an atom. Shells are shown as
whole numbers, followed by subshell letters (s,p,d,f), with
superscripted numbers totaling the number of electrons
residing in each respective subshell.
e An atom's chemical behavior is solely determined by the
electrons in the unfilled shells. Low-level shells that are
completely filled have little or no effect on the chemical
bonding characteristics of elements.
e Elements with completely filled electron shells are almost
entirely unreactive, and are called noble (formerly known
as inert).
Valence and Crystal structure
Valence: The electrons in the outer most shell, or valence
Shell, are known as valence electrons. These valence
electrons are responsible for the chemical properties of the
chemical elements. It is these electrons which participate in
chemical reactions with other elements. An over simplified
chemistry rule applicable to simple reactions is that atoms
try to form a complete outer shell of 8 electrons (two for the L
Shell). Atoms may give away a few electrons to expose an
underlying complete shell. Atoms may accept a few electrons
to complete the shell. These two processes form ions from
atoms. Atoms may even share electrons among atoms in an
attempt to complete the outer shell. This process forms
molecular bonds. That is, atoms associate to form a molecule.
For example group | elements: Li, Na, K, Cu, Ag, and Au have
a single valence electron. (Figure below) These elements all
have similar chemical properties. These atoms readily give
away one electron to react with other elements. The ability to
easily give away an electron makes these elements excellent
conductors.
Periodic table group IA elements: Li, Na, and K, and group IB
elements: Cu, Ag, and Au have one electron in the outer, or
valence, shell, which is readily donated. Inner shell electrons:
For n= 1, 2, 3, 4; 2n? = 2, 8, 18, 32.
Group VIIA elements: FI, Cl, Br, and | all have 7 electrons in
the outer shell. These elements readily accept an electron to
fill up the outer shell with a full 8 electrons. (Figure below) If
these elements do accept an electron, a negative ion is
formed from the neutral atom. These elements which do not
give up electrons are insulators.
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Periodic table group VIIA elements: F, Cl, Br, and | with 7
valence electrons readily accept an electron in reactions with
other elements.
For example, a Cl atom accepts an electron from an Na atom
to become a Cl’ ion as shown in Figure below. An jonisa
charged particle formed from an atom by either donating or
accepting an electron. As the Na atom donates an electron, it
becomes a Na? ion. This is how Na and Cl atoms combine to
form NaCl, table salt, which is actually NatCl,, a pair of ions.
The Nat and Cl carrying opposite charges, attract one other.
Neutral Sodium atom donates an electron to neutral Chlorine
atom forming Nat and Cr ions.
Sodium chloride crystallizes in the cubic structure shown in
Figure below. This model is not to scale to show the three
dimensional structure. The Na*Cl ions are actually packed
similar to layers of stacked marbles. The easily drawn cubic
crystal structure illustrates that a solid crystal may contain
charged particles.
Group VIIIA elements: He, Ne, Ar, Kr, Xe all have 8 electrons
in the valence shell. (Figure below) That is, the valence shell
is complete meaning these elements neither donate nor
accept electrons. Nor do they readily participate in chemical
reactions since group VIIIA elements do not easily combine
with other elements. In recent years chemists have forced Xe
and Kr to form a few compounds, however for the purposes of
our discussion this is not applicable. These elements are
good electrical insulators and are gases at room temperature.
©©©
Group VIIIA elements: He, Ne, Ar, Kr, Xe are largely
unreactive since the valence shell is complete..
Group IVA elements: C, Si, Ge, having 4 electrons in the
valence shell as shown in Figure below form compounds by
Sharing electrons with other elements without forming ions.
This shared electron bonding is known as covalent bonding.
Note that the center atom (and the others by extension) has
completed its valence shell by sharing electrons. Note that
the figure is a 2-d representation of bonding, which is
actually 3-d. It is this group, IVA, that we are interested in for
its semiconducting properties.
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(a) Group IVA elements: C, Si, Ge having 4 electrons in the
valence shell, (b) complete the valence shell by sharing
electrons with other elements.
Crystal structure: Most inorganic substances form their
atoms (or ions) into an ordered array known as a crystal. The
outer electron clouds of atoms interact in an orderly manner.
Even metals are composed of crystals at the microscopic
level. If a metal sample is given an optical polish, then acid
etched, the microscopic microcrystalline structure shows as
in Figure below. It is also possible to purchase, at
considerable expense, metallic single crystal specimens from
specialized suppliers. Polishing and etching such a specimen
discloses no microcrystalline structure. Practically all
industrial metals are polycrystalline. Most modern
semiconductors, on the other hand, are single crystal
devices. We are primarily interested in monocrystalline
structures.
(a) (b)
(a) Metal sample, (b) polished, (c) acid etched to show
microcrystalline structure.
Many metals are soft and easily deformed by the various
metal working techniques. The microcrystals are deformed in
metal working. Also, the valence electrons are free to move
about the crystal lattice, and from crystal to crystal. The
valence electrons do not belong to any particular atom, but
to all atoms.
The rigid crystal structure in Figure below is composed of a
regular repeating pattern of positive Na ions and negative Cl
ions. The Na and Cl atoms form Nat and Cl ions by
transferring an electron from Na to Cl, with no free electrons.
Electrons are not free to move about the crystal lattice, a
difference compared with a metal. Nor are the ions free. lons
are fixed in place within the crystal structure. Though, the
ions are free to move about if the NaCl crystal is dissolved in
water. However, the crystal no longer exists. The regular,
repeating structure is gone. Evaporation of the water
deposits the Nat and Cl ions in the form of new crystals as
the oppositely charged ions attract each other. lonic
materials form crystal structures due to the strong
electrostatic attraction of the oppositely charged ions.
NaCl crystal having a cubic structure.
Semiconductors in Group 14 (formerly part of Group IV) form
a tetrahedral bonding pattern utilizing the s and p orbital
electrons about the atom, sharing electron-pair bonds to four
adjacent atoms. (Figure below(a) ). Group 14 elements have
four outer electrons: two in a spherical s-orbital and two in p-
orbitals. One of the p-orbitals is unoccupied. The three p-
orbitals hybridize with the s-orbital to form four sp?
molecular orbitals. These four electron clouds repel one
another to equidistant tetrahedral spacing about the Si atom,
attracted by the positive nucleus as shown in Figure below.
y ! _- 7
+ + = .-
NA
aa
fig
¢
@,
2 N
Sz Px Py sp® NZ 7
One s-orbital and three p-orbital electrons hybridize, forming
four sp? molecular orbitals.
Every semiconductor atom, Si, Ge, or C (diamond) is
chemically bonded to four other atoms by covalent bonds,
Shared electron bonds. Two electrons may share an orbital if
each have opposite spin quantum numbers. Thus, an
unpaired electron may share an orbital with an electron from
another atom. This corresponds to overlapping Figure
below(a) of the electron clouds, or bonding. Figure below (b)
iS one fourth of the volume of the diamond crystal structure
unit cell shown in Figure below at the origin. The bonds are
particularly strong in diamond, decreasing in strength going
down group IV to silicon, and germanium. Silicon and
germanium both form crystals with a diamond structure.
(a) Tetrahedral bonding of Si atom. (b) leads to 1/4 of the
cubic unit cell
The diamond unit ce// is the basic crystal building block.
Figure below shows four atoms (dark) bonded to four others
within the volume of the cell. This is equivalent to placing
one of Figure above(b) at the origin in Figure below, then
placing three more on adjacent faces to fill the full cube. Six
atoms fall on the middle of each of the six cube faces,
showing two bonds. The other two bonds to adjacent cubes
were omitted for clarity. Out of eight cube corners, four atoms
bond to an atom within the cube. Where are the other four
atoms bonded? The other four bond to adjacent cubes of the
crystal. Keep in mind that even though four corner atoms
show no bonds in the cube, all atoms within the crystal are
bonded in one giant molecule. A semiconductor crystal is
built up from copies of this unit cell.
Face centered atoms
Atom bonded to 4 others
Other atoms bonded to
chain in cube
Atoms bonded outside of
cube
O08 @
Si, Ge, and C (diamond) form interleaved face centered cube.
The crystal is effectively one molecule. An atom covalent
bonds to four others, which in turn bond to four others, and
so on. The crystal lattice is relatively stiff resisting
deformation. Few electrons free themselves for conduction
about the crystal. A property of semiconductors is that once
an electron is freed, a positively charged empty space
develops which also contributes to conduction.
e REVIEW
e Atoms try to form a complete outer, valence, shell of 8-
electrons (2-electrons for the innermost shell). Atoms
may donate a few electrons to expose an underlying shell
of 8, accept a few electrons to complete a shell, or share
electrons to complete a shell.
e Atoms often form ordered arrays of ions or atoms in a
rigid structure known as a crystal.
e Aneutral atom may form a positive ion by donating an
electron.
e Aneutral atom may form a negative ion by accepting an
electron
e The group IVA semiconductors: C, Si, Ge crystallize into a
diamond structure. Each atom in the crystal is part of a
giant molecule, bonding to four other atoms.
e Most semiconductor devices are manufactured from
single crystals.
Band theory of solids
Quantum physics describes the states of electrons in an atom
according to the four-fold scheme of quantum numbers. The
quantum numbers describe the a//lowable states electrons
may assume in an atom. To use the analogy of an
amphitheater, quantum numbers describe how many rows
and seats are available. Individual electrons may be
described by the combination of quantum numbers, like a
spectator in an amphitheater assigned to a particular row
and seat.
Like spectators in an amphitheater moving between seats
and rows, electrons may change their statuses, given the
presence of available spaces for them to fit, and available
energy. Since shell level is closely related to the amount of
energy that an electron possesses, “leaps” between shell
(and even subshell) levels requires transfers of energy. If an
electron is to move into a higher-order shell, it requires that
additional energy be given to the electron from an external
source. Using the amphitheater analogy, it takes an increase
in energy for a person to move into a higher row of seats,
because that person must climb to a greater height against
the force of gravity. Conversely, an electron “leaping” into a
lower shell gives up some of its energy, like a person jumping
down into a lower row of seats, the expended energy
manifesting as heat and sound.
Not all “leaps” are equal. Leaps between different shells
require a substantial exchange of energy, but leaps between
subshells or between orbitals require lesser exchanges.
When atoms combine to form substances, the outermost
Shells, subshells, and orbitals merge, providing a greater
number of available energy levels for electrons to assume.
When large numbers of atoms are close to each other, these
available energy levels form a nearly continuous band
wherein electrons may move as illustrated in Figure below
3 ——_ 3
p i 3p
Overlap
——- 3s
3s ——_ 3s
Single atom Five atoms Multitudes of atoms
in close proximity in close proximity
Electron band overlap in metallic elements.
It is the width of these bands and their proximity to existing
electrons that determines how mobile those electrons will be
when exposed to an electric field. In metallic substances,
empty bands overlap with bands containing electrons,
meaning that electrons of a single atom may move to what
would normally be a higher-level state with little or no
additional energy imparted. Thus, the outer electrons are
Said to be “free,” and ready to move at the beckoning of an
electric field.
Band overlap will not occur in all substances, no matter how
many atoms are close to each other. In some substances, a
substantial gap remains between the highest band
containing electrons (the so-called va/ence band) and the
next band, which is empty (the so-called conduction band).
See Figure below. As a result, valence electrons are “bound”
to their constituent atoms and cannot become mobile within
the substance without a significant amount of imparted
energy. These substances are electrical insulators.
Conduction band
as
‘Energy gap"
Valence band
Multitudes of atoms
in close proximity
Electron band separation in insulating substances.
Materials that fall within the category of semiconductors
have a narrow gap between the valence and conduction
bands. Thus, the amount of energy required to motivate a
valence electron into the conduction band where it becomes
mobile is quite modest. (Figure below)
semiconducting substance
metalic substance for reference
Conduction band
"Energy gap"
Valence band
(a) (b)
Electron band separation in semiconducting substances, (a)
multitudes of semiconducting close atoms still results in a
significant band gap, (b) multitudes of close metal atoms for
reference.
At low temperatures, little thermal energy is available to
push valence electrons across this gap, and the
semiconducting material acts more as an insulator. At higher
temperatures, though, the ambient thermal energy becomes
enough to force electrons across the gap, and the material
will increase conduction of electricity.
It is difficult to predict the conductive properties of a
substance by examining the electron configurations of its
constituent atoms. Although the best metallic conductors of
electricity (silver, copper, and gold) all have outer s subshells
with a single electron, the relationship between conductivity
and valence electron count is not necessarily consistent:
Specific resistance Electron Specific resistance Electron
Element (p) at 20° Celsius configuration Element (p) at 20° Celsius configuration
Silver (Ag) 9.546 O-cmil/ft 4d'°5s'_ || Molybdenum (Mo) 32.12 Q-cmil/ft 4d°5s'
Copper (Cu) 10.09 Q-cmil/ft 3d'°4s' Zinc (Zn) 35.49 Q-cmil/t 3d'°4s*
Gold (Au) 13.32 Q-cmil/ft 5d'°6s' Nickel (Ni) 41.69 Q-cmil/ft 3d°4s*
Aluminum (Al) 15.94 Q-cmil/ft 3p' lron (Fe) 57.81 Q-cmil/tt 3d°4s*
Tungsten (W) 31.76 Q-cmil/ft 5d‘6s* Platinum (Pt) 63.16 Q-cmil/ft 5d°6s'
The electron band configurations produced by compounds of
different elements defies easy association with the electron
configurations of its constituent elements.
e REVIEW:
e Energy is required to remove an electron from the
valence band to a higher unoccupied band, a conduction
band. More energy is required to move between shells,
less between subshells.
e Since the valence and conduction bands overlap in
metals, little energy removes an electron. Metals are
excellent conductors.
e The large gap between the valence and conduction
bands of an insulator requires high energy to remove an
electron. Thus, insulators do not conduct.
e Semiconductors have a small non-overlapping gap
between the valence and conduction bands. Pure
semiconductors are neither good insulators nor
conductors. Semiconductors are semi-conductive.
Electrons and “holes”
Pure semiconductors are relatively good insulators as
compared with metals, though not nearly as good as a true
insulator like glass. To be useful in semiconductor
applications, the intrinsic semiconductor (pure undoped
semiconductor) must have no more than one impurity atom
in 10 billion semiconductor atoms. This is analogous toa
grain of salt impurity in a railroad boxcar of sugar. Impure, or
dirty semiconductors are considerably more conductive,
though not as good as metals. Why might this be? To answer
that question, we must look at the electron structure of such
materials in Figure below.
Figure below (a) shows four electrons in the valence shell of a
semiconductor forming covalent bonds to four other atoms.
This is a flattened, easier to draw, version of Figure above. All
electrons of an atom are tied up in four covalent bonds, pairs
of shared electrons. Electrons are not free to move about the
crystal lattice. Thus, intrinsic, pure, semiconductors are
relatively good insulators as compared to metals.
(a) Intrinsic semiconductor is an insulator having a complete
electron shell. (b) However, thermal energy can create few
electron hole pairs resulting in weak conduction.
Thermal energy may occasionally free an electron from the
crystal lattice as in Figure above (b). This electron is free for
conduction about the crystal lattice. When the electron was
freed, it left an empty spot with a positive charge in the
crystal lattice known as a hole. This hole is not fixed to the
lattice; but, is free to move about. The free electron and hole
both contribute to conduction about the crystal lattice. That
is, the electron is free until it falls into a hole. This is called
recombination. \f an external electric field is applied to the
semiconductor, the electrons and holes will conduct in
opposite directions. Increasing temperature will increase the
number of electrons and holes, decreasing the resistance.
This is opposite of metals, where resistance increases with
temperature by increasing the collisions of electrons with the
crystal lattice. The number of electrons and holes in an
intrinsic semiconductor are equal. However, both carriers do
not necessarily move with the same velocity with the
application of an external field. Another way of stating this is
that the mobility is not the same for electrons and holes.
Pure semiconductors, by themselves, are not particularly
useful. Though, semiconductors must be refined to a high
level of purity as a starting point prior the addition of specific
impurities.
Semiconductor material pure to 1 part in 10 billion, may
have specific impurities added at approximately 1 part per
10 million to increase the number of carriers. The addition of
a desired impurity to a semiconductor is known as doping.
Doping increases the conductivity of a semiconductor so that
it is more comparable to a metal than an insulator.
It is possible to increase the number of negative charge
carriers within the semiconductor crystal lattice by doping
with an electron donor like Phosphorus. Electron donors, also
known as N-type dopants include elements from group VA of
the periodic table: nitrogen, phosphorus, arsenic, and
antimony. Nitrogen and phosphorus are N-type dopants for
diamond. Phosphorus, arsenic, and antimony are used with
silicon.
The crystal lattice in Figure below (b) contains atoms having
four electrons in the outer shell, forming four covalent bonds
to adjacent atoms. This is the anticipated crystal lattice. The
addition of a phosphorus atom with five electrons in the
outer shell introduces an extra electron into the lattice as
compared with the silicon atom. The pentavalent impurity
forms four covalent bonds to four silicon atoms with four of
the five electrons, fitting into the lattice with one electron
left over. Note that this spare electron is not strongly bonded
to the lattice as the electrons of normal Si atoms are. It is free
to move about the crystal lattice, not being bound to the
Phosphorus lattice site. Since we have doped at one part
phosphorus in 10 million silicon atoms, few free electrons
were created compared with the numerous silicon atoms.
However, many electrons were created compared with the
fewer electron-hole pairs in intrinsic silicon. Application of an
external electric field produces strong conduction in the
doped semiconductor in the conduction band (above the
valence band). A heavier doping level produces stronger
conduction. Thus, a poorly conducting intrinsic
semiconductor has been converted into a good electrical
conductor.
.OMmnOOG
a
electron
hole movement movement
(a) Outer shell electron configuration of donor N-type
Phosphorus, Silicon (for reference), and acceptor P-type
Boron. (b) N-type donor impurity creates free electron (c) P-
type acceptor impurity creates hole, a positive charge
carrier.
It is also possible to introduce an impurity lacking an electron
as compared with silicon, having three electrons in the
valence shell as compared with four for silicon. In Figure
above (c), this leaves an empty spot known asa hole, a
positive charge carrier. The boron atom tries to bond to four
silicon atoms, but only has three electrons in the valence
band. In attempting to form four covalent bonds the three
electrons move around trying to form four bonds. This makes
the hole appear to move. Furthermore, the trivalent atom
may borrow an electron from an adjacent (or more distant)
silicon atom to form four covalent bonds. However, this
leaves the silicon atom deficient by one electron. In other
words, the hole has moved to an adjacent (or more distant)
silicon atom. Holes reside in the valence band, a level below
the conduction band. Doping with an electron acceptor, an
atom which may accept an electron, creates a deficiency of
electrons, the same as an excess of holes. Since holes are
positive charge carriers, an electron acceptor dopant is also
known as a P-type dopant. The P-type dopant leaves the
semiconductor with an excess of holes, positive charge
carriers. The P-type elements from group IIIA of the periodic
table include: boron, aluminum, gallium, and indium. Boron
is used as a P-type dopant for silicon and diamond
semiconductors, while indium is used with germanium.
The “marble in a tube” analogy to electron conduction in
Figure below relates the movement of holes with the
movement of electrons. The marble represent electrons ina
conductor, the tube. The movement of electrons from left to
right as in a wire or N-type semiconductor is explained by an
electron entering the tube at the left forcing the exit of an
electron at the right. Conduction of N-type electrons occurs
in the conduction band. Compare that with the movement of
a hole in the valence band.
electron movement
—» hole movement
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(a) (b) 4 electron movement
Marble in a tube analogy: (a) Electrons move right in the
conduction band as electrons enter tube. (b) Hole moves
right in the valence band as electrons move left.
For a hole to enter at the left of Figure above (b), an electron
must be removed. When moving a hole left to right, the
electron must be moved right to left. The first electron is
ejected from the left end of the tube so that the hole may
move to the right into the tube. The electron is moving in the
opposite direction of the positive hole. As the hole moves
farther to the right, electrons must move left to
accommodate the hole. The hole is the absence of an
electron in the valence band due to P-type doping. It has a
localized positive charge. To move the hole in a given
direction, the valence electrons move in the opposite
direction.
Electron flow in an N-type semiconductor is similar to
electrons moving in a metallic wire. The N-type dopant atoms
will yield electrons available for conduction. These electrons,
due to the dopant are known as majority carriers, for they are
in the majority as compared to the very few thermal holes. If
an electric field is applied across the N-type semiconductor
bar in Figure below (a), electrons enter the negative (left)
end of the bar, traverse the crystal lattice, and exit at right to
the (+) battery terminal.
_~ electron enters electron exits ~
(a) N-type
(b) P-type
(a) N-type semiconductor with electrons moving left to right
through the crystal lattice. (b) P-type semiconductor with
holes moving left to right, which corresponds to electrons
moving in the opposite direction.
Current flow in a P-type semiconductor is a little more
difficult to explain. The P-type dopant, an electron acceptor,
yields localized regions of positive charge known as holes.
The majority carrier in a P-type semiconductor is the hole.
While holes form at the trivalent dopant atom sites, they may
move about the semiconductor bar. Note that the battery in
Figure above (b) is reversed from (a). The positive battery
terminal is connected to the left end of the P-type bar.
Electron flow is out of the negative battery terminal, through
the P-type bar, returning to the positive battery terminal. An
electron leaving the positive (left) end of the semiconductor
bar for the positive battery terminal leaves a hole in the
semiconductor, that may move to the right. Holes traverse
the crystal lattice from left to right. At the negative end of
the bar an electron from the battery combines with a hole,
neutralizing it. This makes room for another hole to move in
at the positive end of the bar toward the right. Keep in mind
that as holes move left to right, that it is actually electrons
moving in the opposite direction that is responsible for the
apparant hole movement.
The elements used to produce semiconductors are
summarized in Figure below. The oldest group IVA bulk
semiconductor material germanium is only used to a limited
extent today. Silicon based semiconductors account for about
90% of commercial production of all semiconductors.
Diamond based semiconductors are a research and
development activity with considerable potential at this
time. Compound semiconductors not listed include silicon
germanium (thin layers on Si wafers), silicon carbide and III-V
compounds such as gallium arsenide. III-VI compound
semiconductors include: AIN, GaN, InN, AIP, AlAs, AlSb, GaP,
GaAs, GaSb, InP, InAs, InSb, Al,Gaj,_,As and In,Gaj_,As.
Columns II and VI of periodic table, not shown in the figure,
also form compound semiconductors.
Elemental semiconductors
C(diamond), Si, Ge
13° «OIA 14 IVA 15 VA
B B 5 1c 6 |N 7
P-type dopant for = —~ [| Gown | Coron | core
2 N,P
\ (N-type dopant for C
B, Al, Ga, In si J
P-type dopant for Si 4
/
Al, Ga, In sens | \ P, As, Sb
P-type dopant for Ge ( 74.90 150 / N-type dopant for Si, Ge
i
Group IIIA P-type dopants, group IV basic semiconductor
materials, and group VA N-type dopants.
The main reason for the inclusion of the IIIA and VA groups in
Figure above is to show the dopants used with the group IVA
semiconductors. Group IIIA elements are acceptors, P-type
dopants, which accept electrons leaving a hole in the crystal
lattice, a positive carrier. Boron is the P-type dopant for
diamond, and the most common dopant for silicon
semiconductors. Indium is the P-type dopant for germanium.
Group VA elements are donors, N-type dopants, yielding a
free electron. Nitrogen and Phosphorus are suitable N-type
dopants for diamond. Phosphorus and arsenic are the most
commonly used N-type dopants for silicon; though, antimony
can be used.
e REVIEW:
e Intrinsic semiconductor materials, pure to 1 part in 10
billion, are poor conductors.
e N-type semiconductor is doped with a pentavalent
impurity to create free electrons. Such a material is
conductive. The electron is the majority carrier.
e P-type semiconductor, doped with a trivalent impurity,
has an abundance of free holes. These are positive
charge carriers. The P-type material is conductive. The
hole is the majority carrier.
e Most semiconductors are based on elements from group
IVA of the periodic table, silicon being the most
prevalent. Germanium is all but obsolete. Carbon
(diamond) is being developed.
e Compound semiconductors such as silicon carbide (group
IVA) and gallium arsenide (group III-V) are widely used.
The P-N junction
If a block of P-type semiconductor is placed in contact with a
block of N-type semiconductor in Figure below(a), the result
is of no value. We have two conductive blocks in contact with
each other, showing no unique properties. The problem is
two separate and distinct crystal bodies. The number of
electrons is balanced by the number of protons in both
blocks. Thus, neither block has any net charge.
However, a single semiconductor crystal manufactured with
P-type material at one end and N-type material at the other
in Figure below (b) has some unique properties. The P-type
material has positive majority charge carriers, holes, which
are free to move about the crystal lattice. The N-type
material has mobile negative majority carriers, electrons.
Near the junction, the N-type material electrons diffuse
across the junction, combining with holes in P-type material.
The region of the P-type material near the junction takes ona
net negative charge because of the electrons attracted. Since
electrons departed the N-type region, it takes on a localized
positive charge. The thin layer of the crystal lattice between
these charges has been depleted of majority carriers, thus, is
known as the depletion region. \t becomes nonconductive
intrinsic semiconductor material. In effect, we have nearly an
insulator separating the conductive P and N doped regions.
no charge
separation
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lector { Noretal lattice ‘intrinsic
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separation
(a)
(a) Blocks of P and N semiconductor in contact have no
exploitable properties. (b) Single crystal doped with P and N
type impurities develops a potential barrier.
This separation of charges at the PN junction constitutes a
potential barrier. This potential barrier must be overcome by
an external voltage source to make the junction conduct. The
formation of the junction and potential barrier happens
during the manufacturing process. The magnitude of the
potential barrier is a function of the materials used in
manufacturing. Silicon PN junctions have a higher potential
barrier than germanium junctions.
In Figure below(a) the battery is arranged so that the
negative terminal supplies electrons to the N-type material.
These electrons diffuse toward the junction. The positive
terminal removes electrons from the P-type semiconductor,
creating holes that diffuse toward the junction. If the battery
voltage is great enough to overcome the junction potential
(0.6V in Si), the N-type electrons and P-holes combine
annihilating each other. This frees up space within the lattice
for more carriers to flow toward the junction. Thus, currents
of N-type and P-type majority carriers flow toward the
junction. The recombination at the junction allows a battery
current to flow through the PN junction diode. Sucha
junction is said to be forward biased.
~ depletion region
electrons —> =— holes electrons ~ ae — holes
(a) Forward (b) Reverse
(a) Forward battery bias repels carriers toward junction,
where recombination results in battery current. (b) Reverse
battery bias attracts carriers toward battery terminals, away
from junction. Depletion region thickness increases. No
sustained battery current flows.
If the battery polarity is reversed as in Figure above(b)
majority carriers are attracted away from the junction toward
the battery terminals. The positive battery terminal attracts
N-type majority carriers, electrons, away from the junction.
The negative terminal attracts P-type majority carriers, holes,
away from the junction. This increases the thickness of the
nonconducting depletion region. There is no recombination
of majority carriers; thus, no conduction. This arrangement of
battery polarity is called reverse bias.
The diode schematic symbol is illustrated in Figure below(b)
corresponding to the doped semiconductor bar at (a). The
diode is a unidirectional device. Electron current only flows in
one direction, against the arrow, corresponding to forward
bias. The cathode, bar, of the diode symbol corresponds to N-
type semiconductor. The anode, arrow, corresponds to the P-
type semiconductor. To remember this relationship, Not-
pointing (bar) on the symbol corresponds to N-type
semiconductor. Pointing (arrow) corresponds to P-type.
electrons —~ =— holes
mA
reverse bias
(a)
N-type P-type
t pointin ointin
erp 9) (P 9) breakdown
(b) cathode anode (c) A
(a) Forward biased PN junction, (b) Corresponding diode
schematic symbol (c) Silicon Diode | vs V characteristic
curve.
If a diode is forward biased as in Figure above(a), current will
increase slightly as voltage is increased from O V. In the case
of a silicon diode a measurable current flows when the
voltage approaches 0.6 V in Figure above(c). As the voltage
increases past 0.6 V, current increases considerably after the
knee. Increasing the voltage well beyond 0.7 V may result in
high enough current to destroy the diode. The forward
voltage, Vf, is a characteristic of the semiconductor: 0.6 to
0.7 V for silicon, 0.2 V for germanium, a few volts for Light
Emitting Diodes (LED). The forward current ranges from a few
mA for point contact diodes to 100 mA for small signal diodes
to tens or thousands of amperes for power diodes.
If the diode is reverse biased, only the leakage current of the
intrinsic semiconductor flows. This is plotted to the left of the
Origin in Figure above(c). This current will only be as high as
1 WA for the most extreme conditions for silicon small signal
diodes. This current does not increase appreciably with
increasing reverse bias until the diode breaks down. At
breakdown, the current increases so greatly that the diode
will be destroyed unless a high series resistance limits
current. We normally select a diode with a higher reverse
voltage rating than any applied voltage to prevent this.
Silicon diodes are typically available with reverse break down
ratings of 50, 100, 200, 400, 800 V and higher. It is possible
to fabricate diodes with a lower rating of a few volts for use
as voltage standards.
We previously mentioned that the reverse leakage current of
under a WA for silicon diodes was due to conduction of the
intrinsic semiconductor. This is the leakage that can be
explained by theory. Thermal energy produces few electron
hole pairs, which conduct leakage current until
recombination. In actual practice this predictable current is
only part of the leakage current. Much of the leakage current
is due to surface conduction, related to the lack of
cleanliness of the semiconductor surface. Both leakage
currents increase with increasing temperature, approaching a
UA for small silicon diodes.
For germanium, the leakage current is orders of magnitude
higher. Since germanium semiconductors are rarely used
today, this is not a problem in practice.
e REVIEW:
e PN junctions are fabricated from a monocrystalline piece
of semiconductor with both a P-type and N-type region in
proximity at a junction.
e The transfer of electrons from the N side of the junction
to holes annihilated on the P side of the junction
produces a barrier voltage. This is 0.6 to 0.7 V in silicon,
and varies with other semiconductors.
e A forward biased PN junction conducts a current once the
barrier voltage is overcome. The external applied
potential forces majority carriers toward the junction
where recombination takes place, allowing current flow.
e Areverse biased PN junction conducts almost no current.
The applied reverse bias attracts majority carriers away
from the junction. This increases the thickness of the
nonconducting depletion region.
e Reverse biased PN junctions show a temperature
dependent reverse leakage current. This is less than a yA
In small silicon diodes.
Junction diodes
There were some historic crude, but usable semiconductor
rectifiers before high purity materials were available.
Ferdinand Braun invented a lead sulfide, PoS, based point
contact rectifier in 1874. Cuprous oxide rectifiers were used
as power rectifiers in 1924. The forward voltage drop is 0.2 V.
The linear characteristic curve perhaps is why Cu50 was used
as a rectifier for the AC scale on D'Arsonval based
multimeters. This diode is also photosensitive.
Selenium oxide rectifiers were used before modern power
diode rectifiers became available. These and the Cu5,0
rectifiers were polycrystalline devices. Photoelectric cells
were once made from Selenium.
Before the modern semiconductor era, an early diode
application was as a radio frequency detector, which
recovered audio from a radio signal. The “semiconductor”
was a polycrystalline piece of the mineral galena, lead
sulfide, PbS. A pointed metallic wire known as a cat whisker
was brought in contact with a spot on a crystal within the
polycrystalline mineral. (Figure below) The operator labored
to find a “sensitive” spot on the galena by moving the cat
whisker about. Presumably there were P and N-type spots
randomly distributed throughout the crystal due to the
variability of uncontrolled impurities. Less often the mineral
iron pyrites, fools gold, was used, as was the mineral
carborundum, silicon carbide, SiC, another detector, part of a
foxhole radio, consisted of a sharpened pencil lead bound to
a bent safety pin, touching a rusty blue-blade disposable
razor blade. These all required searching for a sensitive spot,
easily lost because of vibration.
Crystal detector
Replacing the mineral with an N-doped semiconductor
(Figure below(a) ) makes the whole surface sensitive, so that
searching for a sensitive spot was no longer required. This
device was perfected by G.W.Pickard in 1906. The pointed
metal contact produced a localized P-type region within the
semiconductor. The metal point was fixed in place, and the
whole point contact diode encapsulated in a cylindrical body
for mechanical and electrical stability. (Figure below(d) ) Note
that the cathode bar on the schematic corresponds to the bar
on the physical package.
Silicon point contact diodes made an important contribution
to radar in World War II, detecting giga-hertz radio frequency
echo signals in the radar receiver. The concept to be made
clear is that the point contact diode preceded the junction
diode and modern semiconductors by several decades. Even
to this day, the point contact diode is a practical means of
microwave frequency detection because of its low
Capacitance. Germanium point contact diodes were once
more readily available than they are today, being preferred
for the lower 0.2 V forward voltage in some applications like
self-powered crystal radios. Point contact diodes, though
sensitive to a wide bandwidth, have a low current capability
compared with junction diodes.
Anode Anode
Anode
’
Cathode
(a) Cathode 6) Cathode (c) (d)
Silicon diode cross-section: (a) point contact diode, (b)
Junction diode, (c) schematic symbol, (d) small signal diode
package.
Most diodes today are silicon junction diodes. The cross-
section in Figure above(b) looks a bit more complex than a
simple PN junction; though, it is still a PN junction. Starting
at the cathode connection, the Nt indicates this region is
heavily doped, having nothing to do with polarity. This
reduces the series resistance of the diode. The N' region is
lightly doped as indicated by the (-). Light doping produces a
diode with a higher reverse breakdown voltage, important for
high voltage power rectifier diodes. Lower voltage diodes,
even low voltage power rectifiers, would have lower forward
losses with heavier doping. The heaviest level of doping
produce zener diodes designed for a low reverse breakdown
voltage. However, heavy doping increases the reverse
leakage current. The P* region at the anode contact is
heavily doped P-type semiconductor, a good contact
strategy. Glass encapsulated small signal junction diodes are
capable of 10's to 100's of mA of current. Plastic or ceramic
encapsulated power rectifier diodes handle to 1000's of
amperes of current.
e REVIEW:
e Point contact diodes have superb high frequency
characteristics, usable well into the microwave
frequencies.
Junction diodes range in size from small signal diodes to
power rectifiers capable of 1000's of amperes.
The level of doping near the junction determines the
reverse breakdown voltage. Light doping produces a high
voltage diode. Heavy doping produces a lower
breakdown voltage, and increases reverse leakage
current. Zener diodes have a lower breakdown voltage
because of heavy doping.
Bipolar junction transistors
The bipolar junction transistor (BJT) was named because its
operation involves conduction by two carriers: electrons and
holes in the same crystal. The first bipolar transistor was
invented at Bell Labs by William Shockley, Walter Brattain,
and John Bardeen so late in 1947 that it was not published
until 1948. Thus, many texts differ as to the date of
invention. Brattain fabricated a germanium point contact
transistor, bearing some resemblance to a point contact
diode. Within a month, Shockley had a more practical
junction transistor, which we describe in following
paragraphs. They were awarded the Nobel Prize in Physics in
1956 for the transistor.
The bipolar junction transistor shown in Figure below(a) is an
NPN three layer semiconductor sandwich with an emitter and
collector at the ends, and a base in between. It is as if a third
layer were added to a two layer diode. If this were the only
requirement, we would have no more than a pair of back-to-
back diodes. In fact, it is far easier to build a pair of back-to-
back diodes. The key to the fabrication of a bipolar junction
transistor is to make the middle layer, the base, as thin as
possible without shorting the outside layers, the emitter and
collector. We cannot over emphasize the importance of the
thin base region.
The device in Figure below(a) has a pair of junctions, emitter
to base and base to collector, and two depletion regions.
emitter base _ collector
base _ collector
EY pe E Cc
(a) (b) B - [+
(a) NPN junction bipolar transistor. (b) Apply reverse bias to
collector base junction.
It is customary to reverse bias the base-collector junction of a
bipolar junction transistor as shown in (Figure above(b). Note
that this increases the width of the depletion region. The
reverse bias voltage could be a few volts to tens of volts for
most transistors. There is no current flow, except leakage
current, in the collector circuit.
In Figure below(a), a voltage source has been added to the
emitter base circuit. Normally we forward bias the emitter-
base junction, overcoming the 0.6 V potential barrier. This is
similar to forward biasing a junction diode. This voltage
source needs to exceed 0.6 V for majority carriers (electrons
for NPN) to flow from the emitter into the base becoming
minority carriers in the P-type semiconductor.
If the base region were thick, as in a pair of back-to-back
diodes, all the current entering the base would flow out the
base lead. In our NPN transistor example, electrons leaving
the emitter for the base would combine with holes in the
base, making room for more holes to be created at the (+)
battery terminal on the base as electrons exit.
However, the base is manufactured thin. A few majority
carriers in the emitter, injected as minority carriers into the
base, actually recombine. See Figure below(b). Few electrons
injected by the emitter into the base of an NPN transistor fall
into holes. Also, few electrons entering the base flow directly
through the base to the positive battery terminal. Most of the
emitter current of electrons diffuses through the thin base
into the collector. Moreover, modulating the small base
current produces a larger change in collector current. If the
base voltage falls below approximately 0.6 V for a silicon
transistor, the large emitter-collector current ceases to flow.
E Cc
(a) [Ly [8 “4+ (b)
NPN junction bipolar transistor with reverse biased collector-
base: (a) Adding forward bias to base-emitter junction,
results in (b) a small base current and large emitter and
collector currents.
In Figure below we take a closer look at the current
amplification mechanism. We have an enlarged view of an
NPN junction transistor with emphasis on the thin base
region. Though not shown, we assume that external voltage
sources 1) forward bias the emitter-base junction, 2) reverse
bias the base-collector junction. Electrons, majority carriers,
enter the emitter from the (-) battery terminal. The base
current flow corresponds to electrons leaving the base
terminal for the (+) battery terminal. This is but a small
current compared to the emitter current.
Cc (ayn (d)
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- + region Neier
Disposition of electrons entering base: (a) Lost due to
recombination with base holes. (b) Flows out base lead. (c)
Most diffuse from emitter through thin base into base-
collector depletion region, and (d) are rapidly swept by the
strong depletion region electric field into the collector.
pllscist ss
Majority carriers within the N-type emitter are electrons,
becoming minority carriers when entering the P-type base.
These electrons face four possible fates entering the thin P-
type base. A few at Figure above(a) fall into holes in the base
that contributes to base current flow to the (+) battery
terminal. Not shown, holes in the base may diffuse into the
emitter and combine with electrons, contributing to base
terminal current. Few at (b) flow on through the base to the
(+) battery terminal as if the base were a resistor. Both (a)
and (b) contribute to the very small base current flow. Base
current is typically 1% of emitter or collector current for small
signal transistors. Most of the emitter electrons diffuse right
through the thin base (c) into the base-collector depletion
region. Note the polarity of the depletion region surrounding
the electron at (d). The strong electric field sweeps the
electron rapidly into the collector. The strength of the field is
proportional to the collector battery voltage. Thus 99% of the
emitter current flows into the collector. It is controlled by the
base current, which is 1% of the emitter current. This is a
potential current gain of 99, the ratio of I-/Ip , also Known as
beta, B.
This magic, the diffusion of 99% of the emitter carriers
through the base, is only possible if the base is very thin.
What would be the fate of the base minority carriers in a base
100 times thicker? One would expect the recombination rate,
electrons falling into holes, to be much higher. Perhaps 99%,
instead of 1%, would fall into holes, never getting to the
collector. The second point to make is that the base current
may control 99% of the emitter current, only if 99% of the
emitter current diffuses into the collector. If it all flows out
the base, no control is possible.
Another feature accounting for passing 99% of the electrons
from emitter to collector is that real bipolar junction
transistors use a small heavily doped emitter. The high
concentration of emitter electrons forces many electrons to
diffuse into the base. The lower doping concentration in the
base means fewer holes diffuse into the emitter, which would
increase the base current. Diffusion of carriers from emitter to
base is strongly favored.
The thin base and the heavily doped emitter help keep the
emitter efficiency high, 99% for example. This corresponds to
100% emitter current splitting between the base as 1% and
the collector as 99%. The emitter efficiency is known as a =
Ic/le.
Bipolar junction transistors are available as PNP as well as
NPN devices. We present a comparison of these two in Figure
below. The difference is the polarity of the base emitter diode
junctions, as signified by the direction of the schematic
symbol emitter arrow. It points in the same direction as the
anode arrow for a junction diode, against electron current
flow. See diode junction, Figure previous. The point of the
arrow and bar correspond to P-type and N-type
semiconductors, respectively. For NPN and PNP emitters, the
arrow points away and toward the base respectively. There is
no schematic arrow on the collector. However, the base-
collector junction is the same polarity as the base-emitter
junction compared to a diode. Note, we speak of diode, not
power supply, polarity.
Compare NPN transistor at (a) with the PNP transistor at (b).
Note direction of emitter arrow and supply polarity.
The voltage sources for PNP transistors are reversed
compared with an NPN transistors as shown in Figure above.
The base-emitter junction must be forward biased in both
cases. The base on a PNP transistor is biased negative (b)
compared with positive (a) for an NPN. In both cases the
base-collector junction is reverse biased. The PNP collector
power supply is negative compared with positive for an NPN
transistor.
Collector Collector
Base {
(6) Emitter
Base Emitter Collector
P substrate
Bipolar junction transistor: (a) discrete device cross-section,
(b) schematic symbol, (c) integrated circuit cross-section.
Base Emitter (a) (c)
Note that the BJT in Figure above(a) has heavy doping in the
emitter as indicated by the Nt notation. The base has a
normal P-dopant level. The base is much thinner than the
not-to-scale cross-section shows. The collector is lightly
doped as indicated by the N’ notation. The collector needs to
be lightly doped so that the collector-base junction will have
a high breakdown voltage. This translates into a high
allowable collector power supply voltage. Small signal silicon
transistors have a 60-80 V breakdown voltage. Though, it
may run to hundreds of volts for high voltage transistors. The
collector also needs to be heavily doped to minimize ohmic
losses if the transistor must handle high current. These
contradicting requirements are met by doping the collector
more heavily at the metallic contact area. The collector near
the base is lightly doped as compared with the emitter. The
heavy doping in the emitter gives the emitter-base a low
approximate 7 V breakdown voltage in small signal
transistors. The heavily doped emitter makes the emitter-
base junction have zener diode like characteristics in reverse
bias.
The BJT die, a piece of a sliced and diced semiconductor
wafer, is mounted collector down to a metal case for power
transistors. That is, the metal case is electrically connected
to the collector. A small signal die may be encapsulated in
epoxy. In power transistors, aluminum bonding wires connect
the base and emitter to package leads. Small signal
transistor dies may be mounted directly to the lead wires.
Multiple transistors may be fabricated on a single die called
an integrated circuit. Even the collector may be bonded out
to a lead instead of the case. The integrated circuit may
contain internal wiring of the transistors and other integrated
components. The integrated BJT shown in Figure above(c) is
much thinner than the “not to scale” drawing. The P* region
isolates multiple transistors in a single die. An aluminum
metalization layer (not shown) interconnects multiple
transistors and other components. The emitter region is
heavily doped, N+ compared to the base and collector to
improve emitter efficiency.
Discrete PNP transistors are almost as high quality as the
NPN counterpart. However, integrated PNP transistors are not
nearly a good as the NPN variety within the same integrated
circuit die. Thus, integrated circuits use the NPN variety as
much as possible.
e REVIEW:
e Bipolar transistors conduct current using both electrons
and holes in the same device.
e Operation of a bipolar transistor as a current amplifier
requires that the collector-base junction be reverse
biased and the emitter-base junction be forward biased.
e A transistor differs from a pair of back to back diodes in
that the base, the center layer, is very thin. This allows
majority carriers from the emitter to diffuse as minority
carriers through the base into the depletion region of the
base-collector junction, where the strong electric field
collects them.
e Emitter efficiency is improved by heavier doping
compared with the collector. Emitter efficiency: a = Ic/l_,
0.99 for small signal devices
¢ Current gain is B=I-/lg, 100 to 300 for small signal
transistors.
Junction field-effect transistors
The field effect transistor was proposed by Julius Lilienfeld in
US patents in 1926 and 1933 (1,900,018). Moreover,
Shockley, Brattain, and Bardeen were investigating the field
effect transistor in 1947. Though, the extreme difficulties
sidetracked them into inventing the bipolar transistor
instead. Shockley's field effect transistor theory was
published in 1952. However, the materials processing
technology was not mature enough until 1960 when John
Atalla produced a working device.
A field effect transistor (FET) is a unipolar device, conducting
a current using only one kind of charge carrier. If based on an
N-type slab of semiconductor, the carriers are electrons.
Conversely, a P-type based device uses only holes.
At the circuit level, field effect transistor operation is simple.
A voltage applied to the gate, input element, controls the
resistance of the channel, the unipolar region between the
gate regions. (Figure below) In an N-channel device, this is a
lightly doped N-type slab of silicon with terminals at the
ends. The source and drain terminals are analogous to the
emitter and collector, respectively, of a BJT. In an N-channel
device, a heavy P-type region on both sides of the center of
the slab serves as a control electrode, the gate. The gate is
analogous to the base of a BJT.
“Cleanliness is next to godliness” applies to the manufacture
of field effect transistors. Though it is possible to make
bipolar transistors outside of a clean room, it is a necessity
for field effect transistors. Even in such an environment,
manufacture is tricky because of contamination control
issues. The unipolar field effect transistor is conceptually
simple, but difficult to manufacture. Most transistors today
are a metal oxide semiconductor variety (later section) of the
field effect transistor contained within integrated circuits.
However, discrete JFET devices are available.
Junction field effect transistor cross-section.
A properly biased N-channel junction field effect transistor
(JFET) is shown in Figure above. The gate constitutes a diode
junction to the source to drain semiconductor slab. The gate
is reverse biased. If a voltage (or an ohmmeter) were applied
between the source and drain, the N-type bar would conduct
in either direction because of the doping. Neither gate nor
gate bias is required for conduction. If a gate junction is
formed as shown, conduction can be controlled by the
degree of reverse bias.
Figure below(a) shows the depletion region at the gate
junction. This is due to diffusion of holes from the P-type gate
region into the N-type channel, giving the charge separation
about the junction, with a non-conductive depletion region at
the junction. The depletion region extends more deeply into
the channel side due to the heavy gate doping and light
channel doping.
ae
fi!
N-channel JFET: (a) Depletion at gate diode. (b) Reverse
biased gate diode increases depletion region. (c) Increasing
reverse bias enlarges depletion region. (d) Increasing reverse
bias pinches-off the S-D channel.
The thickness of the depletion region can be increased Figure
above(b) by applying moderate reverse bias. This increases
the resistance of the source to drain channel by narrowing
the channel. Increasing the reverse bias at (c) increases the
depletion region, decreases the channel width, and increases
the channel resistance. Increasing the reverse bias Vez at (d)
will pinch-offthe channel current. The channel resistance will
be very high. This Ves at which pinch-off occurs is Vp, the
pinch-off voltage. It is typically a few volts. In summation, the
channel resistance can be controlled by the degree of reverse
biasing on the gate.
The source and drain are interchangeable, and the source to
drain current may flow in either direction for low level drain
battery voltage (< 0.6 V). That is, the drain battery may be
replaced by a low voltage AC source. For a high drain power
supply voltage, to 10's of volts for small signal devices, the
polarity must be as indicated in Figure below(a). This drain
power supply, not shown in previous figures, distorts the
depletion region, enlarging it on the drain side of the gate.
This is a more correct representation for common DC drain
supply voltages, from a few to tens of volts. As drain voltage
Vps increased,the gate depletion region expands toward the
drain. This increases the length of the narrow channel,
increasing its resistance a little. We say "a little" because
large resistance changes are due to changing gate bias.
Figure below(b) shows the schematic symbol for an N-
channel field effect transistor compared to the silicon cross-
section at (a). The gate arrow points in the same direction as
a junction diode. The “pointing” arrow and “non-pointing”
bar correspond to P and N-type semiconductors, respectively.
N-channel JFET electron current flow from source to drain in
(a) cross-section, (b) schematic symbol.
Figure above shows a large electron current flow from (-)
battery terminal, to FET source, out the drain, returning to
the (+) battery terminal. This current flow may be controlled
by varying the gate voltage. A load in series with the battery
sees an amplified version of the changing gate voltage.
P-channel field effect transistors are also available. The
channel is made of P-type material. The gate is a heavily
dopped N-type region. All the voltage sources are reversed in
the P-channel circuit (Figure below) as compared with the
more popular N-channel device. Also note, the arrow points
out of the gate of the schematic symbol (b) of the P-channel
field effect transistor.
(a)
P-channel JFET: (a) N-type gate, P-type channel, reversed
voltage sources compared with N-channel device. (b) Note
reversed gate arrow and voltage sources on schematic.
As the positive gate bias voltage is increased, the resistance
of the P-channel increases, decreasing the current flow in the
drain circuit.
Discrete devices are manufactured with the cross-section
shown in Figure below. The cross-section, oriented so that it
corresponds to the schematic symbol, is upside down with
respect to a semiconductor wafer. That is, the gate
connections are on the top of the wafer. The gate is heavily
doped, P*, to diffuse holes well into the channel for a large
depletion region. The source and drain connections in this N-
channel device are heavily doped, N* to lower connection
resistance. However, the channel surrounding the gate is
lightly doped to allow holes from the gate to diffuse deeply
into the channel. That is the N’ region.
Drain
Gate
Source
(b)
Source Gate Drain
Gate Source (a) (c) |P_ substrate
Junction field effect transistor: (a) Discrete device cross-
section, (b) schematic symbol, (c) integrated circuit device
cross-section.
All three FET terminals are available on the top of the die for
the integrated circuit version so that a metalization layer
(not shown) can interconnect multiple components. (Figure
above(c) ) Integrated circuit FET's are used in analog circuits
for the high gate input resistance.. The N-channel region
under the gate must be very thin so that the intrinsic region
about the gate can control and pinch-off the channel. Thus,
gate regions on both sides of the channel are not necessary.
Cross-section Junction field-effect transistor
(static induction type)
Drain
Schematic symbol
Drain
Gate
Gate
Source
Source (a) (b)
Junction field effect transistor (static induction type): (a)
Cross-section, (b) schematic symbol.
The static induction field effect transistor (SIT) is a short
channel device with a buried gate. (Figure above) Itisa
power device, aS opposed to a small signal device. The low
gate resistance and low gate to source capacitance make for
a fast switching device. The SIT is capable of hundreds of
amps and thousands of volts. And, is said to be capable of an
incredible frequency of 10 gHz.[YYT]
Source Gate Drain
Drain
Gate a
Source (b)
Metal semiconductor field effect transistor (MESFET): (a)
schematic symbol, (b) cross-section.
The Metal semiconductor field effect transistor (MESFET) is
similar to a JFET except the gate is a schottky diode instead
of a junction diode. A schottky diode is a metal rectifying
contact to a semiconductor compared with a more common
ohmic contact. In Figure above the source and drain are
heavily doped (N*). The channel is lightly doped (N).
MESFET's are higher speed than JFET's. The MESET isa
depletion mode device, normally on, like a JFET. They are
used as microwave power amplifiers to 30 gHz. MESFET's can
be fabricated from silicon, gallium arsenide, indium
phosphide, silicon carbide, and the diamond allotrope of
carbon.
e REVIEW:
e The unipolar junction field effect transistor (FET or JFET)
is So called because conduction in the channel is due to
one type of carrier
e The JFET source, gate, and drain correspond to the BJT's
emitter, base, and collector, respectively.
e Application of reverse bias to the gate varies the channel
resistance by expanding the gate diode depletion region.
Insulated-gate field-effect transistors
(MOSFET)
The insulated-gate field-effect transistor (IGFET), also known
as the metal oxide field effect transistor (MOSFET), is a
derivative of the field effect transistor (FET). Today, most
transistors are of the MOSFET type as components of digital
integrated circuits. Though discrete BJT's are more numerous
than discrete MOSFET's. The MOSFET transistor count within
an integrated circuit may approach hundreds of a million.
The dimensions of individual MOSFET devices are under a
micron, decreasing every 18 months. Much larger MOSFET's
are capable of switching nearly 100 amperes of current at
low voltages; some handle nearly 1000 V at lower currents.
These devices occupy a good fraction of a square centimeter
of silicon. MOSFET's find much wider application than JFET's.
However, MOSFET power devices are not as widely used as
bipolar junction transistors at this time.
The MOSFET has source, gate, and drain terminals like the
FET. However, the gate lead does not make a direct
connection to the silicon compared with the case for the FET.
The MOSFET gate is a metallic or polysilicon layer atop a
silicon dioxide insulator. The gate bears a resemblance to a
metal oxide semiconductor (MOS) capacitor in Figure below.
When charged, the plates of the capacitor take on the charge
polarity of the respective battery terminals. The lower plate is
P-type silicon from which electrons are repelled by the
negative (-) battery terminal toward the oxide, and attracted
by the positive (+) top plate. This excess of electrons near
the oxide creates an inverted (excess of electrons) channel
under the oxide. This channel is also accompanied by a
depletion region isolating the channel from the bulk silicon
substrate.
inverted
channel
Poxide
depletion
P
N-channel MOS capacitor: (a) no charge, (b) charged.
In Figure below (a) the MOS capacitor is placed between a
pair of N-type diffusions in a P-type substrate. With no charge
on the capacitor, no bias on the gate, the N-type diffusions,
the source and drain, remain electrically isolated.
Source Gate Drain
N-channel MOSFET (enhancement type): (a) O V gate bias,
(b) positive gate bias.
A positive bias applied to the gate, charges the capacitor
(the gate). The gate atop the oxide takes on a positive
charge from the gate bias battery. The P-type substrate below
the gate takes on a negative charge. An inversion region with
an excess of electrons forms below the gate oxide. This
region now connects the source and drain N-type regions,
forming a continuous N-region from source to drain. Thus, the
MOSFET, like the FET is a unipolar device. One type of charge
Carrier is responsible for conduction. This example is an N-
channel MOSFET. Conduction of a large current from source
to drain is possible with a voltage applied between these
connections. A practical circuit would have a load in series
with the drain battery in Figure above (b).
The MOSFET described above in Figure above is Known as an
enhancement mode MOSFET. The non-conducting, off,
channel is turned on by enhancing the channel below the
gate by application of a bias. This is the most common kind
of device. The other kind of MOSFET will not be described
here. See the Insulated-gate field-effect transistor chapter for
the depletion mode device.
The MOSFET, like the FET, is a voltage controlled device. A
voltage input to the gate controls the flow of current from
source to drain. The gate does not draw a continuous current.
Though, the gate draws a surge of current to charge the gate
Capacitance.
The cross-section of an N-channel discrete MOSFET is shown
in Figure below (a). Discrete devices are usually optimized for
high power switching. The N* indicates that the source and
drain are heavily N-type doped. This minimizes resistive
losses in the high current path from source to drain. The N"
indicates light doping. The P-region under the gate, between
source and drain can be inverted by application of a positive
bias voltage. The doping profile is a cross-section, which may
be laid out in a serpentine pattern on the silicon die. This
greatly increases the area, and consequently, the current
handling ability.
Drain
Drain
inversion Gate |
Source
— = silicon dioxide
(a) Gate Source __ insulator (b)
N-channel MOSFET (enhancement type): (a) Cross-section,
(b) schematic symbol.
The MOSFET schematic symbol in Figure above (b) shows a
“floating” gate, indicating no direct connection to the silicon
substrate. The broken line from source to drain indicates that
this device is off, not conducting, with zero bias on the gate.
A normally “off” MOSFET is an enhancement mode device.
The channel must be enhanced by application of a bias to
the gate for conduction. The “pointing” end of the substrate
arrow corresponds to P-type material, which points toward an
N-type channel, the “non-pointing” end. This is the symbol
for an N-channel MOSFET. The arrow points in the opposite
direction for a P-channel device (not shown). MOSFET's are
four terminal devices: source, gate, drain, and substrate. The
substrate is connected to the source in discrete MOSFET's,
making the packaged part a three terminal device.
MOSFET's, that are part of an integrated circuit, have the
substrate common to all devices, unless purposely isolated.
This common connection may be bonded out of the die for
connection to a ground or power supply bias voltage.
Drain
Drain
inversion _
Gate | ]
Source
mame = Silicon dioxide
insulator (b)
Gate Source
N-channel “V-MOS” transistor: (a) Cross-section, (b)
schematic symbol.
The V-MOS device in (Figure above) is an improved power
MOSFET with the doping profile arranged for lower on-state
source to drain resistance. VMOS takes its name from the V-
Shaped gate region, which increases the cross-sectional area
of the source-drain path. This minimizes losses and allows
switching of higher levels of power. UMOS, a variation using a
U-shaped grove, is more reproducible in manufacture.
e REVIEW:
e MOSFET's are unipoar conduction devices, conduction
with one type of charge carrier, like a FET, but unlike a
BJT.
e A MOSFET is a voltage controlled device like a FET. A
gate voltage input controls the source to drain current.
e The MOSFET gate draws no continuous current, except
leakage. However, a considerable initial surge of current
is required to charge the gate capacitance.
Thyristors
Thyristors are a broad classification of bipolar-conducting
semiconductor devices having four (or more) alternating N-P-
N-P layers. Thyristors include: silicon controlled rectifier
(SCR), TRIAC, gate turn off switch (GTO), silicon controlled
switch (SCS), AC diode (DIAC), unijunction transistor (UJT),
programmable unijunction transistor (PUT). Only the SCR is
examined in this section; though the GTO is mentioned.
Shockley proposed the four layer diode thyristor in 1950. It
was not realized until years later at General Electric. SCR's
are now available to handle power levels spanning watts to
megawatts. The smallest devices, packaged like small-signal
transistors, switch 100's of milliamps at near 100 VAC. The
largest packaged devices are 172 mm in diameter, switching
5600 Amps at 10,000 VAC. The highest power SCR's may
consist of a whole semiconductor wafer several inches in
diameter (100's of mm).
Anode Anode
Gate
(a) Cathode (b) Cathode
Silicon controlled rectifier (SCR): (a) doping profile, (b) B/T
equivalent circuit.
The silicon controlled rectifier is a four layer diode with a
gate connection as in Figure above (a). When turned on, it
conducts like a diode, for one polarity of current. If not
triggered on, it is nonconducting. Operation is explained in
terms of the compound connected transistor equivalent in
Figure above (b). A positive trigger signal is applied between
the gate and cathode terminals. This causes the NPN
equivalent transistor to conduct. The collector of the
conducting NPN transistor pulls low, moving the PNP base
towards its collector voltage, which causes the PNP to
conduct. The collector of the conducting PNP pulls high,
moving the NPN base in the direction of its collector. This
positive feedback (regeneration) reinforces the NPN's already
conducting state. Moreover, the NPN will now conduct even
in the absence of a gate signal. Once an SCR conducts, it
continues for as long as a positive anode voltage Is present.
For the DC battery shown, this is forever. However, SCR's are
most often used with an alternating current or pulsating DC
supply. Conduction ceases with the expiration of the positive
half of the sinewave at the anode. Moreover, most practical
SCR circuits depend on the AC cycle going to zero to cutoff or
commutate the SCR.
Figure below (a) shows the doping profile of an SCR. Note
that the cathode, which corresponds to an equivalent emitter
of an NPN transistor is heavily doped as Nt indicates. The
anode is also heavily doped (P*). It is the equivalent emitter
of a PNP transistor. The two middle layers, corresponding to
base and collector regions of the equivalent transistors, are
less heavily doped: N and P. This profile in high power SCR's
may be spread across a whole semiconductor wafer of
substantial diameter.
Anode schematic symbols
Anode Anode
Gate Xx Gate x
Cathode Cathode
SCR GTO
(a) (b) (c)
Gate Cathode
Thyristors: (a) Cross-section, (b) silicon controlled rectifier
(SCR) symbol, (c) gate turn-off thyristor (GTO) symbol.
The schematic symbols for an SCR and GTO are shown in
Figures above (b & c). The basic diode symbol indicates that
cathode to anode conduction is unidirectional like a diode.
The addition of a gate lead indicates control of diode
conduction. The gate turn off switch (GTO) has bidirectional
arrows about the gate lead, indicating that the conduction
can be disabled by a negative pulse, as well as initiated by a
positive pulse.
In addition to the ubiquitous silicon based SCR's,
experimental silicon carbide devices have been produced.
Silicon carbide (SiC) operates at higher temperatures, and is
more conductive of heat than any metal, second to diamond.
This should allow for either physically smaller or higher
power Capable devices.
e REVIEW:
e SCR's are the most prevalent member of the thyristor
four layer diode family.
A positive pulse applied to the gate of an SCR triggers it
into conduction. Conduction continues even if the gate
pulse is removed. Conduction only ceases when the
anode to cathode voltage drops to zero.
SCR's are most often used with an AC supply (or
pulsating DC) because of the continuous conduction.
A gate turn off switch (GTO) may be turned off by
application of a negative pulse to the gate.
SCR's switch megawatts of power, up to 5600 A and
10,000 V.
Semiconductor manufacturing
techniques
The manufacture of only silicon based semiconductors is
described in this section; most semiconductors are silicon.
Silicon is particularly suitable for integrated circuits because
it readily forms an oxide coating, useful in patterning
integrated components like transistors.
Silicon is the second most common element in the Earth's
crust in the form of silicon dioxide, SiO>, otherwise known as
silica sand. Silicon is freed from silicon dioxide by reduction
with carbon in an electric arc furnace
Si0, + C = C05+ Si
Such metalurgical grade silicon is suitable for use in silicon
steel transformer laminations, but not nearly pure enough for
semiconductor applications. Conversion to the chloride SiCl,
(or SIHCI3) allows purification by fractional distillation.
Reduction by ultrapure zinc or magnesium yields sponge
silicon, requiring further purification. Or, thermal
decomposition on a hot polycrystalline silicon rod heater by
hydrogen yields ultra pure silicon.
Si + 3HCL = SiHCl; + H
SiHCL3 + H> = Si + 3HCL>
The polycrystalline silicon is melted in a fused silica crucible
heated by an induction heated graphite suceptor. The
graphite heater may alternately be directly driven by a low
voltage at high current. In the Czochralski process, the
silicon melt is solidified on to a pencil sized monocrystal
silicon rod of the desired crystal lattice orientation. (Figure
below) The rod is rotated and pulled upward at a rate to
encourage the diameter to expand to several inches. Once
this diameter is attained, the boule is automatically pulled at
a rate to maintain a constant diameter to a length of a few
feet. Dopants may be added to the crucible melt to create,
for example, a P-type semiconductor. The growing apparatus
is enclosed within an inert atmosphere.
1 lift rod
Si boule
fused silica crucible
RF induction coil
Si melt
Czochralski monocrystalline silicon growth.
The finished boule is ground to a precise final diameter, and
the ends trimmed. The boule is sliced into wafers by an
inside diameter diamond saw. The wafers are ground flat and
polished. The wafers could have an N-type epi/taxia/ layer
grown atop the wafer by thermal deposition for higher
quality. Wafers at this stage of manufacture are delivered by
the silicon wafer manufacturer to the semiconductor
manufacturer.
Si boule
U
cut wafers
diamond blade
driven edge ~~
Silicon boule is diamond sawed into wafers.
The processing of semiconductors involves photo
lithography, a process for making metal lithographic printing
plates by acid etching. The electronics based version of this
is the processing of copper printed circuit boards. This is
reviewed in Figure below as an easy introduction to the photo
lithography involved in semiconductor processing.
rx
(a) copper PCB (b) apply photoresist (c) place artwork (d) expose
) remove artwork (f) develop resist (g) etch copper (h) strip resist
Processing of copper printed circuit boards Is similar to the
photo lithographic steps of semiconductor processing.
We start with a copper foil laminated to an epoxy fiberglass
board in Figure above (a). We also need positive artwork with
black lines corresponding to the copper wiring lines and pads
that are to remain on the finished board. Positive artwork is
required because positive acting resist is used. Though,
negative resist is available for both circuit boards and
semiconductor processing. At (b) the liquid positive photo
resist is applied to the copper face of the printed circuit
board (PCB). It is allowed to dry and may be baked in an
oven. The artwork may be a plastic film positive reproduction
of the original artwork scaled to the required size. The
artwork is placed in contact with the circuit board under a
glass plate at (c). The board is exposed to ultraviolet light (d)
to form a /atent image of softened photo resist. The artwork is
removed (e) and the softened resist washed away by an
alkaline solution (f). The rinsed and dried (baked) circuit
board has a hardened resist image atop the copper lines and
pads that are to remain after etching. The board is immersed
in the etchant (g) to remove copper not protected by
hardened resist. The etched board is rinsed and the resist
removed by a solvent.
The major difference in the patterning of semiconductors is
that a silicon dioxide layer atop the wafer takes the place of
the resist during the high temperature processing steps.
Though, the resist is required in low temperature wet
processing to pattern the silicon dioxide.
An N-type doped silicon wafer in Figure below (a) is the
starting material in the manufacture of semiconductor
junctions. A silicon dioxide layer (b) is grown atop the wafer
in the presence of oxygen or water vapor at high temperature
(over 1000° C in a diffusion furnace. A pool of resist is
applied to the center of the cooled wafer, then spun ina
vacuum chuck to evenly distribute the resist. The baked on
resist (c) has a chrome on glass mask applied to the wafer at
(d). This mask contains a pattern of windows which is
exposed to ultraviolet light (e).
(a) N-type wafer (b) grow SiO, c) apply photoresist (d) place mask
an
(e) expose (f) remove mask (g) develop resist (h) HF etch
sa =u
(i) strip resist (j) P-type diffusion
Manufacture of a silicon diode junction.
After the mask is removed in Figure above (f), the positive
resist can be developed (g) in an alkaline solution, opening
windows in the UV softened resist. The purpose of the resist
is to protect the silicon dioxide from the hydrofluoric acid
etch (h), leaving only open windows corresponding to the
mask openings. The remaining resist (i) is stripped from the
wafer before returning to the diffusion furnace. The wafer is
exposed to a gaseous P-type dopant at high temperature ina
diffusion furnace (j). The dopant only diffuses into the silicon
through the openings in the silicon dioxide layer. Each P-
diffusion through an opening produces a PN junction. If
diodes were the desired product, the wafer would be
diamond scribed and broken into individual diode chips.
However, the whole wafer may be processed further into
bipolar junction transistors.
To convert the diodes into transistors, a small N-type
diffusion in the middle of the existing P-region is required.
Repeating the previous steps with a mask having smaller
Openings accomplishes this. Though not shown in Figure
above (j), an oxide layer was probably formed in that step
during the P-diffusion. The oxide layer over the P-diffusion is
shown in Figure below (k). Positive photo resist is applied and
dried (Il). The chrome on glass emitter mask is applied (m),
and UV exposed (n). The mask is removed (0). The UV
softened resist in the emitter opening is removed with an
alkaline solution (p). The exposed silicon dioxide is etched
away with hydrofluoric acid (HF) at (q)
k) grow SiO, (l) apply photoresist (m) place mask n) expose
) remove mask __(p) develop resist ) HF etch r) strip resist
POCI
eee c
) N-type diffusion (t) metalization
Manufacture of a bipolar junction transistor, continuation of
Manufacture of a silicon diode junction.
After the unexposed resist is stripped from the wafer (r), it is
placed in a diffusion furnace (Figure above (s) for high
temperature processing. An N-type gaseous dopant, such
phosphorus oxychloride (POCI) diffuses through the small
emitter window in the oxide (s). This creates NPN layers
corresponding to the emitter, base, and collector of a BJT. It is
important that the N-type emitter not be driven all the way
through the P-type base, shorting the emitter and collector.
The base region between the emitter and collector also needs
to be thin so that the transistor has a useful B. Otherwise, a
thick base region could form a pair of diodes rather than a
transistor. At (t) metalization is shown making contact with
the transistor regions. This requires a repeat of the previous
steps (not shown here) with a mask for contact openings
through the oxide. Another repeat with another mask defines
the metalization pattern atop the oxide and contacting the
transistor regions through the openings.
The metalization could connect numerous transistors and
other components into an integrated circuit. Though, only
one transistor is shown. The finished wafer is diamond
scribed and broken into individual dies for packaging. Fine
gauge aluminum wire bonds the metalized contacts on the
die to a /ead frame, which brings the contacts out of the final
package.
e REVIEW:
e Most semiconductor are based on ultra pure silicon
because it forms a glass oxide atop the wafer. This oxide
can be patterned with photo lithography, making
complex integrated circuits possible.
e Sausage shaped single crystals of silicon are grown by
the Czochralski process, These are diamond sawed into
wafers.
e The patterning of silicon wafers by photo lithography is
similar to patterning copper printed circuit boards. Photo
resist is applied to the wafer, which is exposed to UV
light through a mask. The resist is developed, then the
wafer is etched.
e hydrofluoric acid etching opens windows in the
protective silicon dioxide atop the wafer.
e Exposure to gaseous dopants at high temperature
produces semiconductor junctions as defined by the
openings in the silicon dioxide layer.
e The photo lithography is repeated for more diffusions,
contacts, and metalization.
e The metalization may interconnect multiple components
into an integrated circuit.
Superconducting devices
Superconducting devices, though not widely used, have
some unique characteristics not available in standard
semiconductor devices. High sensitivity with respect to
amplification of electrical signals, detection of magnetic
fields, and detection of light are prized applications. High
speed switching is also possible, though not applied to
computers at this time. Conventional superconducting
devices must be cooled to within a few degrees of 0 Kelvin
(-273 ° C). Though, work is proceeding at this time on high
temperature superconductor based devices, usable at 90 K
and below. This is significant because inexpensive liquid
nitrogen may be used for cooling.
Superconductivity: Heike Onnes discovered
superconductivity in mercury (Hg) in 1911, for which he won
a Nobel prize. Most metals decrease electrical resistance with
decreasing temperature. Though, most do not decrease to
zero resistance as O Kelvin is approached. Mercury is unique
in that its resistance abruptly drops to zero Q at 4.2 K.
Superconductors lose all resistance abruptly when cooled
below their critical temperature, T- A property of
superconductivity is no power loss in conductors. Current
may flow in a loop of superconducting wire for thousands of
years. Super conductors include lead (Pb), aluminum, (Al),
tin (Sn) and niobium (Nb).
Cooper pair: Lossless conduction in superconductors is not
by ordinary electron flow. Electron flow in normal conductors
encounters opposition as collisions with the rigid ionic metal
crystal lattice. Decreasing vibrations of the crystal lattice
with decreasing temperature accounts for decreasing
resistance- up to a point. Lattice vibrations cease at absolute
zero, but not the energy dissipating collisions of electrons
with the lattice. Thus, normal conductors do not lose all
resistance at absolute zero.
Electrons in superconductors form a pair of electrons called a
cooper pair, as temperature drops below the critical
temperature at which superconductivity begins. The cooper
pair exists because it is at a lower energy level than unpaired
electrons. The electrons are attracted to each other due to
the exchange of phonons, very low energy particles related
to vibrations. This cooper pair, quantum mechanical entity
(particle or wave) is not subject to the normal laws of
physics. This entity propagates through the lattice without
encountering the metal ions comprising the fixed lattice.
Thus, it dissipates no energy. The quantum mechanical
nature of the cooper pair only allows it to exchange discrete
amounts of energy, not continuously variable amounts. An
absolute minimum quantum of energy is acceptable to the
cooper pair. If the vibrational energy of the crystal lattice is
less, (due to the low temperature), the cooper pair cannot
accept it, cannot be scattered by the lattice. Thus, under the
critical temperature, the cooper pairs flow unimpeded
through the lattice.
Josephson junctions: Brian Josephson won a Nobel prize
for his 1962 prediction of the /osepheson junction. A
Josephson junction is a pair of superconductors bridged by a
thin insulator, as in Figure below (a), through which electrons
can tunnel. The first Josephson junctions were lead
superconductors bridged by an insulator. These days a tri-
layer of aluminum and niobium is preferred. Electrons can
tunnel through the insulator even with zero voltage applied
across the superconductors.
If a voltage is applied across the junction, the current
decreases and oscillates at a high frequency proportional to
voltage. The relationship between applied voltage and
frequency is so precise that the standard volt is now defined
in terms of Josephson junction oscillation frequency. The
Josephson junction can also serve as a hypersensitive
detector of low level magnetic fields. It is also very sensitive
to electromagnetic radiation from microwaves to gamma
rays.
lead (Pb)
Gaz ‘
‘lead oxide
(a) Josephson junction, (b) Josephson transistor.
Josephson transistor: An electrode close to the oxide of
the Josephson junction can influence the junction by
Capacitive coupling. Such an assembly in Figure above (b) is
a Josephson transistor. A major feature of the Josephson
transistor is low power dissipation applicable to high density
circuitry, for example, computers. This transistor is generally
part of a more complex superconducting device like a SQUID
or RSFQ.
SQUID: A Superconducting quantum interference device or
SQUID is an assembly of Josephson junctions within a
superconducting ring. Only the DC SQUID Is considered in
this discussion. This device is highly sensitive to low level
magnetic fields.
A constant current bias is forced across the ring in parallel
with both Josephson junctions in Figure below. The current
divides equally between the two junctions in the absence of
an applied magnetic field and no voltage is developed across
across the ring. [JBc] While any value of Magnetic flux (®)
may be applied to the SQUID, only a quantized value (a
multiple of the flux quanta) can flow through the opening in
the superconducting ring.[JBa] If the applied flux is not an
exact multiple of the flux quanta, the excess flux is cancelled
by a circulating current around the ring which produces a
fractional flux quanta. The circulating current will flow in that
direction which cancels any excess flux above a multiple of
the flux quanta. It may either add to, or subtract from the
applied flux, up to +(1/2) a flux quanta. If the circulating
current flows clockwise, the current adds to the top
Josepheson junction and subtracts from the lower one.
Changing applied flux linearly causes the circulating current
to vary as a sinusoid.[JBb] This can be measured as a voltage
across the SQUID. As the applied magnetic field is increased,
a voltage pulse may be counted for each increase by a flux
quanta.[HYP]
counter
Superconduction quantum interference device (SQUID):
Josephson junction pair within a superconducting ring. A
change in flux produces a voltage variation across the J/ pair.
A SQUID is said to be sensitive to 107!4 Tesla, It can detect
the magnetic field of neural currents in the brain at 107?3
Tesla. Compare this with the 30 x 10° Tesla strength of the
Earth's magnetic field.
Rapid single flux quantum (RSFQ): Rather than mimic
silicon semiconductor circuits, RSFQ circuits rely upon new
concepts: magnetic flux quantization within a
superconductor, and movement of the flux quanta produces
a picosecond quantized voltage pulse. Magnetic flux can
only exist within a section of superconductor quantized in
discrete multiples. The lowest flux quanta allowed is
employed. The pulses are switched by Josephson junctions
instead of conventional transistors. The superconductors are
based on atriple layer of aluminum and niobium with a
critical temperature of 9.5 K, cooled to 5 K.
RSQF's operate at over 100 gHz with very little power
dissipation. Manufacture is simple with existing
photolithographic techniques. Though, operation requires
refrigeration down to 5 K. Real world commercial
applications include analog-to-digital and digital to analog
converters, toggle flip-flops, shift registers, memory, adders,
and multipliers.[DKB]
High temperature superconductors: High temperature
superconductors are compounds exhibiting
superconductivity above the liquid nitrogen boiling point of
77 K. This is significant because liquid nitrogen is readily
available and inexpensive. Most conventional
superconductors are metals; widely used high temperature
superconductors are cuprates, mixed oxides of copper (Cu),
for example YBa>Cu307_,, critical temperature, T. = 90K.A
list of others is available.[OXFD] Most of the devices
described in this section are being developed in high
temperature superconductor versions for less critical
applications. Though they do not have the performance of
the conventional metal superconductor devices, the liquid
nitrogen cooling is more available.
REVIEW:
Most metals decrease resistance as they approach
absolute 0; though, the resistance does not drop to 0.
Superconductors experience a rapid drop to zero
resistance at their critical temperature on being cooled.
Typically T, is within 10 K of absolute zero.
A Cooper pair, electron pair, a quantum mechanical
entity, moves unimpeded through the metal crystal
lattice.
Electrons are able to tunnel through a Josephson
junction, an insulating gap across a pair of
superconductors.
The addition of a third electrode, or gate, near the
junction constitutes a Josephson transistor.
A SQUID, Superconduction quantum interference device,
is a highly sensitive detector of magnetic fields. It counts
quantum units of a magnetic field within a
superconducting ring.
RSFQ, Rapid single flux quantum is a high speed
switching device based on switching the magnetic
quanta existing withing a superconducting loop.
High temperature superconductors, T, above liquid
nitrogen boiling point, may also be used to build the
superconducting devices in this section.
Quantum devices
Most integrated circuits are digital, based on MOS (CMOS)
transistors. Every couple of years since the late 1960's a
geometry shrink has taken place, increasing the circuit
density- more circuitry at lower cost in the same space. As of
this writing (2006), the MOS transistor gate length is 65-nm
for leading edge production, with 45-nm anticipated within a
year. At 65-nm leakage currents were becoming evident. At
45-nm, heroic innovations were required to minimize this
leakage. The end of shrinkage in MOS transistors is expected
at 20- to 30-nm. Though some think that 1- to 2-nm is the
limit. Photolithography, or other lithographic techniques, will
continue to improve, providing ever smaller geometry.
However, conventional MOS transistors are not expected to
be usable at these smaller geometries below 20- to 30-nm.
Improved photolithography will have to be applied to other
than the conventional transistors, dimensions (under 20- to
30-nm). The objectional MOS leakage currents are due to
quantum mechanical effects-electron tunneling through gate
oxide, and the narrow channel. In summary, quantum
mechanical effects are a hindrance to ever smaller
conventional MOS transistors. The path to ever smaller
geometry devices involves unique active devices which make
practical use of quantum mechanical principles. As physical
geometry becomes very small, electrons may be treated as
the quantum mechanical equivalent: a wave. Devices
making use of quantum mechanical principles include:
resonant tunneling diodes, quantum tunneling transistors,
metal insulator metal diodes, and quantum dot transistors.
Quantum tunneling: is the passing of electrons through an
insulating barrier which is thin compared to the de Broglie
(here) electron wavelength. If the “electron wave” is large
compared to the barrier, there is a possibility that the wave
appears on both sides of the barrier.
Oo f-
Energy
Energy
Energy
Clasical view Quantum mechanical view
Classical view of an electron surmounting a barrier, or not.
Quantum mechanical view allows an electron to tunnel
through a barrier. The probability (green) is related to the
barrier thickness. After Figure 1 [PHA]
In classical physics, an electron must have sufficient energy
to surmount a barrier. Otherwise, it recoils from the barrier.
(Figure above) Quantum mechanics allows for a probability of
the electron being on the other side of the barrier. If treated
as a wave, the electron may look quite large compared to the
thickness of the barrier. Even when treated as a wave, there
IS only a small probability that it will be found on the other
side of a thick barrier. See green portion of curve, Figure
above. Thinning the barrier increases the probability that the
electron is found on the other side of the barrier. [PHA]
Tunnel diode: The unqualified term tunnel diode refers to
the esaki tunnel diode, an early quantum device. A reverse
biased diode forms a depletion region, an insulating region,
between the conductive anode and cathode. This depletion
region is only thin as compared to the electron wavelength
when heavily doped- 1000 times the doping of a rectifier
diode. With proper biasing, quantum tunneling is possible.
See CH 3 for details.
RTD, resonant tunneling diode: This is a quantum device
not to be confused with the Esaki tunnel diode, CH 3, a
conventional heavily doped bipolar semiconductor. Electrons
tunne!/ through two barriers separated by a well in flowing
source to drain in a resonant tunneling diode. Tunneling is
also known as quantum mechanical tunneling. The flow of
electrons is controlled by diode bias. This matches the
energy levels of the electrons in the source to the quantized
level in the well so that electrons can tunnel through the
barriers. The energy level in the well is quantized because
the well is small. When the energy levels are equal, a
resonance occurs, allowing electron flow through the barriers
as shown in Figure below (b). No bias or too much bias, in
Figures below (a) and (c) respectively, yields an energy
mismatch between the source and the well, and no
conduction.
energy
lavel
Resonant tunneling diode (RTD): (a) No bias, source and well
energy levels not matched, no conduction. (b) Small bias
causes matched energy levels (resonance); conduction
results. (c) Further bias mismatches energy levels,
decreasing conduction.
As bias is increased from zero across the RTD, the current
increases and then decreases, corresponding to off, on, and
off states. This makes simplification of conventional
transistor circuits possible by substituting a pair of RTD's for
two transistors. For example, two back-to-back RTD's and a
transistor form a memory cell, using fewer components, less
area and power compared with a conventional circuit. The
potential application of RTD's is to reduce the component
count, area, and power dissipation of conventional transistor
circuits by replacing some, though not all, transistors. [GEP]
RTD's have been shown to oscillate up to 712 gHz. [ERB]
Double-layer tunneling transistor: The De/tt, otherwise
known as the Double-layer tunneling transistor is constructed
of a pair of conductive wells separated by an insulator or
high band gap semiconductor. (Figure below) The wells are
so thin that electrons are confined to two dimensions. These
are known as quantum wells. A pair of these quantum wells
are insulated by a thin GaAlAs, high band gap (does not
easily conduct) layer. Electrons can tunne/ through the
insulating layer if the electrons in the two quantum wells
have the same momentum and energy. The wells are so thin
that the electron may be treated as a wave- the quantum
mechanical duality of particles and waves. The top and
optional bottom control gates may be adjusted to equalize
the energy levels (resonance) of the electrons to allow
conduction from source to drain. Figure below, barrier
diagram red bars show unequal energy levels in the wells, an
“off-state” condition. Proper biasing of the gates equalizes
the energy levels of electrons in the wells, the “on-state”
condition. The bars would be at the same level in the energy
level diagram.
bottom quantum well top depletion top quantum \
contact (drain) gate i, Oe (drain)
| 2a am,
epiton |
depletion
barrier bottom ~~. bottom gate (optional)
diagram depletion gate
Double-layer tunneling transistor (Deltt) is composed of two
electron containing wells separated by a nonconducting
barrier. The gate voltages may be adjusted so that the
energy and momentum of the electrons in the wells are
equal which permits electrons to tunnel through the
nonconductive barrier. (The energy levels are shown as
unequal in the barrier diagram.)
If gate bias is increased beyond that required for tunneling,
the energy levels in the quantum wells no longer match,
tunneling is inhibited, source to drain current decreases. To
summarize, increasing gate bias from zero results in on, off,
on conditions. This allows a pair of Deltt's to be stacked in
the manner of a CMOS complementary pair; though, different
p- and n-type transistors are not required. Power supply
voltage is about 100 mV. Experimental Deltt's have been
produced which operate near 4.2 K, 77 K, and 0° C. Room
temperature versions are expected.[GEP] [IGB] [PFS]
MIIM diode: The metal/-insulator-insulator-metal (MIIM)
diode is a quantum tunneling device, not based on
semiconductors. See “MIIM diode section” Figure below. The
insulator layers must be thin compared to the de Broglie
(here) electron wavelength, for quantum tunneling to be
possible. For diode action, there must be a prefered
tunneling direction, resulting in a sharp bend in the diode
forward characteristic curve. The MIIM diode has a sharper
forward curve than the metal insulator metal (MIM) diode, not
considered here.
quantum
well
Energy
Energy
Energy
Distance Distance Distance
MIIM diode
section No bias Forward bias Reverse bias
Metal insulator insulator metal (MIIM) diode: Cross section of
diode. Energy levels for no bias, forward bias, and reverse
bias. After Figure 1 [PHI].
The energy levels of M1 and M2 are equal in “no bias” Figure
above. However, (thermal) electrons cannot flow due to the
high 11 and 12 barriers. Electrons in metal M2 have a higher
energy level in “reverse bias” Figure above, but still cannot
overcome the insulator barrier. As “forward bias” Figure
above is increased, a quantum well, an area where electrons
may exist, is formed between the insulators. Electrons may
pass through insulator I1 if M1 is bised at the same energy
level as the quantum well. A simple explanation is that the
distance through the insulators is shorter. A longer
explanation is that as bias increases, the probability of the
electron wave overlapping from M1 to the quantum well
increases. For a more detailed explanation see Phiar Corp.
[PHI]
MIIM devices operate at higher frequencies (3.7 THz) than
microwave transistors. [RCJ3] The addition of a third
electrode to a MIIM diode produces a transistor.
Quantum dot transistor: An isolated conductor may take
on a charge, measured in coulombs for large objects. Fora
nano-scale isolated conductor known as a quantum dot, the
charge is measured in electrons. A quantum dot of 1- to 3-nm
may take on an incremental charge of a single electron. This
is the basis of the quantum dot transistor, also Known as a
single electron transistor.
A quantum dot placed atop a thin insulator over an electron
rich source is known as a Single electron box. (Figure below
(a)) The energy required to transfer an electron is related to
the size of the dot and the number of electrons already on
the dot.
A gate electrode above the quantum dot can adjust the
energy level of the dot so that quantum mechanical
tunneling of an electron (as a wave) from the source through
the insulator is possible. (Figure below (b)) Thus, a single
electron may tunnel to the dot.
+
gate
quantum
dot
tunneling
source drain
(a) (b) (c)
(a) Single electron box, an isolated quantum dot separated
from an electron source by an insulator. (b) Positive charge
on the gate polarizes quantum dot, tunneling an electron
from the source to the dot. (c) Quantum transistor: channel
is replaced by quantum dot surrounded by tunneling barrier.
If the quantum dot is surrounded by a tunnel barrier and
embedded between the source and drain of a conventional
FET, as in Figure above (c) , the charge on the dot can
modulate the flow of electrons from source to drain. As gate
voltage increases, the source to drain current increases, up to
a point. A further increase in gate voltage decreases drain
current. This is similar to the behavior of the RTD and Deltt
resonant devices. Only one kind of transistor is required to
build a complementary logic gate.[GEP]
Single electron transistor: If a pair of conductors,
superconductors, or semiconductors are separated by a pair
of tunnel barriers (insulator), Surrounding a tiny conductive
island, like a quantum dot, the flow of a single charge (a
Cooper pair for superconductors) may be controlled by a
gate. This is a single electron transistor similar to Figure
above (c). Increasing the positive charge on the gate, allows
an electron to tunnel to the island. If it is sufficiently small,
the low capacitance will cause the dot potential to rise
substantially due to the single electron. No more electrons
can tunnel to the island due the electron charge. This is
known at the coulomb blockade. The electron which tunneled
to the island, can tunnel to the drain.
Single electron transistors operate at near absolute zero. The
exception is the graphene single electron transistor, having a
graphene island. They are all experimental devices.
Graphene transistor: Graphite, an allotrope of carbon,
does not have the rigid interlocking crystalline structure of
diamond. None the less, it has a crystalline structure- one
atom thick, a so called two-dimensional structure. A graphite
is a three-dimensional crystal. However, it cleaves into thin
sheets. Experimenters, taking this to the extreme, produce
micron sized specks as thin as a single atom known as
graphene. (Figure below (a)) These membranes have unique
electronic properties. Highly conductive, conduction is by
either electrons or holes, without doping of any kind. [AKG]
Graphene sheets may be cut into transistor structures by
lithographic techniques. The transistors bear some
resemblance to a MOSFET. A gate capacitively coupled to a
graphene channel controls conduction.
As silicon transistors scale to smaller sizes, leakage increases
along with power dissipation. And they get smaller every
couple of years. Graphene transistors dissipate little power.
And, they switch at high speed. Graphene might bea
replacement for silicon someday.
Graphene can be fashioned into devices as small as sixty
atoms wide. Graphene quantum dots within a transistor this
small serve as single electron transistors. Previous single
electron transistors fashioned from either superconductors or
conventional semiconductors operate near absolute zero.
Graphene single electron transistors uniquely function at
room temperature.[]WA]
Graphene transistors are laboratory curiosities at this time. If
they are to go into production two decades from now,
graphene wafers must be produced. The first step,
production of graphene by chemical vapor deposition (CVD)
has been accomplished on an experimental scale. Though, no
wafers are available to date.
(a) Graphene: A single sheet of the graphite allotrope of
carbon. The atoms are arranged in a hexagonal pattern with
a carbon at each intersection. (b) Carbon nanotube: A rolled-
up sheet of graphene.
Carbon nanotube transistor: If a 2-D sheet of graphene is
rolled, the resulting 1-D structure is known as a carbon
nanotube. (Figure above (b)) The reason to treat it as 1-
dimensional is that it is highly conductive. Electrons traverse
the carbon nanotube without being scattered by a crystal
lattice. Resistance in normal metals is caused by scattering
of electrons by the metallic crystalline lattice. If electrons
avoid this scattering, conduction is said to be by ballistic
transport. Both metallic (acting) and semiconducting carbon
nanotubes have been produced. [MBR]
Field effect transistors may be fashioned from a carbon
nanotubes by depositing source and drain contacts on the
ends, and capacitively coupling a gate to the nanotube
between the contacts. Both p- and n-type transistors have
been fabricated. Why the interest in carbon nanotube
transistors? Nanotube semiconductors are Smaller, faster,
lower power compared with silicon transistors. [PNG]
Spintronics: Conventional semiconductors control the flow
of electron charge, current. Digital states are represented by
“on” or “off” flow of current. AS semiconductors become more
dense with the move to smaller geometry, the power that
must be dissipated as heat increases to the point that it is
difficult to remove. Electrons have properties other than
charge such as spin. A tentative explanation of e/ectron spin
is the rotation of distributed electron charge about the spin
axis, analogous to diurnal rotation of the Earth. The loops of
current created by charge movement, form a magnetic field.
However, the electron is more like a point charge than a
distributed charge, Thus, the rotating distributed charge
analogy is not a correct explanation of spin. Electron spin
may have one of two states: up or down which may represent
digital states. More precisely the spin (m,;) quantum number
may be +1/2 the angular momentum (Il) quantum number.
[DDA]
Controlling electron spin instead of charge flow considerably
reduces power dissipation and increases switching speed.
Spintronics, an acronym for SPIN TRansport electrONICS, is
not widely applied because of the difficulty of generating,
controlling, and sensing electron spin. However, high
density, non-volatile magnetic spin memory is in production
using modified semiconductor processes. This is related to
the spin valve magnetic read head used in computer
harddisk drives, not mentioned further here.
A simple magnetic tunnel junction (MTJ) is shown in Figure
below (a), consisting of a pair of ferromagnetic, strong
magnetic properties like iron (Fe), layers separated by a thin
insulator. Electrons can tunnel through a sufficiently thin
insulator due to the quantum mechanical properties of
electrons- the wave nature of electrons. The current flow
through the MT] is a function of the magnetization, spin
polarity, of the ferromagnetic layers. The resistance of the
MT] is low if the magnetic spin of the top layer is in the same
direction (polarity) as the bottom layer. If the magnetic spins
of the two layers oppose, the resistance is higher. [WJG]
— contact
2 2 = — —ferromagnet —
— tunneling ——
insulator
~~ — — — — ferromagnet —
contact
antiferromagnet
contact
(a)
(a) Magnetic tunnel junction (MTJ): Pair of ferromagnetic
layers separated by a thin insulator. The resistance varies
with the magnetization polarity of the top layer (b)
Antiferromagnetic bias magnet and pinned bottom
ferromagnetic layer increases resistance sensitivity to
changes in polarity of the top ferromagnetic layer. Adapted
from [W/G] Figure 3.
The change in resistance can be enhanced by the addition of
an antiferromagnet, material having spins aligned but
opposing, below the bottom layer in Figure above (b). This
bias magnet pins the lower ferromagnetic layer spin to a
single unchanging polarity. The top layer magnetization
(spin) may be flipped to represent data by the application of
an external magnetic field not shown in the figure. The
pinned layer is not affected by external magnetic fields.
Again, the MT] resistance is lowest when the spin of the top
ferromagnetic layer is the same sense as the bottom pinned
ferromagnetic layer. [WJG]
The MTJ may be improved further by splitting the pinned
ferromagnetic layer into two layers separated by a buffer
layer in Figure below (a). This isolates the top layer. The
bottom ferromagnetic layer is pinned by the antiferromagnet
as in the previous figure. The ferromagnetic layer atop the
buffer is attracted by the bottom ferromagnetic layer.
Opposites attract. Thus, the spin polarity of the additional
layer iS opposite of that in the bottom layer due to attraction.
The bottom and middle ferromagnetic layers remain fixed.
The top ferromagnetic layer may be set to either spin polarity
by high currents in proximate conductors (not shown). This is
how data are stored. Data are read out by the difference in
current flow through the tunnel junction. Resistance is lowest
if the layers on both sides of the insulting layer are of the
same spin. [WJG]
—top contact
___ ferromagnet
_— tunneling
insulator
— ferromagnet
—- coupling layer
—— ferromagnet
| pinned layers ;;data |
anti-
ferromagnet
(a) bottom contact (b)
(a)Splitting the pinned ferromagnetic layer of (b) by a buffer
layer improves stability and isolates the top ferromagnetic
unpinned layer. Data are stored in the top ferromagnetic
layer based on spin polarity (b) MT] cell embedded in read
lines of a semiconductor die- one of many MT]'s. Adapted
from [IBM]
An array of magnetic tunnel junctions may be embedded ina
silicon wafer with conductors connecting the top and bottom
terminals for reading data bits from the MT]'s with
conventional CMOS circuitry. One such MT] is shown in Figure
above (b) with the read conductors. Not shown, another
crossed array of conductors carrying heavy write currents
switch the magnetic spin of the top ferromagnetic layer to
store data. A current is applied to one of many “X”
conductors and a “Y” conductor. One MT] in the array is
magnetized under the conductors’ cross-over. Data are read
out by sensing the MTJ current with conventional silicon
semiconductor circuitry. [IBM]
The main reason for interest in magnetic tunnel junction
memory is that it is nonvolatile. It does not lose data when
powered “off”. Other types of nonvolatile memory are
capable of only limited storage cycles. MT] memory is also
higher speed than most semiconductor memory types. It is
now (2006) a commercial product. [TLE]
Not a commercial product, or even a laboratory device, is the
theoretical spin transistor which might one day make spin
logic gates possible. The spin transistor is a derivative of the
theoretical spin diode.
It has been known for some time that electrons flowing
through a cobalt-iron ferromagnet become spin polarized.
The ferromagnet acts as a filter passing electrons of one spin
preferentially. These electrons may flow into an adjacent
nonmagnetic conductor (or semiconductor) retaining the
spin polarization for a short time, nano-seconds. Though,
spin polarized electrons may propagate a considerable
distance compared with semiconductor dimensions. The spin
polarized electrons may be detected by a nickel-iron
ferromagnetic layer adjacent to the semiconductor. [DDA]
[RCJ2]
It has also been shown that electron spin polarization occurs
when circularly polarized light illuminates some
semiconductor materials. Thus, it should be possible to inject
spin polarized electrons into a semiconductor diode or
transistor. The interest in spin based transistors and gates is
because of the non-dissipative nature of spin propagation,
compared with dissipative charge flow. As conventional
semiconductors are scaled down in size, power dissipation
increases. At some point the scaling down will no longer be
practical. Researchers are looking for a replacement for the
conventional charge flow based transistor. That device may
be based on spintronics. [RC}]
e REVIEW:
e As MOS gate oxide thins with each generation of smaller
transistors, excessive gate leakage causes unacceptable
power dissipation and heating. The limit of scaling down
conventional semiconductor geometry is within sight.
Resonant tunneling diode (RTD): Quantum mechanical
effects, which degrade conventional semiconductors, are
employed in the RTD. The flow of electrons through a
sufficiently thin insulator, is by the wave nature of the
electron- particle wave duality. The RTD functions as an
amplifier.
¢e Double layer tunneling transistor (Deltt): The Deltt isa
transistor version of the RTD. Gate bias controls the
ability of electrons to tunnel through a thin insulator from
one quantum well to another (source to drain).
e Quantum dot transistor: A quantum dot, capable of
holding a charge, is surrounded by a thin tunnel barrier
replacing the gate of a conventional FET. The charge on
the quantum dot controls source to drain current flow.
e Spintronics: Electrons have two basic properties: charge
and spin. Conventional electronic devices control the
flow of charge, dissipating energy. Spintronic devices
manipulate electron spin, a propagative, non-dissipative
process.
Semiconductor devices in SPICE
The SPICE (simulation program, integrated circuit emphesis)
electronic simulation program provides circuit elements and
models for semiconductors. The SPICE element names begin
with d, g, J, or m correspond to diode, BJT, JFET and MOSFET
elements, respectively. These elements are accompanied by
corresponding “models” These models have extensive lists of
parameters describing the device. Though, we do not list
them here. In this section we provide a very brief listing of
simple spice models for semiconductors, just enough to get
started. For more details on models and an extensive list of
model parameters see Kuphaldt. [TRK] This reference also
gives instructions on using SPICE.
Diode: The diode statement begins with a diode element
name which must begin with “d” plus optional characters.
Some example diode element names include: d1, d2, dtest,
da, db, d101, etc. Two node numbers specify the connection
of the anode and cathode, respectively, to other components.
The node numbers are followed by a model name, referring
to a “.model” statement.
The model statement line begins with “.model”, followed by
the model name matching one or more diode statements.
Next is a “d” indicating that a diode is being modeled. The
remainder of the model statement is a list of optional diode
parameters of the form ParameterName=ParameterValue.
None are shown in the example below. For a list, see
reference, “diodes”. [TRK]
General form: d[name] [ anode] [ cathode] [ model]
.model [modelname] d ( [ parmtri1=x]
[parmtr2=y] .. .)
Example: di 1 2 modl
.model modil d
Models for specific diode part numbers are often furnished by
the semiconductor diode manufacturer. These models
include parameters. Otherwise, the parameters default to so
called “default values”, as in the example.
BJT, bipolar junction transistor: The BJT element
statement begins with an element name which must begin
with “q” with associated circuit symbol designator
characters, example: q1, q2, qa, qgood. The BJT node
numbers (connections) identify the wiring of the collector,
base, emitter respectively. A model name following the node
numbers is associated with a model statement.
General form: q[ name] [collector] [base] [emitter] [ model]
.model [modelname] [npn or pnp] ([{ parmtr1=x]
4
Example: ql 2 3 © modl
.model modl pnp
Example: q2 7 8 9 q2n090
.model q2n090 npn ( bf=75 )
The model statement begins with “.model”, followed by the
model name, followed by one of “npn” or “pnp”. The optional
list of parameters follows, and may continue for a few lines
beginning with line continuation symbol “+”, plus. Shown
above is the forward B parameter set to 75 for the
hypothetical q2n090 model. Detailed transistor models are
often available from semiconductor manufacturers.
FET, field effect transistor The field effect transistor
element statement begins with an element name beginning
with “j” for JFET associated with some unique characters,
example: j101, j2b, jalpha, etc. The node numbers follow for
the drain, gate and source terminals, respectively. The node
numbers define connectivity to other circuit components.
Finally, a model name indicates the JFET model to use.
General form: j[ name] [drain] [ gate] [ source] [ model]
.model [modelname] [njf or pjf] ( [ parmtri1=x]
“)
Example: jl 2 3 © modl
.model modl pjf
j3 4 5 0 mod2
.model mod2 njf ( vto=-4.0 )
The “.model” in the JFET model statement is followed by the
model name to identify this model to the JFET element
statement(s) using it. Following the model name is either pjf
or njf for p-channel or n-channel JFET's respectively. A long
list of JFET parameters may follow. We only show how to set
Vp, pinch off voltage, to -4.0 V for an n-channel JFET model.
Otherwise, this vto parameter defaults to -2.5 V or 2.5V for n-
channel or p-channel devices, respectively.
MOSFET, metal oxide field effect transistor The
MOSFET element name must begin with “m”, and is the first
word in the element statement. Following are the four node
numbers for the drain, gate, source, and substrate,
respectively. Next is the model name. Note that the source
and substrate are both connected to the same node “0” in
the example. Discrete MOSFET's are packaged as three
terminal devices, the source and substrate are the same
physical terminal. Integrated MOSFET's are four terminal
devices; the substrate is a fourth terminal. Integrated
MOSFET's may have numerous devices sharing the same
substrate, separate from the sources. Though, the sources
might still be connected to the common substrate.
General form: ml name] [drain] [gate] [source] [ substrate]
[ model]
.model [modelname] [nmos or pmos] (
[ parmtrl=x] ... )
Example: ml 2 3 @ © modl
m5 5 6 © O mod4
.model modl1 pmos
.model mod4 nmos ( vto=1 )
The MOSFET model statement begins with “.model” followed
by the model name followed by either “pmos” or “nmos”.
Optional MOSFET model parameters follow. The list of
possible parameters is long. See Volume 5, “MOSFET” for
details. [TRK] MOSFET manufacturers provide detailed
models. Otherwise, defaults are in effect.
The bare minimum semiconductor SPICE information is
provided in this section. The models shown here allow
simulation of basic circuits. In particular, these models do not
account for high speed or high frequency operation.
Simulations are shown in the Volume 5 Chapter 7, “Using
SPICE sa".
e REVIEW:
e Semiconductors may be computer simulated with SPICE.
e SPICE provides element statements and models for the
diode, BJT, JFET, and MOSFET.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Maciej Noszczyski (December 2003): Corrected spelling of
Niels Bohr's name.
Bill Heath (September 2002): Pointed out error in
illustration of carbon atom -- the nucleus was shown with
seven protons instead of six.
Bibliography
1. [DDA]David D. Awschalom, Michael E. Flatte, Nitin
Samarth, “Spintronics”, Scientific American, June 2002 at
http://www.sciam.com
2.[JBa]JJohn Bland, “The Fluxoid” in “A Mossbauer
Spectroscopy and Magnetometry Study of Magnetic
Multilayers and Oxides”, Oliver Lodge Laboratory,
Department of Physics, University of Liverpool, 2002, at
http://www.cmp.liv.ac.uk/frink/thesis/thesis/node45.html
3. JBb]John Bland, “Superconducting Quantum
Interference Device
(SQUID)” in “A Mossbauer Spectroscopy and
Magnetometry Study of Magnetic Multilayers and
Oxides”, Oliver Lodge Laboratory, Department of Physics,
University of Liverpool, 2002, at
http://www.cmp.liv.ac.uk/frink/thesis/thesis/node48.html
4. [JBcJJjohn Bland, “SQUID Magnetometer” in “A Mossbauer
Spectroscopy and Magnetometry Study of Magnetic
Multilayers and Oxides”, Oliver Lodge Laboratory,
Department of Physics, University of Liverpool, 2002, at
http://www.cmp.liv.ac.uk/frink/thesis/thesis/node48.html
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Systems”, Hypres, Inc., NY, at
http://www.hypres.com/papers/Brock-RSFQ-CirSys-
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semiconducting record”, Cnet News, December 19, 2003,
at http://news.com.com/2100-1006-5129761.html
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Graphics, May 13, 1998, at
http://www.aip.org/mgr/png/html/tubefet.htm
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as Submillimetre-Wave Sources”, Philosophical
Transactions: Mathematical, Physical and Engineering
Sciences, Vol. 354, No. 1717, The Current Status of
Semiconductor Tunnelling Devices (Oct. 15, 1996), pp.
2365-2381 at http://links jstor.org/sici?sici=1364-
503X(19961015)354%3A1717%3C2365%3ARTDASS%3
E2.0.CO%3B2-Q
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magnetic tunnel junction MRAM at IBM: From first
junctions to a 16-Mb MRAM demonstrator chip”, IBM
Research and Development, Spintronics, Volume 50,
Number 1, 2006, at
http://www.research.iom.com/journal/rd/501/gallagherht
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Technology to Date”, IBM Research, at
http://domino.research.iobm.com/comm/pr.nsf/pages/news
.20030610_ mram.html
[GEP]Linda Geppert “Quantum Transistors: toward
nanoectronic”, IEEE Spectrum, September 2000, at
http://www.ece.osu.edu/~berger/press/quan0900.pdf
[AKG]JA. K. Geim1 and K. S. Novoselov1 , “The rise of
graphene”, Nature Materials, 6, 2007, at
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9.html
13.
14.
15.
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18.
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[IGB]llan Greenberg, “Transistor Technology Takes a
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toward live chips,” EE Times, 11/06/2006, at
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[TRK]Tony R. Kuphaldt, “Lessons in Electricity”,
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[TLE]Tom Lee, “Is nonvolatile MRAM right for your
consumer embedded device application? ”, Freescale
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February 4, 1998, at
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Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—| | 4/l—
Lessons In Electric Circuits --
Volume Ill
Chapter 3
DIODES AND RECTIFIERS
Introduction
Meter check of a diode
Diode ratings
Rectifier circuits
Peak detector
Clipper circuits
Clamper circuits
Voltage multipliers
Inductor commutating circuits
Diode switching circuits
o Logic
o Analog_switch
Zener diodes
Special-purpose diodes
o Schottky diodes
o Tunnel diodes
Light-emitting diodes
Laser diodes
Photodiodes
Solar cells
Varicap or varactor diodes
Snap diode
PIN diodes
IMPATT diode
Gunn diode
o Shockley diode
o Constant-current diodes
Other diode technologies
o SiC diodes
o Polymer diode
SPICE models
Contributors
Bibliography
oOo 0 0 0 G0 0 0 O
Introduction
A diode is an electrical device allowing current to move through it in
one direction with far greater ease than in the other. The most
common kind of diode in modern circuit design is the semiconductor
diode, although other diode technologies exist. Semiconductor diodes
are symbolized in schematic diagrams such as Figure below. The term
“diode” is customarily reserved for small signal devices, | <= 1 A. The
term rectifier is used for power devices, | > 1A.
SS ee ee
Semiconductor diode schematic symbol: Arrows indicate the direction
of electron current flow.
When placed in a simple battery-lamp circuit, the diode will either
allow or prevent current through the lamp, depending on the polarity
of the applied voltage. (Figure below)
Diode operation: (a) Current flow is permitted; the diode is forward
biased. (b) Current flow ts prohibited; the diode Is reversed biased.
When the polarity of the battery is such that electrons are allowed to
flow through the diode, the diode is said to be forward-biased.
Conversely, when the battery is “backward” and the diode blocks
current, the diode is said to be reverse-biased. A diode may be
thought of as like a switch: “closed” when forward-biased and “open”
when reverse-biased.
Oddly enough, the direction of the diode symbol's “arrowhead” points
against the direction of electron flow. This is because the diode
symbol was invented by engineers, who predominantly use
conventional flow notation in their schematics, showing current as a
flow of charge from the positive (+) side of the voltage source to the
negative (-). This convention holds true for all semiconductor symbols
possessing “arrowheads:” the arrow points in the permitted direction
of conventional flow, and against the permitted direction of electron
flow.
Diode behavior is analogous to the behavior of a hydraulic device
called a check valve. A check valve allows fluid flow through it in only
one direction as in Figure below.
= |
Hydraulic
check valve
Flow permitted (b) Flow prohibited
Hydraulic check valve analogy: (a) Electron current flow permitted. (b)
Current flow prohibited.
Check valves are essentially pressure-operated devices: they open
and allow flow if the pressure across them is of the correct “polarity”
to open the gate (in the analogy shown, greater fluid pressure on the
right than on the left). If the pressure is of the opposite “polarity,” the
pressure difference across the check valve will close and hold the gate
so that no flow occurs.
Like check valves, diodes are essentially “pressure-” operated
(voltage-operated) devices. The essential difference between forward-
bias and reverse-bias is the polarity of the voltage dropped across the
diode. Let's take a closer look at the simple battery-diode-lamp circuit
shown earlier, this time investigating voltage drops across the various
components in Figure below.
Diode circuit voltage measurements: (a) Forward biased. (b) Reverse
biased.
A forward-biased diode conducts current and drops a small voltage
across it, leaving most of the battery voltage dropped across the lamp.
If the battery's polarity is reversed, the diode becomes reverse-biased,
and drops a// of the battery's voltage leaving none for the lamp. If we
consider the diode to be a self-actuating switch (closed in the forward-
bias mode and open in the reverse-bias mode), this behavior makes
sense. The most substantial difference is that the diode drops a lot
more voltage when conducting than the average mechanical switch
(0.7 volts versus tens of millivolts).
This forward-bias voltage drop exhibited by the diode is due to the
action of the depletion region formed by the P-N junction under the
influence of an applied voltage. If no voltage applied is across a
semiconductor diode, a thin depletion region exists around the region
of the P-N junction, preventing current flow. (Figure below (a)) The
depletion region is almost devoid of available charge carriers, and
acts as an insulator:
P-N junction representation
Lt Depletion region
Anode Cathode
Schematic symbol
(b) J Stripe marks cathode
—{ | Real component appearance
(c)
Diode representations: PN-junction model, schematic symbol, physical
part.
The schematic symbol of the diode is shown in Figure above (b) such
that the anode (pointing end) corresponds to the P-type
semiconductor at (a). The cathode bar, non-pointing end, at (b)
corresponds to the N-type material at (a). Also note that the cathode
stripe on the physical part (c) corresponds to the cathode on the
symbol.
If a reverse-biasing voltage is applied across the P-N junction, this
depletion region expands, further resisting any current through it.
(Figure below)
- +
T
LU
Reverse-biased — Depletion region
Depletion region expands with reverse bias.
Conversely, if a forward-biasing voltage is applied across the P-N
junction, the depletion region collapses becoming thinner. The diode
becomes less resistive to current through it. In order for a sustained
current to go through the diode; though, the depletion region must be
fully collapsed by the applied voltage. This takes a certain minimum
voltage to accomplish, called the forward voltage as illustrated in
Figure below.
0.4V
Partial forward-biased
(a) LI Depletion region (b) Depletion region fully collapsed
Inceasing forward bias from (a) to (b) decreases depletion region
thickness.
For silicon diodes, the typical forward voltage is 0.7 volts, nominal. For
germanium diodes, the forward voltage is only 0.3 volts. The chemical
constituency of the P-N junction comprising the diode accounts for its
nominal forward voltage figure, which is why silicon and germanium
diodes have such different forward voltages. Forward voltage drop
remains approximately constant for a wide range of diode currents,
meaning that diode voltage drop is not like that of a resistor or even a
normal (closed) switch. For most simplified circuit analysis, the
voltage drop across a conducting diode may be considered constant
at the nominal figure and not related to the amount of current.
Actually, forward voltage drop is more complex. An equation describes
the exact current through a diode, given the voltage dropped across
the junction, the temperature of the junction, and several physical
constants. It is commonly known as the diode equation:
Ip =I (o'Y™ - 1)
Where,
I, = Diode current in amps
I, = Saturation current.in amps
(typically 1 x 10°'* amps)
e = Euler’s constant (~ 2.718281828)
q = charge of electron (1.6 x 10°’? coulombs)
Vp = Voltage applied across diode in volts
N = "Nonideality" or "emission" coefficient
(typically between 1 and 2)
k = Boltzmann's constant (1.38 x 10°)
T = Junction temperature in Kelvins
The term kT/q describes the voltage produced within the P-N junction
due to the action of temperature, and is called the thermal! voltage, or
V, of the junction. At room temperature, this is about 26 millivolts.
Knowing this, and assuming a “nonideality” coefficient of 1, we may
simplify the diode equation and re-write it as such:
=k eVv/0.02 26 1)
Where,
I, = Diode current in amps
I, = Saturation current, in amps
(typically 1 x 10°'* amps)
e = Euler’s Number (~ 2.718281 828)
Vp = Voltage applied across diode in volts
You need not be familiar with the “diode equation” to analyze simple
diode circuits. Just understand that the voltage dropped across a
current-conducting diode does change with the amount of current
going through it, but that this change is fairly small over a wide range
of currents. This is why many textbooks simply say the voltage drop
across a conducting, semiconductor diode remains constant at 0.7
volts for silicon and 0.3 volts for germanium. However, some circuits
intentionally make use of the P-N junction's inherent exponential
current/voltage relationship and thus can only be understood in the
context of this equation. Also, since temperature is a factor in the
diode equation, a forward-biased P-N junction may also be used as a
temperature-sensing device, and thus can only be understood if one
has a conceptual grasp on this mathematical relationship.
A reverse-biased diode prevents current from going through it, due to
the expanded depletion region. In actuality, a very small amount of
current can and does go through a reverse-biased diode, called the
leakage current, but it can be ignored for most purposes. The ability of
a diode to withstand reverse-bias voltages is limited, as it is for any
insulator. If the applied reverse-bias voltage becomes too great, the
diode will experience a condition known as breakdown (Figure below),
which is usually destructive. A diode's maximum reverse-bias voltage
rating is Known as the Peak Inverse Voltage, or PIV, and may be
obtained from the manufacturer. Like forward voltage, the PIV rating
of a diode varies with temperature, except that PIV increases with
increased temperature and decreases as the diode becomes cooler --
exactly opposite that of forward voltage.
Ip
forward |
reverse-bias forward-bias
Vp
breakdown! reverse |
Diode curve: showing knee at 0.7 V forward bias for Si, and reverse
breakdown.
Typically, the PIV rating of a generic “rectifier” diode is at least 50
volts at room temperature. Diodes with PIV ratings in the many
thousands of volts are available for modest prices.
e REVIEW:
e A diode is an electrical component acting as a one-way valve for
current.
e When voltage is applied across a diode in such a way that the
diode allows current, the diode is said to be forward-biased.
e When voltage is applied across a diode in such a way that the
diode prohibits current, the diode is said to be reverse-biased.
e The voltage dropped across a conducting, forward-biased diode is
called the forward voltage. Forward voltage for a diode varies only
slightly for changes in forward current and temperature, and is
fixed by the chemical composition of the P-N junction.
Silicon diodes have a forward voltage of approximately 0.7 volts.
Germanium diodes have a forward voltage of approximately 0.3
volts.
The maximum reverse-bias voltage that a diode can withstand
without “breaking down” is called the Peak Inverse Voltage, or PIV
rating.
Meter check of a diode
Being able to determine the polarity (cathode versus anode) and basic
functionality of a diode is a very important skill for the electronics
hobbyist or technician to have. Since we know that a diode is
essentially nothing more than a one-way valve for electricity, it makes
sense we should be able to verify its one-way nature using a DC
(battery-powered) ohmmeter as in Figure below. Connected one way
across the diode, the meter should show a very low resistance at (a).
Connected the other way across the diode, it should show a very high
resistance at (b) (“OL’ on some digital meter models).
Anode ¥
Cathode
Cath k
Anode
Determination of diode polarity: (a) Low resistance indicates forward
bias, black lead is cathode and red lead anode (for most meters) (b)
Reversing leads shows high resistance indicating reverse bias.
Of course, to determine which end of the diode is the cathode and
which is the anode, you must know with certainty which test lead of
the meter is positive (+) and which is negative (-) when set to the
“resistance” or “QO” function. With most digital multimeters I've seen,
the red lead becomes positive and the black lead negative when set
to measure resistance, in accordance with standard electronics color-
code convention. However, this is not guaranteed for all meters. Many
analog multimeters, for example, actually make their black leads
positive (+) and their red leads negative (-) when switched to the
“resistance” function, because it is easier to manufacture it that way!
One problem with using an ohmmeter to check a diode is that the
readings obtained only have qualitative value, not quantitative. In
other words, an ohmmeter only tells you which way the diode
conducts; the low-value resistance indication obtained while
conducting is useless. If an ohmmeter shows a value of “1.73 ohms”
while forward-biasing a diode, that figure of 1.73 QO doesn't represent
any real-world quantity useful to us as technicians or circuit
designers. It neither represents the forward voltage drop nor any
“bulk” resistance in the semiconductor material of the diode itself, but
rather is a figure dependent upon both quantities and will vary
substantially with the particular ohmmeter used to take the reading.
For this reason, some digital multimeter manufacturers equip their
meters with a special “diode check” function which displays the
actual forward voltage drop of the diode in volts, rather than a
“resistance” figure in ohms. These meters work by forcing a small
current through the diode and measuring the voltage dropped
between the two test leads. (Figure below)
O54
f '
a“) Anode
’
Cathode
Meter with a “Diode check” function displays the forward voltage drop
of 0.548 volts instead of a low resistance.
The forward voltage reading obtained with such a meter will typically
be less than the “normal” drop of 0.7 volts for silicon and 0.3 volts for
germanium, because the current provided by the meter is of trivial
proportions. If a multimeter with diode-check function isn't available,
or you would like to measure a diode's forward voltage drop at some
non-trivial current, the circuit of Figure below may be constructed
using a battery, resistor, and voltmeter
Resistor
Measuring forward voltage of a diode without“diode check” meter
function: (a) Schematic diagram. (b) Pictorial diagram.
Connecting the diode backwards to this testing circuit will simply
result in the voltmeter indicating the full voltage of the battery.
If this circuit were designed to provide a constant or nearly constant
current through the diode despite changes in forward voltage drop, it
could be used as the basis of a temperature-measurement instrument,
the voltage measured across the diode being inversely proportional to
diode junction temperature. Of course, diode current should be kept
to a minimum to avoid self-heating (the diode dissipating substantial
amounts of heat energy), which would interfere with temperature
measurement.
Beware that some digital multimeters equipped with a “diode check”
function may output a very low test voltage (less than 0.3 volts) when
set to the regular “resistance” (Q) function: too low to fully collapse
the depletion region of a PN junction. The philosophy here is that the
“diode check” function is to be used for testing semiconductor
devices, and the “resistance” function for anything else. By using a
very low test voltage to measure resistance, it is easier for a
technician to measure the resistance of non-semiconductor
components connected to semiconductor components, since the
semiconductor component junctions will not become forward-biased
with such low voltages.
Consider the example of a resistor and diode connected in parallel,
soldered in place on a printed circuit board (PCB). Normally, one
would have to unsolder the resistor from the circuit (disconnect it from
all other components) before measuring its resistance, otherwise any
parallel-connected components would affect the reading obtained.
When using a multimeter which outputs a very low test voltage to the
probes in the “resistance” function mode, the diode's PN junction will
not have enough voltage impressed across it to become forward-
biased, and will only pass negligible current. Consequently, the meter
“sees” the diode as an open (no continuity), and only registers the
resistor's resistance. (Figure below)
Ohmmeter equipped with a low test voltage (<0.7 V) does not see
diodes allowing it to measure parallel resistors.
If such an ohmmeter were used to test a diode, it would indicate a
very high resistance (many mega-ohms) even if connected to the
diode in the “correct” (forward-biased) direction. (Figure below)
Ohmmeter equipped with a low test voltage, too low to forward bias
diodes, does not see diodes.
Reverse voltage strength of a diode is not as easily tested, because
exceeding a normal diode's PIV usually results in destruction of the
diode. Special types of diodes, though, which are designed to “break
down” in reverse-bias mode without damage (called zener diodes),
which are tested with the same voltage source / resistor / voltmeter
circuit, provided that the voltage source is of high enough value to
force the diode into its breakdown region. More on this subject in a
later section of this chapter.
e REVIEW:
e An ohmmeter may be used to qualitatively check diode function.
There should be low resistance measured one way and very high
resistance measured the other way. When using an ohmmeter for
this purpose, be sure you know which test lead is positive and
which is negative! The actual polarity may not follow the colors of
the leads as you might expect, depending on the particular design
of meter.
e Some multimeters provide a “diode check” function that displays
the actual forward voltage of the diode when its conducting
current. Such meters typically indicate a slightly lower forward
voltage than what is “nominal” for a diode, due to the very small
amount of current used during the check.
Diode ratings
In addition to forward voltage drop (V;) and peak inverse voltage
(PIV), there are many other ratings of diodes important to circuit
design and component selection. Semiconductor manufacturers
provide detailed specifications on their products -- diodes included --
in publications known as datasheets. Datasheets for a wide variety of
semiconductor components may be found in reference books and on
the internet. | prefer the internet as a source of component
specifications because all the data obtained from manufacturer
websites are up-to-date.
A typical diode datasheet will contain figures for the following
parameters:
Maximum repetitive reverse voltage = Vary, the maximum amount of
voltage the diode can withstand in reverse-bias mode, in repeated
pulses. Ideally, this figure would be infinite.
Maximum DC reverse voltage = Vp or Voc, the maximum amount of
voltage the diode can withstand in reverse-bias mode on a continual
basis. Ideally, this figure would be infinite.
Maximum forward voltage = Vr, usually specified at the diode's rated
forward current. Ideally, this figure would be zero: the diode providing
no opposition whatsoever to forward current. In reality, the forward
voltage is described by the “diode equation.”
Maximum (average) forward current = Ir;ay), the maximum average
amount of current the diode is able to conduct in forward bias mode.
This is fundamentally a thermal limitation: how much heat can the PN
junction handle, given that dissipation power is equal to current (1)
multiplied by voltage (V or E) and forward voltage is dependent upon
both current and junction temperature. Ideally, this figure would be
infinite.
Maximum (peak or surge) forward current = Icy OF igsurgey, the
maximum peak amount of current the diode is able to conduct in
forward bias mode. Again, this rating is limited by the diode junction's
thermal capacity, and is usually much higher than the average
current rating due to thermal inertia (the fact that it takes a finite
amount of time for the diode to reach maximum temperature for a
given current). Ideally, this figure would be infinite.
Maximum total dissipation = Pp, the amount of power (in watts)
allowable for the diode to dissipate, given the dissipation (P=IE) of
diode current multiplied by diode voltage drop, and also the
dissipation (P=I?R) of diode current squared multiplied by bulk
resistance. Fundamentally limited by the diode's thermal capacity
(ability to tolerate high temperatures).
Operating junction temperature = T), the maximum allowable
temperature for the diode's PN junction, usually given in degrees
Celsius (°C). Heat is the “Achilles' heel” of semiconductor devices:
they must be kept cool to function properly and give long service life.
Storage temperature range = Tot, the range of allowable
temperatures for storing a diode (unpowered). Sometimes given in
conjunction with operating junction temperature (T)), because the
maximum storage temperature and the maximum operating
temperature ratings are often identical. If anything, though, maximum
storage temperature rating will be greater than the maximum
operating temperature rating.
Thermal resistance = R(O), the temperature difference between
junction and outside air (R(O)),) or between junction and leads
(R(©),) for a given power dissipation. Expressed in units of degrees
Celsius per watt (°C/W). Ideally, this figure would be zero, meaning
that the diode package was a perfect thermal conductor and radiator,
able to transfer all heat energy from the junction to the outside air (or
to the leads) with no difference in temperature across the thickness of
the diode package. A high thermal resistance means that the diode
will build up excessive temperature at the junction (where its critical)
despite best efforts at cooling the outside of the diode, and thus will
limit its maximum power dissipation.
Maximum reverse Current = Ip, the amount of current through the
diode in reverse-bias operation, with the maximum rated inverse
voltage applied (Vpc). Sometimes referred to as /eakage current.
Ideally, this figure would be zero, as a perfect diode would block all
current when reverse-biased. In reality, it is very small compared to
the maximum forward current.
Typical junction capacitance = C), the typical amount of capacitance
intrinsic to the junction, due to the depletion region acting asa
dielectric separating the anode and cathode connections. This is
usually a very small figure, measured in the range of picofarads (pF).
Reverse recovery time = t,,, the amount of time it takes for a diode to
“turn off” when the voltage across it alternates from forward-bias to
reverse-bias polarity. Ideally, this figure would be zero: the diode
halting conduction immediately upon polarity reversal. For a typical
rectifier diode, reverse recovery time is in the range of tens of
microseconds; for a “fast switching” diode, it may only be a few
nanoseconds.
Most of these parameters vary with temperature or other operating
conditions, and so a single figure fails to fully describe any given
rating. Therefore, manufacturers provide graphs of component ratings
plotted against other variables (such as temperature), so that the
circuit designer has a better idea of what the device is capable of.
Rectifier circuits
Now we come to the most popular application of the diode:
rectification. Simply defined, rectification is the conversion of
alternating current (AC) to direct current (DC). This involves a device
that only allows one-way flow of electrons. As we have seen, this is
exactly what a semiconductor diode does. The simplest kind of
rectifier circuit is the ha/Fwave rectifier. It only allows one half of an
AC waveform to pass through to the load. (Figure below)
AC
voltage
source
Half-wave rectifier circuit.
For most power applications, half-wave rectification is insufficient for
the task. The harmonic content of the rectifier's output waveform is
very large and consequently difficult to filter. Furthermore, the AC
power source only supplies power to the load one half every full cycle,
meaning that half of its capacity is unused. Half-wave rectification is,
however, a very simple way to reduce power to a resistive load. Some
two-position lamp dimmer switches apply full AC power to the lamp
filament for “full” brightness and then half-wave rectify it for a lesser
light output. (Figure below)
Bright
AC
voltage (\) (})
source
Half-wave rectifier application: Two level lamp dimmer.
In the “Dim” switch position, the incandescent lamp receives
approximately one-half the power it would normally receive operating
on full-wave AC. Because the half-wave rectified power pulses far more
rapidly than the filament has time to heat up and cool down, the lamp
does not blink. Instead, its filament merely operates at a lesser
temperature than normal, providing less light output. This principle of
“oulsing” power rapidly to a slow-responding load device to control
the electrical power sent to it is common in the world of industrial
electronics. Since the controlling device (the diode, in this case) is
either fully conducting or fully nonconducting at any given time, it
dissipates little heat energy while controlling load power, making this
method of power control very energy-efficient. This circuit is perhaps
the crudest possible method of pulsing power to a load, but it suffices
as a proof-of-concept application.
If we need to rectify AC power to obtain the full use of both half-cycles
of the sine wave, a different rectifier circuit configuration must be
used. Such a circuit is called a full-wave rectifier. One kind of full-wave
rectifier, called the center-tap design, uses a transformer with a
center-tapped secondary winding and two diodes, as in Figure below.
Sa EI
Full-wave rectifier, center-tapped design.
AC
voltage
source
This circuit's operation is easily understood one half-cycle at a time.
Consider the first half-cycle, when the source voltage polarity is
positive (+) on top and negative (-) on bottom. At this time, only the
top diode is conducting; the bottom diode is blocking current, and the
load “sees” the first half of the sine wave, positive on top and
negative on bottom. Only the top half of the transformer's secondary
winding carries current during this half-cycle as in Figure below.
Full-wave center-tap rectifier: Top half of secondary winding conducts
during positive half-cycle of input, delivering positive half-cycle to
load..
During the next half-cycle, the AC polarity reverses. Now, the other
diode and the other half of the transformer's secondary winding carry
current while the portions of the circuit formerly carrying current
during the last half-cycle sit idle. The load still “sees” half of a sine
wave, of the same polarity as before: positive on top and negative on
bottom. (Figure below)
Full-wave center-tap rectifier: During negative input half-cycle, bottom
half of secondary winding conducts, delivering a positive half-cycle to
the load.
One disadvantage of this full-wave rectifier design is the necessity of
a transformer with a center-tapped secondary winding. If the circuit in
question is one of high power, the size and expense of a Suitable
transformer is significant. Consequently, the center-tap rectifier
design is only seen in low-power applications.
The full-wave center-tapped rectifier polarity at the load may be
reversed by changing the direction of the diodes. Furthermore, the
reversed diodes can be paralleled with an existing positive-output
rectifier. The result is dual-polarity full-wave center-tapped rectifier in
Figure below. Note that the connectivity of the diodes themselves is
the same configuration as a bridge.
AC voltage source
Dual polarity full-wave center tap rectifier
Another, more popular full-wave rectifier design exists, and it is built
around a four-diode bridge configuration. For obvious reasons, this
design is called a full-wave bridge. (Figure below)
Full-wave bridge rectifier.
Current directions for the full-wave bridge rectifier circuit are as shown
in Figure below for positive half-cycle and Figure below for negative
half-cycles of the AC source waveform. Note that regardless of the
polarity of the input, the current flows in the same direction through
the load. That is, the negative half-cycle of source is a positive half-
cycle at the load. The current flow is through two diodes in series for
both polarities. Thus, two diode drops of the source voltage are lost
(0.7:2=1.4 V for Si) in the diodes. This is a disadvantage compared
with a full-wave center-tap design. This disadvantage is only a
problem in very low voltage power supplies.
Full-wave bridge rectifier: Electron flow for positive half-cycles.
> >
Full-wave bridge rectifier: Electron flow for negative half=cycles.
Remembering the proper layout of diodes in a full-wave bridge
rectifier circuit can often be frustrating to the new student of
electronics. I've found that an alternative representation of this circuit
is easier both to remember and to comprehend. It's the exact same
circuit, except all diodes are drawn in a horizontal attitude, all
“pointing” the same direction. (Figure below)
na
Vi A
AC
voltage
source
Alternative layout style for Full-wave bridge rectifier.
One advantage of remembering this layout for a bridge rectifier circuit
is that it expands easily into a polyphase version in Figure below.
3-phase
AC source =
Load
Three-phase full-wave bridge rectifier circuit.
Each three-phase line connects between a pair of diodes: one to route
power to the positive (+) side of the load, and the other to route
power to the negative (-) side of the load. Polyphase systems with
more than three phases are easily accommodated into a bridge
rectifier scheme. Take for instance the six-phase bridge rectifier circuit
in Figure below.
6-phase
AC source
Load
Six-phase full-wave bridge rectifier circuit.
When polyphase AC is rectified, the phase-shifted pulses overlap each
other to produce a DC output that is much “smoother” (has less AC
content) than that produced by the rectification of single-phase AC.
This is a decided advantage in high-power rectifier circuits, where the
sheer physical size of filtering components would be prohibitive but
low-noise DC power must be obtained. The diagram in Figure below
shows the full-wave rectification of three-phase AC.
1 2 3
TIME —~
Resultant DC waveform
J XXAKXKKKAKK
Three-phase AC and 3-phase full-wave rectifier output.
In any case of rectification -- single-phase or polyphase -- the amount
of AC voltage mixed with the rectifier's DC output is called ripple
voltage. In most cases, since “pure” DC is the desired goal, ripple
voltage is undesirable. If the power levels are not too great, filtering
networks may be employed to reduce the amount of ripple in the
output voltage.
Sometimes, the method of rectification is referred to by counting the
number of DC “pulses” output for every 360° of electrical “rotation.” A
single-phase, half-wave rectifier circuit, then, would be called a 1-
pulse rectifier, because it produces a single pulse during the time of
one complete cycle (360°) of the AC waveform. A single-phase, full-
wave rectifier (regardless of design, center-tap or bridge) would be
called a 2-pulse rectifier, because it outputs two pulses of DC during
one AC cycle's worth of time. A three-phase full-wave rectifier would
be called a 6-pulse unit.
Modern electrical engineering convention further describes the
function of a rectifier circuit by using a three-field notation of phases,
ways, and number of pulses. A single-phase, half-wave rectifier circuit
is given the somewhat cryptic designation of 1Ph1W1P (1 phase, 1
way, 1 pulse), meaning that the AC supply voltage is single-phase,
that current on each phase of the AC supply lines moves in only one
direction (way), and that there is a single pulse of DC produced for
every 360° of electrical rotation. A single-phase, full-wave, center-tap
rectifier circuit would be designated as 1Ph1W2P in this notational
system: 1 phase, 1 way or direction of current in each winding half,
and 2 pulses or output voltage per cycle. A single-phase, full-wave,
bridge rectifier would be designated as 1Ph2W2P: the same as for the
center-tap design, except current can go both ways through the AC
lines instead of just one way. The three-phase bridge rectifier circuit
shown earlier would be called a 3Ph2W6P rectifier.
Is it possible to obtain more pulses than twice the number of phases in
a rectifier circuit? The answer to this question is yes: especially in
polyphase circuits. Through the creative use of transformers, sets of
full-wave rectifiers may be paralleled in such a way that more than six
pulses of DC are produced for three phases of AC. A 30° phase shift is
introduced from primary to secondary of a three-phase transformer
when the winding configurations are not of the same type. In other
words, a transformer connected either Y-A or A-Y will exhibit this 30°
phase shift, while a transformer connected Y-Y or A-A will not. This
phenomenon may be exploited by having one transformer connected
Y-Y feed a bridge rectifier, and have another transformer connected Y-
A feed a second bridge rectifier, then parallel the DC outputs of both
rectifiers. (Figure below) Since the ripple voltage waveforms of the two
rectifiers' outputs are phase-shifted 30° from one another, their
superposition results in less ripple than either rectifier output
considered separately: 12 pulses per 360° instead of just six:
3Ph2W12P rectifier circuit
Primary
3-phase
AC input
Polyphase rectifier circuit: 3-phase 2-way 12-pulse (3Ph2W12P)
REVIEW:
Rectification is the conversion of alternating current (AC) to direct
current (DC).
e A half-wave rectifier is a circuit that allows only one half-cycle of
the AC voltage waveform to be applied to the load, resulting in
one non-alternating polarity across it. The resulting DC delivered
to the load “pulsates” significantly.
e A full-wave rectifier is a circuit that converts both half-cycles of
the AC voltage waveform to an unbroken series of voltage pulses
of the same polarity. The resulting DC delivered to the load
doesn't “pulsate” as much.
e Polyphase alternating current, when rectified, gives a much
“smoother” DC waveform (less ripp/e voltage) than rectified
single-phase AC.
Peak detector
A peak detector is a series connection of a diode and a capacitor
outputting a DC voltage equal to the peak value of the applied AC
Signal. The circuit is shown in Figure below with the corresponding
SPICE net list. An AC voltage source applied to the peak detector,
charges the capacitor to the peak of the input. The diode conducts
positive “half cycles,” charging the capacitor to the waveform peak.
When the input waveform falls below the DC “peak” stored on the
Capacitor, the diode is reverse biased, blocking current flow from
capacitor back to the source. Thus, the capacitor retains the peak
value even as the waveform drops to zero. Another view of the peak
detector is that it is the same as a half-wave rectifier with a filter
capacitor added to the output.
-KSPICE 03441.eps
C1 2 0 O.1u
R1 13 1.0k
V1 10 SIN(O 5 1k)
V(2) D1 3 2 diode
= .model diode d
.tran 0.01m 50mm
.end
Peak detector: Diode conducts on positive half cycles charging
capacitor to the peak voltage (less diode forward drop).
It takes a few cycles for the capacitor to charge to the peak as in
Figure below due to the series resistance (RC “time constant”). Why
does the capacitor not charge all the way to 5 V? It would charge to 5
V if an “ideal diode” were obtainable. However, the silicon diode has a
forward voltage drop of 0.7 V which subtracts from the 5 V peak of the
input.
Peak detector: Capacitor charges to peak within a few cycles.
The circuit in Figure above could represent a DC power supply based
on a half-wave rectifier. The resistance would be a few Ohms instead
of 1 kQ due to a transformer secondary winding replacing the voltage
source and resistor. A larger “filter” capacitor would be used. A power
supply based on a 60 Hz source with a filter of a few hundred uF could
supply up to 100 mA. Half-wave supplies seldom supply more due to
the difficulty of filtering a half-wave.
The peak detector may be combined with other components to build a
crystal radio 03442.png.
Clipper circuits
A circuit which removes the peak of a waveform is known as a Clipper.
A negative clipper is shown in Figure below. This schematic diagram
was produced with Xcircuit schematic capture program. Xcircuit
produced the SPICE net list Figure below, except for the second, and
next to last pair of lines which were inserted with a text editor.
-KSPICE 03437.eps
1 - * A K ModelName
5V A D1 © 2 diode
(A) ov? V(2) ‘|IIR1 2 1 1.0k
ae output |/V¥1 1 @ SIN(@ 5 1k)
0 Y .model diode d
.tran .05m 3m
.end
Clipper: clips negative peak at -0.7 V.
During the positive half cycle of the 5 V peak input, the diode is
reversed biased. The diode does not conduct. It is as if the diode were
not there. The positive half cycle is unchanged at the output V(2) in
Figure below. Since the output positive peaks actually overlays the
input sinewave V(1), the input has been shifted upward in the plot for
clarity. In Nutmeg, the SPICE display module, the command “plot
v(1)+1)” accomplishes this.
Yom u(1)+1 — v2)
V(1)+1 Is actually V(1), a 10 Vptp sinewave, offset by 1 V for display
Clarity. V/2) output Is clipped at -0.7 V, by diode D1.
During the negative half cycle of sinewave input of Figure above, the
diode is forward biased, that is, conducting. The negative half cycle of
the sinewave Is shorted out. The negative half cycle of V(2) would be
clipped at O V for an ideal diode. The waveform is clipped at -0.7 V
due to the forward voltage drop of the silicon diode. The spice model
defaults to 0.7 V unless parameters in the model statement specify
otherwise. Germanium or Schottky diodes clip at lower voltages.
Closer examination of the negative clipped peak (Figure above)
reveals that it follows the input for a slight period of time while the
sinewave is moving toward -0.7 V. The clipping action is only effective
after the input sinewave exceeds -0.7 V. The diode is not conducting
for the complete half cycle, though, during most of it.
The addition of an anti-parallel diode to the existing diode in Figure
above yields the symmetrical clipper in Figure below.
-KSPICE 03438.eps
D1 0 2 diode
D2 2 0 diode
Rl 2 1 1.0k
V1 10 SIN(O 5 1k)
.model diode d
.tran 0.05m 3m
end
Symmetrical clipper: Anti-parallel diodes clip both positive and
negative peak, leaving a + 0.7 V output.
Diode D1 clips the negative peak at -0.7 V as before. The additional
diode D2 conducts for positive half cycles of the sine wave as it
exceeds 0.7 V, the forward diode drop. The remainder of the voltage
drops across the series resistor. Thus, both peaks of the input
sinewave are clipped in Figure below. The net list is in Figure above
Diode D1 clips at -0.7 V as it conducts during negative peaks. D2
conducts for positive peaks, clipping at 0.7V.
The most general form of the diode clipper is shown in Figure below.
For an ideal diode, the clipping occurs at the level of the clipping
voltage, V1 and V2. However, the voltage sources have been adjusted
to account for the 0.7 V forward drop of the real silicon diodes. D1
clips at 1.3V +0.7V=2.0V when the diode begins to conduct. D2 clips
at -2.3V -0.7V=-3.0V when D2 conducts.
-KSPICE 03439.eps
V1 301.3
V2 40 -2.3
D1 2 3 diode
D2 4 2 diode
Rl 2 1 1.0k
V3 10 SIN(O 5 1k)
.model diode d
.tran 0.05m 3m
.end
D1 clips the input sinewave at 2V. D2 clips at -3V.
The clipper in Figure above does not have to clip both levels. To clip at
one level with one diode and one voltage source, remove the other
diode and source.
The net list is in Figure above. The waveforms in Figure below show
the clipping of v(1) at output v(2).
Yoo ¥(2) — ¥(41)
D1 clips the sinewave at 2V. D2 clips at -3V.
There is also a zener diode clipper circuit in the “Zener diode” section.
A zener diode replaces both the diode and the DC voltage source.
A practical application of a clipper is to prevent an amplified speech
signal from overdriving a radio transmitter in Figure below. Over
driving the transmitter generates spurious radio signals which causes
interference with other stations. The clipper is a protective measure.
har aerean q
transmitter
Clipper prevents over driving radio transmitter by voice peaks.
microphone
A sinewave may be squared up by overdriving a clipper. Another
clipper application is the protection of exposed inputs of integrated
circuits. The input of the IC is connected to a pair of diodes as at node
“2” of Figure above. The voltage sources are replaced by the power
supply rails of the IC. For example, CMOS IC's use OV and +5 V. Analog
amplifiers might use +12V for the V1 and V2 sources.
e REVIEW
e A resistor and diode driven by an AC voltage source clips the
signal observed across the diode.
¢ A pair of anti-parallel Si diodes clip symmetrically at +0.7V
e The grounded end of a clipper diode(s) can be disconnected and
wired to a DC voltage to clip at an arbitrary level.
e A clipper can serve as a protective measure, preventing a signal
from exceeding the clip limits.
Clamper circuits
The circuits in Figure below are Known as clampers or DC restorers.
The corresponding netlist is in Figure below. These circuits clamp a
peak of a waveform to a specific DC level compared with a
Capacitively coupled signal which swings about its average DC level
(usually OV). If the diode is removed from the clamper, it defaults to a
simple coupling capacitor- no clamping.
What is the clamp voltage? And, which peak gets clamped? In Figure
below (a) the clamp voltage is 0 V ignoring diode drop, (more exactly
0.7 V with Si diode drop). In Figure below, the positive peak of V(1) is
clamped to the 0 V (0.7 V) clamp level. Why is this? On the first
positive half cycle, the diode conducts charging the capacitor left end
to +5 V (4.3 V). This is -5 V (-4.3 V) on the right end at V(1,4). Note
the polarity marked on the capacitor in Figure below (a). The right end
of the capacitor is -5 V DC (-4.3 V) with respect to ground. It also has
an AC 5 V peak sinewave coupled across it from source V(4) to node 1.
The sum of the two is a 5 V peak sine riding on a - 5 V DC (-4.3 V)
level. The diode only conducts on successive positive excursions of
source V(4) if the peak V(4) exceeds the charge on the capacitor. This
only happens if the charge on the capacitor drained off due to a load,
not shown. The charge on the capacitor is equal to the positive peak
of V(4) (less 0.7 diode drop). The AC riding on the negative end, right
end, is shifted down. The positive peak of the waveform is clamped to
0 V (0.7 V) because the diode conducts on the positive peak.
1000 pF
4 2
+ 0
(a) (b)
Clampers: (a) Positive peak clamped to O V. (b) Negative peak
clamped to O V. (c) Negative peak clamped to 5 V.
y — C4) ¥(1,4)
— ¥(2) — V3)
-KSPICE 03443.eps
V1 605
D1 6 3 diode
C1 4 3 1000p
D2 0 2 diode
C2 4 2 1000p
C3 4 1 1000p
D3 1 0 diode
V2 4 0 SIN(O 5 1k)
.model diode d
.tran 0.01m 5m
end
V(4) source voltage 5 V peak used in all clampers. V(1) clamper
output from Figure above (a). V(1,4) DC voltage on capacitor in Figure
(a). V(2) clamper output from Figure (b). V(3) clamper output from
Figure (c).
Suppose the polarity of the diode is reversed as in Figure above (b)?
The diode conducts on the negative peak of source V(4). The negative
peak is clamped to 0 V (-0.7 V). See V(2) in Figure above.
The most general realization of the clamper is shown in Figure above
(c) with the diode connected to a DC reference. The capacitor still
charges during the negative peak of the source. Note that the
polarities of the AC source and the DC reference are series aiding.
Thus, the capacitor charges to the sum to the two, 10 V DC (9.3 V).
Coupling the 5 V peak sinewave across the capacitor yields Figure
above V(3), the sum of the charge on the capacitor and the sinewave.
The negative peak appears to be clamped to 5 V DC (4.3V), the value
of the DC clamp reference (less diode drop).
Describe the waveform if the DC clamp reference is changed from 5 V
to 10 V. The clamped waveform will shift up. The negative peak will be
clamped to 10 V (9.3). Suppose that the amplitude of the sine wave
source is increased from 5 V to 7 V? The negative peak clamp level
will remain unchanged. Though, the amplitude of the sinewave output
will increase.
An application of the clamper circuit is as a “DC restorer” in
“composite video” circuitry in both television transmitters and
receivers. An NTSC (US video standard) video signal “white level”
corresponds to minimum (12.5%) transmitted power. The video “black
level” corresponds to a high level (75% of transmitter power. There is
a “blacker than black level” corresponding to 100% transmitted power
assigned to synchronization signals. The NTSC signal contains both
video and synchronization pulses. The problem with the composite
video is that its average DC level varies with the scene, dark vs light.
The video itself is supposed to vary. However, the sync must always
peak at 100%. To prevent the sync signals from drifting with changing
scenes, a “DC restorer” clamps the top of the sync pulses to a voltage
corresponding to 100% transmitter modulation. [ATCO]
e REVIEW:
e A capacitively coupled signal alternates about its average DC
level (0 V).
e The signal out of a clamper appears the have one peak clamped
to a DC voltage. Example: The negative peak is clamped to 0 VDC,
the waveform appears to be shifted upward. The polarity of the
diode determines which peak is clamped.
e An application of a clamper, or DC restorer, is in clamping the
sync pulses of composite video to a voltage corresponding to
100% of transmitter power.
Voltage multipliers
A voltage multiplier is a specialized rectifier circuit producing an
output which is theoretically an integer times the AC peak input, for
example, 2, 3, or 4 times the AC peak input. Thus, it is possible to get
200 VDC from a 100 Vea, AC source using a doubler, 400 VDC from a
quadrupler. Any load in a practical circuit will lower these voltages.
A voltage doubler application is a DC power supply capable of using
either a 240 VAC or 120 VAC source. The supply uses a switch
selected full-wave bridge to produce about 300 VDC from a 240 VAC
source. The 120 V position of the switch rewires the bridge asa
doubler producing about 300 VDC from the 120 VAC. In both cases,
300 VDC is produced. This is the input to a switching regulator
producing lower voltages for powering, say, a personal computer.
The half-wave voltage doubler in Figure below (a) is composed of two
circuits: a clamper at (b) and peak detector (half-wave rectifier) in
Figure prior, which is shown in modified form in Figure below (c). C2
has been added to a peak detector (half-wave rectifier).
1000 pF , DI ; [+ sv vag [+ 5v >| D1 Cl
+
mae
10V
Half-wave voltage doubler (a) is composed of (b) a clamper and (c) a
half-wave rectifier.
Referring to Figure above (b), C2 charges to 5 V (4.3 V considering the
diode drop) on the negative half cycle of AC input. The right end is
grounded by the conducting D2. The left end is charged at the
negative peak of the AC input. This is the operation of the clamper.
During the positive half cycle, the half-wave rectifier comes into play
at Figure above (c). Diode D2 is out of the circuit since it is reverse
biased. C2 is now in series with the voltage source. Note the polarities
of the generator and C2, series aiding. Thus, rectifier D1 sees a total of
10 V at the peak of the sinewave, 5 V from generator and 5 V from C2.
D1 conducts waveform v(1) (Figure below), charging C1 to the peak of
the sine wave riding on 5 V DC (Figure below v(2)). Waveform v(2) is
the output of the doubler, which stabilizes at 10 V (8.6 V with diode
drops) after a few cycles of sinewave input.
y — ¥(2) — v(4)
v(1)
-KSPICE 03255.eps
C1 2 0 1000p
D1 1 2 diode
C2 4 1 1000p
D2 0 1 diode
V1 4 0 SIN(O 5 Ik)
.model diode d
.tran 0.01m 5m
.end
Voltage doubler: v(4) input. v(1) clamper stage. v(2) half-wave
rectifier stage, which is the doubler output.
The full-wave voltage doubler is composed of a pair of series stacked
half-wave rectifiers. (Figure below) The corresponding netlist is in
Figure below. The bottom rectifier charges C1 on the negative half
cycle of input. The top rectifier charges C2 on the positive halfcycle.
Each capacitor takes on a charge of 5 V (4.3 V considering diode
drop). The output at node 5 is the series total of Cl + C2 or 10 V (8.6
V with diode drops).
| ll
*KSPICE 03273.eps
*R1 3 0 100k
— |iFR2 5 3 100k
1000pF |llp1 © 2 diode
D2 2 5 diode
C1 3 0 1000p
1000 pF C2 5 3 1000p
V1 2 3 SIN(® 5 1k)
.model diode d
.tran 0.01m 5m
.end
Full-wave voltage doubler consists of two half-wave rectifiers
operating on alternating polarities.
Note that the output v(5) Figure below reaches full value within one
cycle of the input v(2) excursion.
Full-wave voltage doubler: v(2) input, v(3)voltage at mid point, v(5)
voltage at output
Figure below illustrates the derivation of the full-wave doubler from a
pair of opposite polarity half-wave rectifiers (a). The negative rectifier
of the pair is redrawn for clarity (b). Both are combined at (c) sharing
the same ground. At (d) the negative rectifier is re-wired to share one
voltage source with the positive rectifier. This yields a +5 V (4.3 V
with diode drop) power supply; though, 10 V is measurable between
the two outputs. The ground reference point is moved so that +10 V is
available with respect to ground.
+5V
(a) = (6) (Cc) (d) (e)
Full-wave doubler: (a) Pair of doublers, (b) redrawn, (c) sharing the
ground, (d) share the same voltage source. (e) move the ground
point.
A voltage tripler (Figure below) is built from a combination of a
doubler and a half wave rectifier (C3, D3). The half-wave rectifier
produces 5 V (4.3 V) at node 3. The doubler provides another 10 V
(8.4 V) between nodes 2 and 3. for a total of 15 V (12.9 V) at the
output node 2 with respect to ground. The netlist is in Figure below.
- L000 pF. :
ci |
10V
TtpoccecvcccccecVecssccececreresssessePovesrs 15V
‘| Single stage rectitier che
1000 pF == = 5V
D3 Y
Voltage tripler composed of doubler stacked atop a single stage
rectifier.
Note that V(3) in Figure below rises to 5 V (4.3 V) on the first negative
half cycle. Input v(4) is shifted upward by 5 V (4.3 V) due to 5 V from
the half-wave rectifier. And 5 V more at v(1) due to the clamper (C2,
D2). D1 charges Cl (waveform v(2)) to the peak value of v(1).
-KSPICE 03283.eps
C3 3 0 1000p
D3 0 4 diode
: j C1 2 3 1000p
Se D1 1 2 diode
C2 4 1 1000p
D2 3 1 diode
V1 4 3 SIN(O 5 1k)
.model diode d
.tran 0.01m 5m
.end
Voltage tripler: v(3) half-wave rectifier, v(4) inout+ 5 V, v(1) clamper,
v(2) final output.
A voltage quadrupler is a stacked combination of two doublers shown
in Figure below. Each doubler provides 10 V (8.6 V) for a series total at
node 2 with respect to ground of 20 V (17.2 V). The netlist is in Figure
below.
Voltage quadrupler, composed of two doublers stacked in series, with
output at node 2.
The waveforms of the quadrupler are shown in Figure below. Two DC
outputs are available: v(3), the doubler output, and v(2) the
quadrupler output. Some of the intermediate voltages at clampers
illustrate that the input sinewave (not shown), which swings by 5 V, is
successively clamped at higher levels: at v(5), v(4) and v(1). Strictly
v(4) is not a clamper output. It is simply the AC voltage source in
series with the v(3) the doubler output. None the less, v(1) is a
clamped version of v(4)
y — v(4) (5)
= yt) a= yt3) KFSPICE 03441. eps
fh tesuiedaontn iT eM aessoamsncnnmecae SPICE 03286.eps
: : v(2)3 C22 4 5 1000p
: C11 3 0 1000p
D11 0 5 diode
D22 5 3 diode
C1 2 3 1000p
D1 1 2 diode
C2 4 1 1000p
D2 3 1 diode
V1 4 3 SIN(O 5 Ik)
.model diode d
.tran 0.01m 5m
end
Voltage quadrupler: DC voltage available at v(3) and v(2).
Intermediate waveforms: Clampers: v(5), v(4), v(1).
Some notes on voltage multipliers are in order at this point. The
circuit parameters used in the examples (V= 5 V 1 kHz, C=1000 pf)
do not provide much current, microamps. Furthermore, load resistors
have been omitted. Loading reduces the voltages from those shown. If
the circuits are to be driven by a kHz source at low voltage, as in the
examples, the capacitors are usually 0.1 to 1.0 UF so that milliamps of
current are available at the output. If the multipliers are driven from
50/60 Hz, the capacitor are a few hundred to a few thousand
microfarads to provide hundreds of milliamps of output current. If
driven from line voltage, pay attention to the polarity and voltage
ratings of the capacitors.
Finally, any direct line driven power supply (no transformer) is
dangerous to the experimenter and line operated test equipment.
Commercial direct driven supplies are safe because the hazardous
circuitry is in an enclosure to protect the user. When breadboarding
these circuits with electrolytic capacitors of any voltage, the
Capacitors will explode if the polarity is reversed. Such circuits should
be powered up behind a safety shield.
A voltage multiplier of cascaded half-wave doublers of arbitrary length
is Known as a Cockcroft-Walton multiplier as shown in Figure below.
This multiplier is used when a high voltage at low current is required.
The advantage over a conventional supply is that an expensive high
voltage transformer is not required- at least not as high as the output.
1000pF 1000 pF 1000 pF 1000 pF
99 1 3 5
1000 pF 1000 pF 1000 pF 1000 pF
Cockcroft-Walton x8 voltage multiplier; output at v(8).
The pair of diodes and capacitors to the left of nodes 1 and 2 in Figure
above constitute a half-wave doubler. Rotating the diodes by 45°
counterclockwise, and the bottom capacitor by 90° makes it look like
Figure prior (a). Four of the doubler sections are cascaded to the right
for a theoretical x8 multiplication factor. Node 1 has a clamper
waveform (not shown), a sinewave shifted up by 1x (5 V). The other
odd numbered nodes are sinewaves clamped to successively higher
voltages. Node 2, the output of the first doubler, is a 2x DC voltage
v(2) in Figure below. Successive even numbered nodes charge to
successively higher voltages: v(4), v(6), v(8)
' sue Ses D1 7 8 diode
=o} = vid) vA) WHIC1 8 6 1000p
as D2 6 7 diode
. C2 5 7 1000p
D3 5 6 diode
C3 4 6 1000p
D4 4 5 diode
C4 3 5 1000p
D5 3 4 diode
C5 2 4 1000p
D6 2 3 diode
D7 1 2 diode
C6 1 3 1000p
C7 2 0 1000p
C8 99 1 1000p
D8 0 1 diode
V1 99 © SIN(O 5 1k)
.model diode d
end
|
Cockcroft-Walton (x8) waveforms. Output Is v(8).
.tran 0.01m 50m
Without diode drops, each doubler yields 2Vin or 10 V, considering
two diode drops (10-1.4)=8.6 V is realistic. For a total of 4 doublers
one expects 4:8.6=34.4 V out of 40 V. Consulting Figure above, v(2) is
about right;however, v(8) is <30 V instead of the anticipated 34.4 V.
The bane of the Cockcroft-Walton multiplier is that each additional
stage adds less than the previous stage. Thus, a practical limit to the
number of stages exist. It is possible to overcome this limitation with a
modification to the basic circuit. [ABR] Also note the time scale of 40
msec compared with 5 ms for previous circuits. It required 40 msec for
the voltages to rise to a terminal value for this circuit. The netlist in
Figure above has a “.tran 0.010m 50m” command to extend the
simulation time to 50 msec; though, only 40 msec is plotted.
The Cockcroft-Walton multiplier serves as a more efficient high
voltage source for photomultiplier tubes requiring up to 2000 V. [ABR]
Moreover, the tube has numerous dynodes, terminals requiring
connection to the lower voltage “even numbered” nodes. The series
string of multiplier taps replaces a heat generating resistive voltage
divider of previous designs.
An AC line operated Cockcroft-Walton multiplier provides high voltage
to “ion generators” for neutralizing electrostatic charge and for air
purifiers.
e REVIEW:
e Avoltage multiplier produces a DC multiple (2,3,4, etc) of the AC
peak input voltage.
The most basic multiplier is a half-wave doubler.
The full-wave double is a superior circuit as a doubler.
A tripler is a half-wave doubler and a conventional rectifier stage
(peak detector).
A quadrupler is a pair of half-wave doublers
A long string of half-wave doublers is known as a Cockcroft-Walton
multiplier.
Inductor commutating circuits
A popular use of diodes is for the mitigation of inductive “kickback:”
the pulses of high voltage produced when direct current through an
inductor is interrupted. Take, for example, this simple circuit in Figure
below with no protection against inductive kickback.
off on
Inductive kickback: (a) Switch open. (b) Switch closed, electron
current flows from battery through coil which has polarity matching
battery. Magnetic field stores energy. (c) Switch open, Current still
flows in coil due to collapsing magnetic field. Note polarity change on
coil. (d) Coil voltage vs time.
When the pushbutton switch is actuated, current goes through the
inductor, producing a magnetic field around it. When the switch is de-
actuated, its contacts open, interrupting current through the inductor,
and causing the magnetic field to rapidly collapse. Because the
voltage induced in a coil of wire is directly proportional to the rate of
change over time of magnetic flux (Faraday's Law: e = Nd@®/dt), this
rapid collapse of magnetism around the coil produces a high voltage
“spike”.
If the inductor in question is an electromagnet coil, such as ina
solenoid or relay (constructed for the purpose of creating a physical
force via its magnetic field when energized), the effect of inductive
“kickback” serves no useful purpose at all. In fact, it is quite
detrimental to the switch, as it causes excessive arcing at the
contacts, greatly reducing their service life. Of the practical methods
for mitigating the high voltage transient created when the switch is
opened, none so simple as the so-called commutating diode in Figure
below.
=.
Inductive kickback with protection: (a) Switch open. (b)Switch closed,
storing energy in magnetic field. (c) Switch open, inductive kickback
is shorted by diode.
In this circuit, the diode is placed in parallel with the coil, such that it
will be reverse-biased when DC voltage Is applied to the coil through
the switch. Thus, when the coil is energized, the diode conducts no
current in Figure above (b).
However, when the switch is opened, the coil's inductance responds to
the decrease in current by inducing a voltage of reverse polarity, in an
effort to maintain current at the same magnitude and in the same
direction. This sudden reversal of voltage polarity across the coil
forward-biases the diode, and the diode provides a current path for
the inductor's current, so that its stored energy is dissipated slowly
rather than suddenly in Figure above (c).
As a result, the voltage induced in the coil by its collapsing magnetic
field is quite low: merely the forward voltage drop of the diode, rather
than hundreds of volts as before. Thus, the switch contacts experience
a voltage drop equal to the battery voltage plus about 0.7 volts (if the
diode is silicon) during this discharge time.
In electronics parlance, commutation refers to the reversal of voltage
polarity or current direction. Thus, the purpose of a commutating
diode is to act whenever voltage reverses polarity, for example, on an
inductor coil when current through it is interrupted. A less formal term
for a commutating diode is snubber, because it “Snubs” or
“squelches” the inductive kickback.
A noteworthy disadvantage of this method is the extra time it imparts
to the coil's discharge. Because the induced voltage is clamped to a
very low value, its rate of magnetic flux change over time is
comparatively slow. Remember that Faraday's Law describes the
magnetic flux rate-of-change (d®/dt) as being proportional to the
induced, instantaneous voltage (eor v). If the instantaneous voltage
is limited to some low figure, then the rate of change of magnetic flux
over time will likewise be limited to a low (slow) figure.
If an electromagnet coil is “snubbed” with a commutating diode, the
magnetic field will dissipate at a relatively slow rate compared to the
original scenario (no diode) where the field disappeared almost
instantly upon switch release. The amount of time in question will
most likely be less than one second, but it will be measurably slower
than without a commutating diode in place. This may be an
intolerable consequence if the coil is used to actuate an
electromechanical relay, because the relay will possess a natural
“time delay” upon coil de-energization, and an unwanted delay of
even a fraction of a second may wreak havoc in some circuits.
Unfortunately, one cannot eliminate the high-voltage transient of
inductive kickback and maintain fast de-magnetization of the coil:
Faraday's Law will not be violated. However, if slow de-magnetization
is unacceptable, a compromise may be struck between transient
voltage and time by allowing the coil's voltage to rise to some higher
level (but not so high as without a commutating diode in place). The
schematic in Figure below shows how this can be done.
aoe V
-— (9)
off on (e)
| (c)
(a) (b) off
(a) Commutating diode with series resistor. (b) Voltage waveform. (Cc)
Level with no diode. (d) Level with diode, no resistor. (e) Compromise
level with diode and resistor.
A resistor placed in series with the commutating diode allows the
coil's induced voltage to rise to a level greater than the diode's
forward voltage drop, thus hastening the process of de-magnetization.
This, of course, will place the switch contacts under greater stress, and
so the resistor must be sized to limit that transient voltage at an
acceptable maximum level.
Diode switching circuits
Diodes can perform switching and digital logic operations. Forward
and reverse bias switch a diode between the low and high impedance
states, respectively. Thus, it serves as a switch.
Logic
Diodes can perform digital logic functions: AND, and OR. Diode logic
was uSed in early digital computers. It only finds limited application
today. Sometimes it is convenient to fashion a single logic gate from a
few diodes.
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momo
—ooolK<
1 =f
fe
(a)
Diode AND gate
An AND gate is shown in Figure above. Logic gates have inputs and an
output (Y) which is a function of the inputs. The inputs to the gate are
high (logic 1), say 10 V, or low, 0 V (logic 0). In the figure, the logic
levels are generated by switches. If a switch is up, the input is
effectively high (1). If the switch is down, it connects a diode cathode
to ground, which is low (0). The output depends on the combination of
inputs at A and B. The inputs and output are customarily recorded in a
“truth table” at (c) to describe the logic of a gate. At (a) all inputs are
high (1). This is recorded in the last line of the truth table at (c). The
output, Y, is high (1) due to the Vt on the top of the resistor. It is
unaffected by open switches. At (b) switch A pulls the cathode of the
connected diode low, pulling output Y low (0.7 V). This is recorded in
the third line of the truth table. The second line of the truth table
describes the output with the switches reversed from (b). Switch B
pulls the diode and output low. The first line of the truth table
recordes the Output=0 for both input low (0). The truth table
describes a logical AND function. Summary: both inputs A and B high
yields a high (1) out.
A two input OR gate composed of a pair of diodes is shown in Figure
below. If both inputs are logic low at (a) as simulated by both switches
“downward,” the output Y is pulled low by the resistor. This logic zero
is recorded in the first line of the truth table at (c). If one of the inputs
is high as at (b), or the other input is high, or both inputs high, the
diode(s) conduct(s), pulling the output Y high. These results are
reordered in the second through fourth lines of the truth table.
Summary: any input “high” is a high out at Y.
line
Y=0 operated
0 A I power
= suppl
0 B 0 pply
=
backup ~~~
battery
OR gate: (a) First line, truth table (TT). (b) Third line TT. (d) Logical OR
of power line supply and back-up battery.
A backup battery may be OR-wired with a line operated DC power
supply in Figure above (d) to power a load, even during a power
failure. With AC power present, the line supply powers the load,
assuming that it is a higher voltage than the battery. In the event of a
power failure, the line supply voltage drops to 0 V; the battery powers
the load. The diodes must be in series with the power sources to
prevent a failed line supply from draining the battery, and to prevent
it from over charging the battery when line power is available. Does
your PC computer retain its BIOS setting when powered off? Does your
VCR (video cassette recorder) retain the clock setting after a power
failure? (PC Yes, old VCR no, new VCR yes.)
Analog switch
Diodes can switch analog signals. A reverse biased diode appears to
be an open circuit. A forward biased diode is a low resistance
conductor. The only problem is isolating the AC signal being switched
from the DC control signal. The circuit in Figure below is a parallel
resonant network: resonant tuning inductor paralleled by one (or
more) of the switched resonator capacitors. This parallel LC resonant
circuit could be a preselector filter for a radio receiver. It could be the
frequency determining network of an oscillator (not shown). The
digital control lines may be driven by a microprocessor interface.
switching
diode
switched
resonator
capacitor
resonant
tuning
inductor
large value
DC blocking
= capacitor
aT os
decoupling
capacitor ———
VY digital control
Diode switch: A digital control signal (low) selects a resonator
capacitor by forward biasing the switching diode.
iH
ae
The large value DC blocking capacitor grounds the resonant tuning
inductor for AC while blocking DC. It would have a low reactance
compared to the parallel LC reactances. This prevents the anode DC
voltage from being shorted to ground by the resonant tuning inductor.
A switched resonator capacitor is selected by pulling the
corresponding digital control low. This forward biases the switching
diode. The DC current path is from +5 V through an RF choke (RFC), a
switching diode, and an RFC to ground via the digital control. The
purpose of the RFC at the +5 V is to keep AC out of the +5 V supply.
The RFC in series with the digital control is to keep AC out of the
external control line. The decoupling capacitor shorts the little AC
leaking through the RFC to ground, bypassing the external digital
control line.
With all three digital control lines high (=+5 V), no switched resonator
Capacitors are selected due to diode reverse bias. Pulling one or more
lines low, selects one or more switched resonator capacitors,
respectively. As more capacitors are switched in parallel with the
resonant tuning inductor, the resonant frequency decreases.
The reverse biased diode capacitance may be substantial compared
with very high frequency or ultra high frequency circuits. PIN diodes
may be used as switches for lower capacitance.
Zener diodes
If we connect a diode and resistor in series with a DC voltage source
so that the diode is forward-biased, the voltage drop across the diode
will remain fairly constant over a wide range of power supply voltages
as in Figure below (a).
According to the “diode equation” here, the current through a forward-
biased PN junction is proportional to e raised to the power of the
forward voltage drop. Because this is an exponential function, current
rises quite rapidly for modest increases in voltage drop. Another way
of considering this is to say that voltage dropped across a forward-
biased diode changes little for large variations in diode current. In the
circuit shown in Figure below (a), diode current is limited by the
voltage of the power supply, the series resistor, and the diode's
voltage drop, which as we know doesn't vary much from 0.7 volts. If
the power supply voltage were to be increased, the resistor's voltage
drop would increase almost the same amount, and the diode's voltage
drop just a little. Conversely, a decrease in power supply voltage
would result in an almost equal decrease in resistor voltage drop, with
just a little decrease in diode voltage drop. In a word, we could
summarize this behavior by saying that the diode is regulating the
voltage drop at approximately 0.7 volts.
Voltage regulation is a useful diode property to exploit. Suppose we
were building some kind of circuit which could not tolerate variations
in power supply voltage, but needed to be powered by a chemical
battery, whose voltage changes over its lifetime. We could form a
circuit as shown and connect the circuit requiring steady voltage
across the diode, where it would receive an unchanging 0.7 volts.
This would certainly work, but most practical circuits of any kind
require a power supply voltage in excess of 0.7 volts to properly
function. One way we could increase our voltage regulation point
would be to connect multiple diodes in series, so that their individual
forward voltage drops of 0.7 volts each would add to create a larger
total. For instance, if we had ten diodes in series, the regulated
voltage would be ten times 0.7, or 7 volts in Figure below (b).
(a) (b)
Forward biased Si reference: (a) single diode, 0.7V, (b) 10-diodes in
series 7.0V.
So long as the battery voltage never sagged below 7 volts, there
would always be about 7 volts dropped across the ten-diode “stack.”
If larger regulated voltages are required, we could either use more
diodes in series (an inelegant option, in my opinion), or try a
fundamentally different approach. We know that diode forward
voltage is a fairly constant figure under a wide range of conditions,
but so is reverse breakdown voltage, and breakdown voltage is
typically much, much greater than forward voltage. If we reversed the
polarity of the diode in our single-diode regulator circuit and
increased the power supply voltage to the point where the diode
“broke down” (could no longer withstand the reverse-bias voltage
impressed across it), the diode would similarly regulate the voltage at
that breakdown point, not allowing it to increase further as in Figure
below (a).
Zener diode
Cathode
(b) Anode
(a) Reverse biased Si small-signal diode breaks down at about 100V.
(b) Symbol for Zener diode.
Unfortunately, when normal rectifying diodes “break down,” they
usually do so destructively. However, it is possible to build a special
type of diode that can handle breakdown without failing completely.
This type of diode is called a zener diode, and its symbol looks like
Figure above (b).
When forward-biased, zener diodes behave much the same as
standard rectifying diodes: they have a forward voltage drop which
follows the “diode equation” and is about 0.7 volts. In reverse-bias
mode, they do not conduct until the applied voltage reaches or
exceeds the so-called zener voltage, at which point the diode is able
to conduct substantial current, and in doing so will try to limit the
voltage dropped across it to that zener voltage point. So long as the
power dissipated by this reverse current does not exceed the diode's
thermal limits, the diode will not be harmed.
Zener diodes are manufactured with zener voltages ranging anywhere
from a few volts to hundreds of volts. This zener voltage changes
slightly with temperature, and like common carbon-composition
resistor values, may be anywhere from 5 percent to 10 percent in error
from the manufacturer's specifications. However, this stability and
accuracy is generally good enough for the zener diode to be used as a
voltage regulator device in common power supply circuit in Figure
below.
+
pain
Zener diode regulator circuit, Zener voltage = 12.6V).
Please take note of the zener diode's orientation in the above circuit:
the diode is reverse-biased, and intentionally so. If we had oriented
the diode in the “normal” way, so as to be forward-biased, it would
only drop 0.7 volts, just like a regular rectifying diode. If we want to
exploit this diode's reverse breakdown properties, we must operate it
in its reverse-bias mode. So long as the power supply voltage remains
above the zener voltage (12.6 volts, in this example), the voltage
dropped across the zener diode will remain at approximately 12.6
volts.
Like any semiconductor device, the zener diode is sensitive to
temperature. Excessive temperature will destroy a zener diode, and
because it both drops voltage and conducts current, it produces its
own heat in accordance with Joule's Law (P=IE). Therefore, one must
be careful to design the regulator circuit in such a way that the
diode's power dissipation rating is not exceeded. Interestingly
enough, when zener diodes fail due to excessive power dissipation,
they usually fail shorted rather than open. A diode failed in this
manner is readily detected: it drops almost zero voltage when biased
either way, like a piece of wire.
Let's examine a zener diode regulating circuit mathematically,
determining all voltages, currents, and power dissipations. Taking the
same form of circuit shown earlier, we'll perform calculations
assuming a zener voltage of 12.6 volts, a power supply voltage of 45
volts, and a series resistor value of 1000 Q (we'll regard the zener
voltage to be exact/y 12.6 volts so as to avoid having to qualify all
figures as “approximate” in Figure below (a)
If the zener diode's voltage is 12.6 volts and the power supply's
voltage is 45 volts, there will be 32.4 volts dropped across the resistor
(45 volts - 12.6 volts = 32.4 volts). 32.4 volts dropped across 1000 Q
gives 32.4 mA of current in the circuit. (Figure below (b))
(a) Zener Voltage regulator with 1000 Q resistor. (b) Calculation of
voltage drops and current.
Power is calculated by multiplying current by voltage (P=IE), so we
can calculate power dissipations for both the resistor and the zener
diode quite easily:
P
P
= (32.4 mA)(32.4 V)
= 1.0498 W
resistor
resistor
Piiaie = (32.4 mA)(12.6 V)
Paiaie = 408.24 mW
A zener diode with a power rating of 0.5 watt would be adequate, as
would a resistor rated for 1.5 or 2 watts of dissipation.
If excessive power dissipation is detrimental, then why not design the
circuit for the least amount of dissipation possible? Why not just size
the resistor for a very high value of resistance, thus severely limiting
current and keeping power dissipation figures very low? Take this
circuit, for example, with a 100 kQ resistor instead of a 1 kQ resistor.
Note that both the power supply voltage and the diode's zener
voltage in Figure below are identical to the last example:
100 kQ
Zener regulator with 100 kQ resistor.
With only 1/100 of the current we had before (324 UA instead of 32.4
mA), both power dissipation figures should be 100 times smaller:
P
P
= (324 WA)(32.4 V)
resistor
resistor
Paiaie = (324 MA)(12.6 V)
Piiaie = 4.0824 mW
Seems ideal, doesn't it? Less power dissipation means lower operating
temperatures for both the diode and the resistor, and also less wasted
energy in the system, right? A higher resistance value does reduce
power dissipation levels in the circuit, but it unfortunately introduces
another problem. Remember that the purpose of a regulator circuit is
to provide a stable voltage for another circuit. In other words, we're
eventually going to power something with 12.6 volts, and this
something will have a current draw of its own. Consider our first
regulator circuit, this time with a 500 Q load connected in parallel
with the zener diode in Figure below.
Zener regulator with 1000 Q series resistor and 500 Q load.
If 12.6 volts is maintained across a 500 Q load, the load will draw 25.2
mA of current. In order for the 1 kQO series “dropping” resistor to drop
32.4 volts (reducing the power supply's voltage of 45 volts down to
12.6 across the zener), it still must conduct 32.4 mA of current. This
leaves 7.2 mA of current through the zener diode.
Now consider our “power-saving” regulator circuit with the 100 kQ
dropping resistor, delivering power to the same 500 Q load. What it is
supposed to do is maintain 12.6 volts across the load, just like the last
circuit. However, as we will see, it cannot accomplish this task. (Figure
below)
100 kQ2 <— 447.76 pA
<— 447.76pA =< —
Rice
500 Q
— 447.76nA —>
Zener non-regulator with 100 KQ series resistor with 500 Q load.>
With the larger value of dropping resistor in place, there will only be
about 224 mV of voltage across the 500 O load, far less than the
expected value of 12.6 volts! Why is this? If we actually had 12.6 volts
across the load, it would draw 25.2 mA of current, as before. This load
current would have to go through the series dropping resistor as it did
before, but with a new (much larger!) dropping resistor in place, the
voltage dropped across that resistor with 25.2 mA of current going
through it would be 2,520 volts! Since we obviously don't have that
much voltage supplied by the battery, this cannot happen.
The situation is easier to comprehend if we temporarily remove the
zener diode from the circuit and analyze the behavior of the two
resistors alone in Figure below.
44.776 V
100 kQ <— 447.76 pa
— w7.76paA <—
Ryu
500 Q
J u47.76pa — 447.76 pA —>
Non-regulator with Zener removed.
Both the 100 kQ dropping resistor and the 500 Q load resistance are
in series with each other, giving a total circuit resistance of 100.5 kQ.
With a total voltage of 45 volts and a total resistance of 100.5 kQ,
Ohm's Law (I=E/R) tells us that the current will be 447.76 UA. Figuring
voltage drops across both resistors (E=IR), we arrive at 44.776 volts
and 224 mV, respectively. If we were to re-install the zener diode at
this point, it would “see” 224 mV across it as well, being in parallel
with the load resistance. This is far below the zener breakdown
voltage of the diode and so it will not “break down” and conduct
current. For that matter, at this low voltage the diode wouldn't
conduct even if it were forward-biased! Thus, the diode ceases to
regulate voltage. At least 12.6 volts must be dropped across to
“activate” it.
The analytical technique of removing a zener diode from a circuit and
seeing whether or not enough voltage is present to make it conduct is
a sound one. Just because a zener diode happens to be connected in a
circuit doesn't guarantee that the full zener voltage will always be
dropped across it! Remember that zener diodes work by /imiting
voltage to some maximum level; they cannot make up for a lack of
voltage.
In summary, any zener diode regulating circuit will function so long as
the load's resistance is equal to or greater than some minimum value.
If the load resistance is too low, it will draw too much current,
dropping too much voltage across the series dropping resistor, leaving
insufficient voltage across the zener diode to make it conduct. When
the zener diode stops conducting current, it can no longer regulate
voltage, and the load voltage will fall below the regulation point.
Our regulator circuit with the 100 kQ dropping resistor must be good
for some value of load resistance, though. To find this acceptable load
resistance value, we can use a table to calculate resistance in the two-
resistor series circuit (no diode), inserting the known values of total
voltage and dropping resistor resistance, and calculating for an
expected load voltage of 12.6 volts:
With 45 volts of total voltage and 12.6 volts across the load, we
should have 32.4 volts across Rgropping:
Raroppi ng Rioad Total
With 32.4 volts across the dropping resistor, and 100 kQ worth of
resistance in it, the current through it will be 324 UA:
R aroppi ng Ryoad Total
Ohm's Law
[=E
R
Being a series circuit, the current is equal through all components at
any given time:
Raroppi ng Rioad Total
Volts
Amps
| took | | Ohms
Rule of series circuits:
Fro = 1, =1,=-.. 1,
Calculating load resistance is now a simple matter of Ohm's Law (R =
E/l), giving us 38.889 kQ:
Raropping Rioad Total
Amps
100k [38889 |__| Ohms
Ohm's Law
Bact
I
XJ - m
Thus, if the load resistance is exactly 38.889 kQ, there will be 12.6
volts across it, diode or no diode. Any load resistance smaller than
38.889 kQ will result in a load voltage less than 12.6 volts, diode or no
diode. With the diode in place, the load voltage will be regulated to a
maximum of 12.6 volts for any load resistance greater than 38.889
kQ.
With the original value of 1 kQ for the dropping resistor, our regulator
circuit was able to adequately regulate voltage even for a load
resistance as low as 500 Q. What we see is a tradeoff between power
dissipation and acceptable load resistance. The higher-value dropping
resistor gave us less power dissipation, at the expense of raising the
acceptable minimum load resistance value. If we wish to regulate
voltage for low-value load resistances, the circuit must be prepared to
handle higher power dissipation.
Zener diodes regulate voltage by acting as complementary loads,
drawing more or less current as necessary to ensure a constant
voltage drop across the load. This is analogous to regulating the speed
of an automobile by braking rather than by varying the throttle
position: not only is it wasteful, but the brakes must be built to handle
all the engine's power when the driving conditions don't demand it.
Despite this fundamental inefficiency of design, zener diode regulator
circuits are widely employed due to their sheer simplicity. In high-
power applications where the inefficiencies would be unacceptable,
other voltage-regulating techniques are applied. But even then, small
zener-based circuits are often used to provide a “reference” voltage to
drive a more efficient amplifier circuit controlling the main power.
Zener diodes are manufactured in standard voltage ratings listed in
Table below. The table “Common zener diode voltages” lists common
voltages for 0.3W and 1.3W parts. The wattage corresponds to die and
package size, and is the power that the diode may dissipate without
damage.
Common zener diode voltages
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Zener diode clipper: A clipping circuit which clips the peaks of
waveform at approximately the zener voltage of the diodes. The
circuit of Figure below has two zeners connected series opposing to
symmetrically clip a waveform at nearly the Zener voltage. The
resistor limits current drawn by the zeners to a safe value.
*SPICE 03445.eps
D1 4 0 diode
D2 4 2 diode
R1 2 1 1.0k
1 7 V1 1 0 SIN(O 20 1k)
{ ‘model diode d bv=10
V(2) .tran 0.001m 2m
20V, output |||-end
OV
offset
Zener diode clipper:
The zener breakdown voltage for the diodes is set at 10 V by the
diode model parameter “bv=10” in the spice net list in Figure above.
This causes the zeners to clip at about 10 V. The back-to-back diodes
clip both peaks. For a positive half-cycle, the top zener is reverse
biased, breaking down at the zener voltage of 10 V. The lower zener
drops approximately 0.7 V since it is forward biased. Thus, a more
accurate clipping level is 10+0.7 =10.7V. Similar negative half-cycle
clipping occurs a -10.7 V. (Figure below) shows the clipping level at a
little over +10 V.
y — ¥(2) — y(t)
Zener diode clipper: v(1) input is clipped at waveform v(2).
e REVIEW:
e Zener diodes are designed to be operated in reverse-bias mode,
providing a relatively low, stable breakdown, or zener voltage at
which they begin to conduct substantial reverse current.
e A zener diode may function as a voltage regulator by acting as an
accessory load, drawing more current from the source if the
voltage is too high, and less if it is too low.
Special-purpose diodes
Schottky diodes
Schottky diodes are constructed of a meta/-to-N junction rather than a
P-N semiconductor junction. Also known as hot-carrier diodes,
Schottky diodes are characterized by fast switching times (low
reverse-recovery time), low forward voltage drop (typically 0.25 to 0.4
volts for a metal-silicon junction), and low junction capacitance.
The schematic symbol for a schottky diode is shown in Figure below.
Anode
Cathode ¥
Schottky diode schematic symbol.
The forward voltage drop (V-), reverse-recovery time (t,,), and junction
capacitance (C)) of Schottky diodes are closer to ideal than the
average “rectifying” diode. This makes them well suited for high-
frequency applications. Unfortunately, though, Schottky diodes
typically have lower forward current (If) and reverse voltage (Vary and
Voc) ratings than rectifying diodes and are thus unsuitable for
applications involving substantial amounts of power. Though they are
used in low voltage switching regulator power supplies.
Schottky diode technology finds broad application in high-speed
computer circuits, where the fast switching time equates to high
speed capability, and the low forward voltage drop equates to less
power dissipation when conducting.
Switching regulator power supplies operating at 100's of KHz cannot
use conventional silicon diodes as rectifiers because of their slow
switching speed . When the signal applied to a diode changes from
forward to reverse bias, conduction continues for a short time, while
carriers are being swept out of the depletion region. Conduction only
ceases after this t, reverse recovery time has expired. Schottky diodes
have a shorter reverse recovery time.
Regardless of switching speed, the 0.7 V forward voltage drop of
silicon diodes causes poor efficiency in low voltage supplies. This is
not a problem in, say, a 10 V supply. Ina 1 V supply the 0.7 V drop isa
substantial portion of the output. One solution is to use a schottky
power diode which has a lower forward drop.
Tunnel diodes
Tunnel diodes exploit a strange quantum phenomenon called resonant
tunneling to provide a negative resistance forward-bias
characteristics. When a small forward-bias voltage is applied across a
tunnel diode, it begins to conduct current. (Figure below(b)) As the
voltage is increased, the current increases and reaches a peak value
called the peak current (Ip). If the voltage is increased a little more,
the current actually begins to decrease until it reaches a low point
called the va/ley current (ly). If the voltage is increased further yet,
the current begins to increase again, this time without decreasing into
another Bad ” The schematic symbol for the tunnel diode shown in
Figure below(a)
Tunnel diode
Anode Forward
y current
me cae
| ae alle
(a) (b) Ve Ww Forward voltage
Tunnel diode (a) Schematic symbol. (b) Current vs voltage plot (c)
Oscillator.
The forward voltages necessary to drive a tunnel diode to its peak and
valley currents are Known as peak voltage (Vp) and valley voltage
(V\), respectively. The region on the graph where current is decreasing
while applied voltage is increasing (between Vp and Vy on the
horizontal scale) is known as the region of negative resistance.
Tunnel diodes, also Known as Esaki diodes in honor of their Japanese
inventor Leo Esaki, are able to transition between peak and valley
current levels very quickly, “switching” between high and low states
of conduction much faster than even Schottky diodes. Tunnel diode
characteristics are also relatively unaffected by changes in
temperature.
1000
Breakdown voltage (V)
10'4 10'° 10° 10'7 10'*
Doping concentration (cm*)
Reverse breakdown voltage versus doping level. After Sze [SGG]
Tunnel diodes are heavily doped in both the P and N regions, 1000
times the level in a rectifier. This can be seen in Figure above.
Standard diodes are to the far left, zener diodes near to the left, and
tunnel diodes to the right of the dashed line. The heavy doping
produces an unusually thin depletion region. This produces an
unusually low reverse breakdown voltage with high leakage. The thin
depletion region causes high capacitance. To overcome this, the
tunnel diode junction area must be tiny. The forward diode
characteristic consists of two regions: a normal forward diode
characteristic with current rising exponentially beyond V-, 0.3 V for
Ge, 0.7 V for Si. Between 0 V and V; is an additional “negative
resistance” characteristic peak. This is due to quantum mechanical
tunneling involving the dual particle-wave nature of electrons. The
depletion region is thin enough compared with the equivalent
wavelength of the electron that they can tunnel through. They do not
have to overcome the normal forward diode voltage V-. The energy
level of the conduction band of the N-type material overlaps the level
of the valence band in the P-type region. With increasing voltage,
tunneling begins; the levels overlap; current increases, up to a point.
As current increases further, the energy levels overlap less; current
decreases with increasing voltage. This is the “negative resistance”
portion of the curve.
Tunnel diodes are not good rectifiers, as they have relatively high
“leakage” current when reverse-biased. Consequently, they find
application only in special circuits where their unique tunnel effect
has value. To exploit the tunnel effect, these diodes are maintained at
a bias voltage somewhere between the peak and valley voltage levels,
always in a forward-biased polarity (anode positive, and cathode
negative).
Perhaps the most common application of a tunnel diode is in simple
high-frequency oscillator circuits as in Figure above(c), where it allows
a DC voltage source to contribute power to an LC “tank” circuit, the
diode conducting when the voltage across it reaches the peak
(tunnel) level and effectively insulating at all other voltages. The
resistors bias the tunnel diode at a few tenths of a volt centered on
the negative resistance portion of the characteristic curve. The L-C
resonant circuit may be a section of waveguide for microwave
operation. Oscillation to 5 GHz is possible.
At one time the tunnel diode was the only solid-state microwave
amplifier available. Tunnel diodes were popular starting in the 1960's.
They were longer lived than traveling wave tube amplifiers, an
important consideration in satellite transmitters. Tunnel diodes are
also resistant to radiation because of the heavy doping. Today various
transistors operate at microwave frequencies. Even small signal
tunnel diodes are expensive and difficult to find today. There is one
remaining manufacturer of germanium tunnel diodes, and none for
silicon devices. They are sometimes used in military equipment
because they are insensitive to radiation and large temperature
changes.
There has been some research involving possible integration of silicon
tunnel diodes into CMOS integrated circuits. They are thought to be
capable of switching at 100 GHz in digital circuits. The sole
manufacturer of germanium devices produces them one at a time. A
batch process for silicon tunnel diodes must be developed, then
integrated with conventional CMOS processes. [SZL]
The Esaki tunnel diode should not be confused with the resonant
tunneling diode CH 2, of more complex construction from compound
semiconductors. The RTD is a more recent development capable of
higher speed.
Light-emitting diodes
Diodes, like all semiconductor devices, are governed by the principles
described in quantum physics. One of these principles is the emission
of specific-frequency radiant energy whenever electrons fall from a
higher energy level to a lower energy level. This is the same principle
at work in a neon lamp, the characteristic pink-orange glow of ionized
neon due to the specific energy transitions of its electrons in the midst
of an electric current. The unique color of a neon lamp's glow is due to
the fact that its neon gas inside the tube, and not due to the
particular amount of current through the tube or voltage between the
two electrodes. Neon gas glows pinkish-orange over a wide range of
ionizing voltages and currents. Each chemical element has its own
“signature” emission of radiant energy when its electrons “jump”
between different, quantized energy levels. Hydrogen gas, for
example, glows red when ionized; mercury vapor glows blue. This is
what makes spectrographic identification of elements possible.
Electrons flowing through a PN junction experience similar transitions
in energy level, and emit radiant energy as they do so. The frequency
of this radiant energy is determined by the crystal structure of the
semiconductor material, and the elements comprising it. Some
semiconductor junctions, composed of special chemical combinations,
emit radiant energy within the spectrum of visible light as the
electrons change energy levels. Simply put, these junctions glow
when forward biased. A diode intentionally designed to glow like a
lamp is called a /ight-emitting diode, or LED.
Forward biased silicon diodes give off heat as electron and holes from
the N-type and P-type regions, respectively, recombine at the
junction. In a forward biased LED, the recombination of electrons and
holes in the active region in Figure below (c) yields photons. This
process is known as e/ectro/uminescence. To give off photons, the
potential barrier through which the electrons fall must be higher than
for a silicon diode. The forward diode drop can range to a few volts for
some color LEDs.
Diodes made from a combination of the elements gallium, arsenic, and
phosphorus (called ga//ium-arsenide-phosphide) glow bright red, and
are some of the most common LEDs manufactured. By altering the
chemical constituency of the PN junction, different colors may be
obtained. Early generations of LEDs were red, green, yellow, orange,
and infra-red, later generations included blue and ultraviolet, with
violet being the latest color added to the selection. Other colors may
be obtained by combining two or more primary-color (red, green, and
blue) LEDs together in the same package, sharing the same optical
lens. This allowed for multicolor LEDs, such as tricolor LEDs
(commercially available in the 1980's) using red and green (which can
create yellow) and later RGB LEDs (red, green, and blue), which cover
the entire color spectrum.
The schematic symbol for an LED is a regular diode shape inside of a
circle, with two small arrows pointing away (indicating emitted light),
shown in Figure below.
\\ j—P-ype
| \ LY) — active region
Anode ong , CO er— n-type
: 26 — substrate
* Cathode short +
— flat
x x) @electron
(a) (b) (c) : ohole
LED, Light Emitting Diode: (a) schematic symbol. (b) Flat side and
short lead of device correspond to cathode, as well as the internal
arrangement of the cathode. (c) Cross section of Led die.
This notation of having two small arrows pointing away from the
device is common to the schematic symbols of all light-emitting
semiconductor devices. Conversely, if a device is light-activated
(meaning that incoming light stimulates it), then the symbol will have
two small arrows pointing toward it. LEDs can sense light. They
generate a small voltage when exposed to light, much like a solar cell
on a small scale. This property can be gainfully applied in a variety of
light-sensing circuits.
Because LEDs are made of different chemical substances than silicon
diodes, their forward voltage drops will be different. Typically, LEDs
have much larger forward voltage drops than rectifying diodes,
anywhere from about 1.6 volts to over 3 volts, depending on the color.
Typical operating current for a standard-sized LED is around 20 mA.
When operating an LED from a DC voltage source greater than the
LED's forward voltage, a series-connected “dropping” resistor must be
included to prevent full source voltage from damaging the LED.
Consider the example circuit in Figure below (a) using a 6 V source.
Ran. pping Ran ipping
Red LED, + 1.12 kQ
V; = 1.6 V typical —— 24V “
¥ I, = 20 mA typical =
(a) (b)
Setting LED current at 20 ma. (a) for a 6 V source, (b) fora 24 V
source.
With the LED dropping 1.6 volts, there will be 4.4 volts dropped across
the resistor. Sizing the resistor for an LED current of 20 mA is as
simple as taking its voltage drop (4.4 volts) and dividing by circuit
current (20 mA), in accordance with Ohm's Law (R=E/I). This gives us
a figure of 220 Q. Calculating power dissipation for this resistor, we
take its voltage drop and multiply by its current (P=IE), and end up
with 88 mW, well within the rating of a 1/8 watt resistor. Higher
battery voltages will require larger-value dropping resistors, and
possibly higher-power rating resistors as well. Consider the example in
Figure above (b) for a supply voltage of 24 volts:
Here, the dropping resistor must be increased to a size of 1.12 kQ to
drop 22.4 volts at 20 mA so that the LED still receives only 1.6 volts.
This also makes for a higher resistor power dissipation: 448 mW,
nearly one-half a watt of power! Obviously, a resistor rated for 1/8
watt power dissipation or even 1/4 watt dissipation will overheat if
used here.
Dropping resistor values need not be precise for LED circuits. Suppose
we were to use a 1 kQ resistor instead of a 1.12 kQ resistor in the
circuit shown above. The result would be a slightly greater circuit
current and LED voltage drop, resulting in a brighter light from the
LED and slightly reduced service life. A dropping resistor with too
much resistance (say, 1.5 kQ instead of 1.12 kQ) will result in less
circuit current, less LED voltage, and a dimmer light. LEDs are quite
tolerant of variation in applied power, so you need not strive for
perfection in sizing the dropping resistor.
Multiple LEDs are sometimes required, say in lighting. If LEDs are
operated in parallel, each must have its own current limiting resistor
as in Figure below (a) to ensure currents dividing more equally.
However, it is more efficient to operate LEDs in series (Figure below
(b)) with a single dropping resistor. As the number of series LEDs
increases the series resistor value must decrease to maintain current,
to a point. The number of LEDs in series (V;) cannot exceed the
capability of the power supply. Multiple series strings may be
employed as in Figure below (c).
In spite of equalizing the currents in multiple LEDs, the brightness of
the devices may not match due to variations in the individual parts.
Parts can be selected for brightness matching for critical applications.
Multiple LEDs: (a) In parallel, (b) in series, (c) series-paralle!
Also because of their unique chemical makeup, LEDs have much,
much lower peak-inverse voltage (PIV) ratings than ordinary rectifying
diodes. A typical LED might only be rated at 5 volts in reverse-bias
mode. Therefore, when using alternating current to power an LED,
connect a protective rectifying diode anti-parallel with the LED to
prevent reverse breakdown every other half-cycle as in Figure below
(a).
Rg roppi ng,
Red LED,
L.12kQ V,_= 1.6 V typical
aay 1, =20 mA typical
Vp =5 V maximum
24V
rectifying diode
Driving an LED with AC
The anti-parallel diode in Figure above can be replaced with an anti-
parallel LED. The resulting pair of anti-parallel LED's illuminate on
alternating half-cycles of the AC sinewave. This configuration draws
20 ma, splitting it equally between the LED's on alternating AC half
cycles. Each LED only receives 10 mA due to this sharing. The same is
true of the LED anti-parallel combination with a rectifier. The LED only
receives 10 ma. If 20 mA was required for the LED(s), The resistor
value could be halved.
The forward voltage drop of LED's is inversely proportional to the
wavelength (A). As wavelength decreases going from infrared to
visible colors to ultraviolet, V; increases. While this trend is most
obvious in the various devices from a single manufacturer, The
voltage range for a particular color LED from various manufacturers
varies. This range of voltages is shown in Table below.
Optical and electrical properties of LED's
LED A nm (= 10 -9m)/|V;(from)
infrared 940 1.2
red 660 15
orange 602-620 2.1
yellow, green 560-595 17
white, blue, violet}- 3
ultraviolet 370 4.2
As lamps, LEDs are superior to incandescent bulbs in many ways. First
and foremost is efficiency: LEDs output far more light power per watt
of electrical input than an incandescent lamp. This is a significant
advantage if the circuit in question is battery-powered, efficiency
translating to longer battery life. Second is the fact that LEDs are far
more reliable, having a much greater service life than incandescent
lamps. This is because LEDs are “cold” devices: they operate at much
cooler temperatures than an incandescent lamp with a white-hot
metal filament, susceptible to breakage from mechanical and thermal
shock. Third is the high speed at which LEDs may be turned on and
off. This advantage is also due to the “cold” operation of LEDs: they
don't have to overcome thermal inertia in transitioning from off to on
or vice versa. For this reason, LEDs are used to transmit digital (on/off)
information as pulses of light, conducted in empty space or through
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fiber-optic cable, at very high rates of speed (millions of pulses per
second).
LEDs excel in monochromatic lighting applications like traffic signals
and automotive tail lights. Incandescents are abysmal in this
application since they require filtering, decreasing efficiency. LEDs do
not require filtering.
One major disadvantage of using LEDs as sources of illumination is
their monochromatic (single-color) emission. No one wants to read a
book under the light of a red, green, or blue LED. However, if used in
combination, LED colors may be mixed for a more broad-spectrum
glow. A new broad spectrum light source is the white LED. While small
white panel indicators have been available for many years,
illumination grade devices are still in development.
Efficiency of lighting
Caine see Efficiency Life
P typ lumen/watt hrs
White LED 35 100,000 costly
White LED, future |100 100,000 R&D target
Incandescent [12 [1000 __|inexpensive |
Halogen 15-17 2000 aaa |
alles 50-100 10,000 cost effective |
uorescent
[Sodium vapor, Ip {70-200 20,000 |outdoor
Mercury vapor 13-48 18,000 outdoor
A white LED is a blue LED exciting a phosphor which emits yellow
light. The blue plus yellow approximates white light. The nature of the
phosphor determines the characteristics of the light. A red phosphor
may be added to improve the quality of the yellow plus blue mixture
at the expense of efficiency. Table above compares white illumination
LEDs to expected future devices and other conventional lamps.
Efficiency is measured in lumens of light output per watt of input
power. If the 50 lumens/watt device can be improved to 100
lumens/watt, white LEDs will be comparable to compact fluorescent
lamps in efficiency.
LEDs in general have been a major subject of R&D since the 1960's.
Because of this it is impractical to cover all geometries, chemistries,
and characteristics that have been created over the decades. The
early devices were relatively dim and took moderate currents. The
efficiencies have been improved in later generations to the point it is
hazardous to look closely and directly into an illuminated LED. This
can result in eye damage, and the LEDs only required a minor increase
in dropping voltage (Vf) and current. Modern high intensity devices
have reached 180 lumens using 0.7 Amps (82 lumens/watt, Luxeon
Rebel series cool white), and even higher intensity models can use
even higher currents with a corresponding increase in brightness.
Other developments, such as quantum dots, are the subject of current
research, so expect to see new things for these devices in the future
Laser diodes
The /aser diode is a further development upon the regular light-
emitting diode, or LED. The term “laser” itself is actually an acronym,
despite the fact its often written in lower-case letters. “Laser” stands
for Light Amplification by Stimulated Emission of Radiation, and
refers to another strange quantum process whereby characteristic
light emitted by electrons falling from high-level to low-level energy
states in a material stimulate other electrons in a substance to make
similar “jumps,” the result being a synchronized output of light from
the material. This synchronization extends to the actual phase of the
emitted light, so that all light waves emitted from a “lasing” material
are not just the same frequency (color), but also the same phase as
each other, so that they reinforce one another and are able to travel in
a very tightly-confined, nondispersing beam. This is why laser light
stays so remarkably focused over long distances: each and every light
wave coming from the laser is in step with each other.
(a)
&
(a) White light of many wavelengths. (b) Mono-chromatic LED light, a
single wavelength. (c) Phase coherent laser light.
Incandescent lamps produce “white” (mixed-frequency, or mixed-
color) light as in Figure above (a). Regular LEDs produce
monochromatic light: same frequency (color), but different phases,
resulting in similar beam dispersion in Figure above (b). Laser LEDs
produce coherent light: light that is both monochromatic (single-color)
and monophasic (single-phase), resulting in precise beam
confinement as in Figure above (c).
Laser light finds wide application in the modern world: everything
from surveying, where a straight and nondispersing light beam is very
useful for precise sighting of measurement markers, to the reading
and writing of optical disks, where only the narrowness of a focused
laser beam is able to resolve the microscopic “pits” in the disk's
surface comprising the binary 1's and 0's of digital information.
Some laser diodes require special high-power “pulsing” circuits to
deliver large quantities of voltage and current in short bursts. Other
laser diodes may be operated continuously at lower power. In the
continuous laser, laser action occurs only within a certain range of
diode current, necessitating some form of current-regulator circuit. As
laser diodes age, their power requirements may change (more current
required for less output power), but it should be remembered that low-
power laser diodes, like LEDs, are fairly long-lived devices, with typical
service lives in the tens of thousands of hours.
Photodiodes
A photodiode is a diode optimized to produce an electron current flow
in response to irradiation by ultraviolet, visible, or infrared light.
Silicon is most often used to fabricate photodiodes; though,
germanium and gallium arsenide can be used. The junction through
which light enters the semiconductor must be thin enough to pass
most of the light on to the active region (depletion region) where light
is converted to electron hole pairs.
In Figure below a shallow P-type diffusion into an N-type wafer
produces a PN junction near the surface of the wafer. The P-type layer
needs to be thin to pass as much light as possible. A heavy N+
diffusion on the back of the wafer makes contact with metalization.
The top metalization may be a fine grid of metallic fingers on the top
of the wafer for large cells. In small photodiodes, the top contact
might be a sole bond wire contacting the bare P-type silicon top.
+ top metal contact —_
f p diffusion
depletion region ____ | @
n type
- n+ contact region —R
bottom metal contact
Photodiode: Schematic symbol and cross section.
Light entering the top of the photodiode stack fall off exponentially in
with depth of the silicon. A thin top P-type layer allows most photons
to pass into the depletion region where electron-hole pairs are formed.
The electric field across the depletion region due to the built in diode
potential causes electrons to be swept into the N-layer, holes into the
P-layer. Actually electron-hole pairs may be formed in any of the
semiconductor regions. However, those formed in the depletion region
are most likely to be separated into the respective N and P-regions.
Many of the electron-hole pairs formed in the P and N-regions
recombine. Only a few do so in the depletion region. Thus, a few
electron-hole pairs in the N and P-regions, and most in the depletion
region contribute to photocurrent, that current resulting from light
falling on the photodiode.
The voltage out of a photodiode may be observed. Operation in this
photovoltaic (PV) mode is not linear over a large dynamic range,
though it is sensitive and has low noise at frequencies less than 100
kHz. The preferred mode of operation is often photocurrent (PC) mode
because the current is linearly proportional to light flux over several
decades of intensity, and higher frequency response can be achieved.
PC mode is achieved with reverse bias or zero bias on the photodiode.
A current amplifier (transimpedance amplifier) should be used with a
photodiode in PC mode. Linearity and PC mode are achieved as long
as the diode does not become forward biased.
High speed operation is often required of photodiodes, as opposed to
solar cells. Speed is a function of diode capacitance, which can be
minimized by decreasing cell area. Thus, a sensor for a high speed
fiber optic link will use an area no larger than necessary, say 1 mm.
Capacitance may also be decreased by increasing the thickness of the
depletion region, in the manufacturing process or by increasing the
reverse bias on the diode.
PIN diode The p-/-n diode or PIN diode is a photodiode with an
intrinsic layer between the P and N-regions as in Figure below. The P-
Intrinsic-N structure increases the distance between the P and N
conductive layers, decreasing capacitance, increasing speed. The
volume of the photo sensitive region also increases, enhancing
conversion efficiency. The bandwidth can extend to 10's of gHz. PIN
photodiodes are the preferred for high sensitivity, and high speed at
moderate cost.
top metal contact ___
p diffusion
intrinsic region
(larger depletion
region)
n type
n+ contact region —
bottom metal contact
PIN photodiode: The intrinsic region increases the thickness of the
depletion region.
Avalanche photo diode:An avalanche photodiode (APD)designed to
operate at high reverse bias exhibits an electron multiplier effect
analogous to a photomultiplier tube. The reverse bias can run from
10's of volts to nearly 2000 V. The high level of reverse bias
accelerates photon created electron-hole pairs in the intrinsic region
to a high enough velocity to free additional carriers from collisions
with the crystal lattice. Thus, many electrons per photon result. The
motivation for the APD is to achieve amplification within the
photodiode to overcome noise in external amplifiers. This works to
some extent. However, the APD creates noise of its own. At high speed
the APD is superior to a PIN diode amplifier combination, though not
for low speed applications. APD's are expensive, roughly the price of a
photomultiplier tube. So, they are only competitive with PIN
photodiodes for niche applications. One such application is single
photon counting as applied to nuclear physics.
Solar cells
A photodiode optimized for efficiently delivering power to a load is the
solar cell. |t operates in photovoltaic mode (PV) because it is forward
biased by the voltage developed across the load resistance.
Monocrystalline solar cells are manufactured in a process similar to
semiconductor processing. This involves growing a single crystal
boule from molten high purity silicon (P-type), though, not as high
purity as for semiconductors. The boule is diamond sawed or wire
sawed into wafers. The ends of the boule must be discarded or
recycled, and silicon is lost in the saw kerf. Since modern cells are
nearly square, silicon is lost in squaring the boule. Cells may be
etched to texture (roughen) the surface to help trap light within the
cell. Considerable silicon is lost in producing the 10 or 15 cm square
wafers. These days (2007) it is common for solar cell manufacturer to
purchase the wafers at this stage from a supplier to the semiconductor
industry.
P-type Wafers are loaded back-to-back into fused silica boats exposing
only the outer surface to the N-type dopant in the diffusion furnace.
The diffusion process forms a thin n-type layer on the top of the cell.
The diffusion also shorts the edges of the cell front to back. The
periphery must be removed by plasma etching to unshort the cell.
Silver and or aluminum paste is screened on the back of the cell, and
a silver grid on the front. These are sintered in a furnace for good
electrical contact. (Figure below)
The cells are wired in series with metal ribbons. For charging 12 V
batteries, 36 cells at approximately 0.5 V are vacuum laminated
between glass, and a polymer metal back. The glass may have a
textured surface to help trap light.
top metal contact ___-<
N diffusion
depletion region ___
P type wafer
bottom metal
contact
Silicon Solar cell
The ultimate commercial high efficiency (21.5%) single crystal silicon
solar cells have all contacts on the back of the cell. The active area of
the cell is increased by moving the top (-) contact conductors to the
back of the cell. The top (-) contacts are normally made to the N-type
silicon on top of the cell. In Figure below the (-) contacts are made to
Nt diffusions on the bottom interleaved with (+) contacts. The top
surface is textured to aid in trapping light within the cell.. [VSW]
Antireflectrive coating
Silicon dioxide passivation ———
N-type diffusion = —————
P-type wafer
N* diffusion —
- contact ~
P* diffusion
+ contact a
N* diffusion
- contact
—
High efficiency solar cell with all contacts on the back. Adapted from
Figure 1 [VSW]
Multicyrstalline silicon cells start out as molten silicon cast into a
rectangular mold. As the silicon cools, it crystallizes into a few large
(mm to cm sized) randomly oriented crystals instead of a single one.
The remainder of the process is the same as for single crystal cells.
The finished cells show lines dividing the individual crystals, as if the
cells were cracked. The high efficiency is not quite as high as single
crystal cells due to losses at crystal grain boundaries. The cell surface
cannot be roughened by etching due to the random orientation of the
crystals. However, an antireflectrive coating improves efficiency.
These cells are competitive for all but space applications.
Three layer cell: The highest efficiency solar cell is a stack of three
cells tuned to absorb different portions of the solar spectrum. Though
three cells can be stacked atop one another, a monolithic single
crystal structure of 20 semiconductor layers is more compact. At 32 %
efficiency, it is now (2007) favored over silicon for space application.
The high cost prevents it from finding many earth bound applications
other than concentrators based on lenses or mirrors.
Intensive research has recently produced a version enhanced for
terrestrial concentrators at 400 - 1000 suns and 40.7% efficiency. This
requires either a big inexpensive Fresnel lens or reflector and a small
area of the expensive semiconductor. This combination is thought to
be competitive with inexpensive silicon cells for solar power plants.
[RRK] [LZy]
Metal organic chemical vapor deposition (MOCVD) deposits the layers
atop a P-type germanium substrate. The top layers of N and P-type
gallium indium phosphide (GalnP) having a band gap of 1.85 eV,
absorbs ultraviolet and visible light. These wavelengths have enough
energy to exceed the band gap. Longer wavelengths (lower energy)
do not have enough energy to create electron-hole pairs, and pass on
through to the next layer. A gallium arsenide layers having a band
gap of 1.42 eV, absorbs near infrared light. Finally the germanium
layer and substrate absorb far infrared. The series of three cells
produce a voltage which is the sum of the voltages of the three cells.
The voltage developed by each material is 0.4 V less than the band
gap energy listed in Table below. For example, for GalnP: 1.8 eV/e -
0.4 V = 1.4 V. For all three the voltage is 1.4V+1.0V+0.3 V =2.7
V. [BRB]
High efficiency triple layer solar cell.
Layer Band gap|Light absorbed
Gallium indium phosphide}1.8 eV UV, visible
Gallium arsenide 1.4 eV near infrared
Germanium 0.7 eV far infrared
Crystalline solar cell arrays have a long usable life. Many arrays are
guaranteed for 25 years, and believed to be good for 40 years. They
do not suffer initial degradation compared with amorphous silicon.
=
Both single and multicrystalline solar cells are based on silicon wafers.
The silicon is both the substrate and the active device layers. Much
silicon is consumed. This kind of cell has been around for decades,
and takes approximately 86% of the solar electric market. For further
information about crystalline solar cells see Honsberg. [CHS]
Amorphous silicon thin film solar cells use tiny amounts of the
active raw material, silicon. Approximately half the cost of
conventional crystalline solar cells is the solar cell grade silicon. The
thin film deposition process reduces this cost. The downside is that
efficiency is about half that of conventional crystalline cells. Moreover,
efficiency degrades by 15-35% upon exposure to sunlight. A 7%
efficient cell soon ages to 5% efficiency. Thin film amorphous silicon
cells work better than crystalline cells in dim light. They are put to
good use in solar powered calculators.
Non-silicon based solar cells make up about 7% of the market. These
are thin-film polycrystalline products. Various compound
semiconductors are the subject of research and development. Some
non-silicon products are in production. Generally, the efficiency is
better than amorphous silicon, but not nearly as good as crystalline
silicon.
Cadmium telluride as a polycrystalline thin film on metal or glass
can have a higher efficiency than amorphous silicon thin films. If
deposited on metal, that layer is the negative contact to the cadmium
telluride thin film. The transparent P-type cadmium sulfide atop the
cadmium telluride serves as a buffer layer. The positive top contact is
transparent, electrically conductive fluorine doped tin oxide. These
layers may be laid down on a Sacrificial foil in place of the glass in the
process in the following pargraph. The sacrificial foil is removed after
the cell is mounted to a permanent substrate.
~ |— glass substrate
_4—nTinoxide ——___
— cadmium suflide ——
—p cadmium telluride
(phosphorus doped) —
™. p+ lead telluride ——&
metal substrate ——-| ~
metal contact
Cadmium telluride solar cell on glass or metal.
A process for depositing cadmium telluride on glass begins with the
deposition of N-type transparent, electrically conducive, tin oxide ona
glass substrate. The next layer is P-type cadmium telluride; though, N-
type or intrinsic may be used. These two layers constitute the NP
junction. A Pt (heavy P-type) layer of lead telluride aids in
establishing a low resistance contact. A metal layer makes the final
contact to the lead telluride. These layers may be laid down by
vacuum deposition, chemical vapor deposition (CVD), screen printing,
electrodeposition, or atmospheric pressure chemical vapor deposition
(APCVD) in helium. [KWM]
A variation of cadmium telluride is mercury cadmium telluride. Having
lower bulk resistance and lower contact resistance improves efficiency
over cadmium telluride.
7 top contact
> N-type transparent
conductor
™ buffer layer
—P type
— bottom contact
Tin oxide
Zinc oxide —————_
Cadmium suflide
CIGS Cadmium Indium
Gallium diSelenide ——
Molybdenum
Polyimide substrate __ |
Cadmium Indium Gallium diSelenide solar cell (CIGS)
Cadmium Indium Gallium diSelenide: A most promising thin film
solar cell at this time (2007) is manufactured on a ten inch wide roll of
flexible polyimide- Cadmium Indium Gallium diSelenide (CIGS). It has
a spectacular efficiency of 10%. Though, commercial grade crystalline
silicon cells surpassed this decades ago, CIGS should be cost
competitive. The deposition processes are at a low enough
temperature to use a polyimide polymer as a substrate instead of
metal or glass. (Figure above) The CIGS is manufactured in a roll to
roll process, which should drive down costs. GIGS cells may also be
produced by an inherently low cost electrochemical process. [EET]
e REVIEW:
e Most solar cells are silicon single crystal or multicrystal because of
their good efficiency and moderate cost.
e Less efficient thin films of various amorphous or polycrystalline
materials comprise the rest of the market.
e Table below compares selected solar cells.
Solar cell properties
Silicon, single crystal
Silicon, single crystal PERL,
terrestrial, space
Silicon, single crystal,
commercial terrestrial
Cypress Semiconductor,
Sunpower, silicon single
Gallium Indium Phosphide/
Gallium Arsenide/
Germanium, single crystal,
Advanced terrestrial version
of above.
Silicon, multicrystalline 18.5%
Thin films,
=
3%
Cadmium telluride, 16%
polycrystalline
Copper indium arsenide
Silicon, amorphous
18%
diselenide, polycrystalline
Maximum
Solar cell type efficiency
Selenium, polycrystalline 0.7%
25%
24%
21.5%
7 me
Practical
efficiency
1883, Charles
Fritts
A% 1950 s, first
silicon solar cell
solar Cars,
cost=100x
commercial
14-17% $5-$10/peak
watt
. all contacts on
se back
Preferred for
space.
Uses optical
concentrator.
40.7%
Fd
7
|
Degrades in sun
light. Good
indoors for
calculators or
cloudy outdoors.
glass or metal
substrate
10 inch flexible
10%
5-7%
[NTH]
1 (0)
Organic polymer, 100% A 5%
plastic
Varicap or varactor diodes
polymer web.
R&D project
A variable capacitance diode is known as a varicap diode or as a
varactor. If a diode is reverse biased, an insulating depletion region
forms between the two semiconductive layers. In many diodes the
width of the depletion region may be changed by varying the reverse
bias. This varies the capacitance. This effect is accentuated in varicap
diodes. The schematic symbols is shown in Figure below, one of which
is packaged as common cathode dual diode.
v
c=
ty
symbol voltage varicap diode
capacitance
Varicap diode: Capacitance varies with reverse bias. This varies the
frequency of a resonant network.
If a varicap diode is part of a resonant circuit as in Figure above, the
frequency may be varied with a control voltage, Veontro). A large
Capacitance, low X,, in series with the varicap prevents V ontro from
being shorted out by inductor L. As long as the series capacitor is
large, it has minimal effect on the frequency of resonant circuit.
Coptional May be used to set the center resonant frequency. Veontro) Can
then vary the frequency about this point. Note that the required
active circuitry to make the resonant network oscillate is not shown.
For an example of a varicap diode tuned AM radio receiver see
“electronic varicap diode tuning,” Ch 9
Some varicap diodes may be referred to as abrupt, hyperabrupt, or
super hyper abrupt. These refer to the change in junction capacitance
with changing reverse bias as being abrupt or hyper-abrupt, or super
hyperabrupt. These diodes offer a relatively large change in
capacitance. This is useful when oscillators or filters are swept over a
large frequency range. Varying the bias of abrupt varicaps over the
rated limits, changes capacitance by a 4:1 ratio, hyperabrupt by 10:1,
super hyperabrupt by 20:1.
Varactor diodes may be used in frequency multiplier circuits. See
“Practical analog semiconductor circuits,” Varactor multiplier
Snap diode
The snap diode, also known as the step recovery diode is designed for
use in high ratio frequency multipliers up to 20 gHz. When the diode
is forward biased, charge is stored in the PN junction. This charge is
drawn out as the diode is reverse biased. The diode looks like a low
impedance current source during forward bias. When reverse bias is
applied it still looks like a low impedance source until all the charge is
withdrawn. It then “snaps” to a high impedance state causing a
voltage impulse, rich in harmonics. An applications is a comb
generator, a generator of many harmonics. Moderate power 2x and 4x
multipliers are another application.
PIN diodes
A PIN diode is a fast low capacitance switching diode. Do not confuse
a PIN switching diode with a PIN photo diode here. A PIN diode is
manufactured like a silicon switching diode with an intrinsic region
added between the PN junction layers. This yields a thicker depletion
region, the insulating layer at the junction of a reverse biased diode.
This results in lower capacitance than a reverse biased switching
diode.
top metal contact —_
p+ contact region —
p diffusion
intrinsic region
(larger depletion
region)
n type
n+ contact region —
bottom metal contact
Pin diode: Cross section aligned with schematic symbol.
PIN diodes are used in place of switching diodes in radio frequency
(RF) applications, for example, a T/R switch here. The 1n4007 1000 V,
1 A general purpose power diode is reported to be usable as a PIN
switching diode. The high voltage rating of this diode is achieved by
the inclusion of an intrinsic layer dividing the PN junction. This
intrinsic layer makes the 1n4007 a PIN diode. Another PIN diode
application is as the antenna switch here for a direction finder
receiver.
PIN diodes serve as variable resistors when the forward bias is varied.
One such application is the voltage variable attenuator here. The low
Capacitance characteristic of PIN diodes, extends the frequency flat
response of the attenuator to microwave frequencies.
IMPATT diode
IMPact Avalanche Transit Time diode is a high power radio frequency
(RF) generator operating from 3 to 100 gHz. IMPATT diodes are
fabricated from silicon, gallium arsenide, or silicon carbide.
An IMPATT diode is reverse biased above the breakdown voltage. The
high doping levels produce a thin depletion region. The resulting high
electric field rapidly accelerates carriers which free other carriers in
collisions with the crystal lattice. Holes are swept into the P, region.
Electrons drift toward the N regions. The cascading effect creates an
avalanche current which increases even as voltage across the junction
decreases. The pulses of current lag the voltage peak across the
junction. A “negative resistance” effect in conjunction with a resonant
circuit produces oscillations at high power levels (high for
semiconductors).
avalanche
resonant circuit
IMPATT diode: Oscillator circuit and heavily doped P and N layers.
The resonant circuit in the schematic diagram of Figure above is the
lumped circuit equivalent of a waveguide section, where the IMPATT
diode is mounted. DC reverse bias is applied through a choke which
keeps RF from being lost in the bias supply. This may be a section of
waveguide known as a bias Tee. Low power RADAR transmitters may
use an IMPATT diode as a power source. They are too noisy for use in
the receiver. [YMCW]
Gunn diode
Diode, gunn Gunn diode
A gunn diode is solely composed of N-type semiconductor. As such, it
is not a true diode. Figure below shows a lightly doped N_ layer
surrounded by heavily doped N+ layers. A voltage applied across the
N-type gallium arsenide gunn diode creates a strong electric field
across the lightly doped N’ layer.
I
r —- iv
!
l Vv
resonant circuit
Gunn diode: Oscillator circuit and cross section of only N-type
semiconductor diode.
As voltage is increased, conduction increases due to electrons in a low
energy conduction band. As voltage is increased beyond the threshold
of approximately 1 V, electrons move from the lower conduction band
to the higher energy conduction band where they no longer
contribute to conduction. In other words, as voltage increases, current
decreases, a negative resistance condition. The oscillation frequency
is determined by the transit time of the conduction electrons, which is
inversely related to the thickness of the N layer.
The frequency may be controlled to some extent by embedding the
gunn diode into a resonant circuit. The lumped circuit equivalent
shown in Figure above is actually a coaxial transmission line or
waveguide. Gallium arsenide gunn diodes are available for operation
from 10 to 200 gHz at 5 to 65 mw power. Gunn diodes may also serve
as amplifiers. [CHW] [IAP]
Shockley diode
The Shockley diodeis a 4-layer thyristor used to trigger larger
thyristors. It only conducts in one direction when triggered by a
voltage exceeding the breakover voltage, about 20 V. See
“Thyristors,” The Shockley Diode. The bidirectional version is called a
diac. See “Thyristors,” The DIAC.
Constant-current diodes
A constant-current diode, also known as a current-limiting diode, or
current-regulating diode, does exactly what its name implies: it
regulates current through it to some maximum level. The constant
current diode is a two terminal version of a JFET. If we try to force more
current through a constant-current diode than its current-regulation
point, it simply “fights back” by dropping more voltage. If we were to
build the circuit in Figure below(a) and plot diode current against
diode voltage, we'd get a graph that rises at first and then levels off at
the current regulation point as in Figure below(b).
Raroppine
constant-current
diode
r
Vidiode
Constant current diode: (a) Test circuit, (b) current vs voltage
characteristic.
One application for a constant-current diode is to automatically limit
current through an LED or laser diode over a wide range of power
supply voltages as in Figure below.
constant-current
diode
LED or laser
» diode
Constant current diode application: driving laser diode.
Of course, the constant-current diode's regulation point should be
chosen to match the LED or laser diode's optimum forward current.
This is especially important for the laser diode, not so much for the
LED, as regular LEDs tend to be more tolerant of forward current
variations.
Another application is in the charging of small secondary-cell
batteries, where a constant charging current leads to predictable
charging times. Of course, large secondary-cell battery banks might
also benefit from constant-current charging, but constant-current
diodes tend to be very small devices, limited to regulating currents in
the milliamp range.
Other diode technologies
SiC diodes
Diodes manufactured from silicon carbide are capable of high
temperature operation to 400°C. This could be in a high temperature
environment: down hole oil well logging, gas turbine engines, auto
engines. Or, operation in a moderate environment at high power
dissipation. Nuclear and space applications are promising as SiC is
100 times more resistant to radiation compared with silicon. SiC is a
better conductor of heat than any metal. Thus, SiC is better than
silicon at conducting away heat. Breakdown voltage is several kV. SiC
power devices are expected to reduce electrical energy losses in the
power industry by a factor of 100.
Polymer diode
Diodes based on organic chemicals have been produced using low
temperature processes. Hole rich and electron rich conductive
polymers may be ink jet printed in layers. Most of the research and
development is of the organic LED (OLED). However, development of
inexpensive printable organic RFID (radio frequency identification)
tags is on going. In this effort, a pentacene organic rectifier has been
operated at 50 MHz. Rectification to 800 MHz is a development goal.
An inexpensive metal insulator metal (MIM) diode acting like a back-
to-back zener diode clipper has been delveloped. Also, a tunnel diode
like device has been fabricated.
SPICE models
The SPICE circuit simulation program provides for modeling diodes in
circuit simulations. The diode model is based on characterization of
individual devices as described in a product data sheet and
manufacturing process characteristics not listed. Some information
has been extracted from a 1N4004 data sheet in Figure below.
_ 100
c _
@ Ww
5 2 30
0 o
ae) o
S § 10
£ 3
5 8
4 c
= i
= rs}
” Cc
£ s |
= O
06 O08 10 12 14 ~= «1.6 I 10 100
V; instaneous forward voltage (V) Vp reverse voltage (V)
Max avg rectified current!, (A) 1 Forward voltage drop V-(V) 1
Peak repetitive reverse voltage V,4., (V) 400 @I-(A) 1
Peak forward surge current I-5,, (A) 30 Max reverse current, (uA) 5
Total capacitance C;(pF) 15 @ V,, (V) 400
Data sheet 1N4004 excerpt, after [DI4].
The diode statement begins with a diode element name which must
begin with “d” plus optional characters. Example diode element
names include: d1, d2, dtest, da, db, d101. Two node numbers specify
the connection of the anode and cathode, respectively, to other
components. The node numbers are followed by a model name,
referring to a subsequent “.model” statement.
The model statement line begins with “.model,” followed by the model
name matching one or more diode statements. Next, a “d” indicates a
diode is being modeled. The remainder of the model statement is a list
of optional diode parameters of the form
ParameterName=ParameterValue. None are used in Example below.
Example2 has some parameters defined. For a list of diode
parameters, see Table below.
General form: d[ name] [ anode] [ cathode] [ modelname]
.model ([modelname] d [ parmtri=x] [parmtr2=y] .. .)
Example: d1 1 2 modl
.model mod1 d
Example2: D2 1 2 Da1iN4004
.model Da1N4004 D (IS=18.8n RS=0 BV=400 IBV=5.00u
CJ0=30 M=0.333 N=2)
Parameter [Units |Default|
Saturation current (diode equation) A [le-14
Parsitic resistance (series resistance) Qa |
N Emission coefficient, 1 to 2 -
TT firansittime = = |s fo |
Zero-bias junction capacitance F |
Junction potential Vit
M Junction grading coefficient - _fjo.5
0.33 for linearly graded junction en
0.5 for abrupt junction a
EG Activation energy:
- Ge: 0.67 a
- Schottky: 0.69 a
W]
<
=
oy
=
=
9
=
be)
W
7)
Uy
a)
ae
=
2
O
oO
S
TE
XTI
IS temperature exponent
pn junction: 3.0 -
Schottky: 2.0
KF
Flicker noise coefficient
AF
Flicker noise exponent
Forward bias depletion capacitance |_
coefficient
BV
[Reverse bre
akdown voltage
IBV
[Reverse bre
akdown current
If diode parameters are not specified as in “Example” model above,
the parameters take on the default values listed in Table above and
Table below. These defaults model integrated circuit diodes. These are
certainly adequate for preliminary work with discrete devices For more
critical work, use SPICE models supplied by the manufacturer [DIn],
SPICE vendors, and other sources. [smi]
SPICE parameters for selected diodes; sk=schottky Ge=germanium;
else silicon.
[Part | 1S | RS | N | TT | GO| M |W [EG XTIBV[IBV
pers Pp pe psf pay bm
aad 315n |2.8 2.03 1.44n2.00p 0.333 0.69|2, 70 10u
BT ap 2m hop pa pe peop |
ene 200p [84m_ 2.19 |144n |4.82p|0.333/0.75 0.67} [60 15u
[IN4148|35p 64m [1.24 |5.0n |4.0p jo.285jo.6 | | (75 - |
[LN3891/63n [9.6m |2 ~~ [110n|114p jo.255j0.6 | | (250; |
a 844n |2.06ml2.06 |4.32u\277p |0.333|- __(|- | [400 10u
TA 76.9n|42.2m|1.45 4.32u 39.87 0.333 EL Jaoolsu |
1N4004/18.8nl- D Z 30p |l0.333|- — |- Pore
data
sheet | =| | | | [| | | | | tJ
Otherwise, derive some of the parameters from the data sheet. First
select a value for spice parameter N between 1 and 2. It is required for
the diode equation (n). Massobrio [PAGM] pp 9, recommends ".. n, the
emission coefficient is usually about 2." In Table above, we see that
power rectifiers 1N3891 (12 A), and 10A04 (10 A) both use about 2.
The first four in the table are not relevant because they are schottky,
schottky, germanium, and silicon small signal, respectively. The
saturation current, IS, is derived from the diode equation, a value of
(Vp, Ip) on the graph in Figure above, and N=2 (n in the diode
equation).
Ip a I<(e¥o/™\r -1)
Vz = 26 mV at 25°C n = 2.0 Vp = 0.925 V at 1A from
graph
1lA= T,(@ (9-925 V)/(2) (26 mV) -1)
Is = 18.8E-9
The numerical values of IS=18.8n and N=2 are entered in last line of
Table above for comparison to the manufacturers model for 1N4004,
which is considerably different. RS defaults to O for now. It will be
estimated later. The important DC static parameters are N, IS, and RS.
Rashid [MHR] suggests that TT, Tp, the transit time, be approximated
from the reverse recovery stored charge Qprp, a data sheet parameter
(not available on our data sheet) and I-, forward current.
Ip = I,(e¥o/™\r -1)
Tp = Qprr/Ip
We take the TT=0 default for lack of Qar. Though it would be
reasonable to take TT for a similar rectifier like the 10A04 at 4.32u.
The 1N3891 TT is not a valid choice because it is a fast recovery
rectifier. CJO, the zero bias junction capacitance is estimated from the
Vp vs C, graph in Figure above. The capacitance at the nearest to zero
voltage on the graph is 30 pF at 1 V. If simulating high speed transient
response, as in switching regulator power supplies, TT and CJO
parameters must be provided.
The junction grading coefficient M is related to the doping profile of
the junction. This is not a data sheet item. The default is 0.5 for an
abrupt junction. We opt for M=0.333 corresponding to a linearly
graded junction. The power rectifiers in Table above use lower values
for M than 0.5.
We take the default values for VJ and EG. Many more diodes use
VJ=0.6 than shown in Table above. However the 10A04 rectifier uses
the default, which we use for our 1N4004 model (Da1N4001 in Table
above). Use the default EG=1.11 for silicon diodes and rectifiers. Table
above lists values for schottky and germanium diodes. Take the XTI=3,
the default IS temperature coefficient for silicon devices. See Table
above for XTI for schottky diodes.
The abbreviated data sheet, Figure above, lists lp = 5 UA @ Vp = 400
V, corresponding to IBV=5u and BV=400 respectively. The 1n4004
SPICE parameters derived from the data sheet are listed in the last
line of Table above for comparison to the manufacturer's model listed
above it. BV is only necessary if the simulation exceeds the reverse
breakdown voltage of the diode, as is the case for zener diodes. IBV,
reverse breakdown current, is frequently omitted, but may be entered
if provided with BV.
Figure below shows a circuit to compare the manufacturers model, the
model derived from the datasheet, and the default model using
default parameters. The three dummy O V sources are necessary for
diode current measurement. The 1 V source is swept from 0 to 1.4 V in
0.2 mV steps. See .DC statement in the netlist in Table below.
DI1N4004 is the manufacturer's diode model, DalN4004 is our
derived diode model.
; D1 D2 D3
b ndud
SPICE circuit for comparison of manufacturer model (D1), calculated
datasheet model (D2), and default model (D3).
SPICE netlist parameters: (D1) DIIN4004 manufacturer's model, (D2)
Da1N40004 datasheet derived, (D3) default diode model.
*SPICE circuit <03468.eps> from XCircuit v3.20
D1 15 DI1N4004
v1 500
D2 1 3 DalN4004
V2 300
D3 1 4 Default
V3 400
V41041
.DC V4 0 1400mV 0.2m
.model Da1N4004 D (IS=18.8n RS=0 BV=400 IBV=5.00u CJ0=30
+M=0.333 N=2.0 TT=0)
.MODEL DI1N4004 D (IS=76.9n RS=42.0m BV=400 IBV=5.00u CJ0=39.8p
+M=0.333 N=1.45 TT=4.32u)
.MODEL Default D
.end
We compare the three models in Figure below. and to the datasheet
graph data in Table below. VD is the diode voltage versus the diode
currents for the manufacturer's model, our calculated datasheet
model and the default diode model. The last column “1N4004 graph”
is from the datasheet voltage versus current curve in Figure above
which we attempt to match. Comparison of the currents for the three
model to the last column shows that the default model is good at low
currents, the manufacturer's model is good at high currents, and our
calculated datasheet model is best of all up to 1 A. Agreement is
almost perfect at 1 A because the IS calculation is based on diode
voltage at 1 A. Our model grossly over states current above 1 A.
A — yv2#branch™ v3#branch
— yl#branch
datasheet
10,0 eee : FOOCUC ORR R ER EN OHH E REED, 2 Ptr Pe 3
First trial of manufacturer model, calculated datasheet model, and
default model.
Comparison of manufacturer model, calculated datasheet model, and
default model to 1N4004 datasheet graph of V vs I.
1N4004
index
graph
3500
0.01
4001
0.13
4500
VD
.000000e-01
.002000e-01
.000000e-01
.250000e-01
.000000e- 00
. 100000e+00
. 200000e+00
. 300000e+00
-400000e+00
model
manufacturer
1;
oe
5s
Ls
1.
612924e+00
346832e+00
310740e+00
.823654e+00
. 3959530400
.548779e+00
.174489e+01
397087e+01
621861e+01
I
1s
model
datasheet
.416211e-02
.825960e-02
.764928e-01
.096870e+00
.675526e+00
.231452e+01
.233392e+02
-5943591e+03
066840e+04
model
default
.674683e-03
.731709e-01
.294824e+01
.404037e+01
.185078e+02
.954471e+04
.411283e+06
.741379e+07
.220203e+09
The solution is to increase RS from the default RS=0. Changing RS
from 0 to 8m in the datasheet model causes the curve to intersect 10
A (not shown) at the same voltage as the manufacturer's model.
Increasing RS to 28.6m shifts the curve further to the right as shown
in Figure below. This has the effect of more closely matching our
datasheet model to the datasheet graph (Figure above). Table below
Shows that the current 1.224470e+01 A at 1.4 V matches the graph
at 12 A. However, the current at 0.925 V has degraded from
1.09687 0e+00 above to 7 .318536e-01.
A — y2#branch™ v3#branch
— vil#branch
10,0 POP CO eee eee eeeneeenes, 2 VOC Cee ee rere ener enes, 2 er te s
datasheet
BO [rvrrsrsneseans deessarnonenfo Hoe Bo
manufacturer
Second trial to improve calculated datasheet model compared with
manufacturer model and default model.
Changing Da1N4004 model statement RS=0 to RS=28.6m decreases
the current at VD=1.4 V to 12.2 A.
.model Da1N4004 D (IS=18.8n RS=28.6m BV=400 IBV=5.00u CJ0=30
+M=0 . 333 N=2.0 TT=0)
model model 1N4001
index VD manufacturer datasheet graph
3505 7.010000e-01 1.628276e+00 1.432463e-02 0.01
4000 8.000000e-01 3.343072e+00 9.297594e-02 0.13
4500 9.000000e-01 5.310740e+00 5.102139e-01 0.7
4625 9.250000e-01 5 .823654e+00 7.318536e-01 1.0
5000 1.000000e-00 7.395953e+00 1.763520e+00 2.0
5500 1.100000e+00 9.548779e+00 3.848553e+00 au3
6000 1.200000e+00 1.174489e+01 6.419621e+00 5.3
6500 1.300000e+00 1.397087e+01 9.254581e+00 8.0
7000 1.400000e+00 1.621861e+01 1.224470e+01 12.
Suggested reader exercise: decrease N so that the current at
VD=0.925 V is restored to 1 A. This may increase the current (12.2 A)
at VD=1.4 V requiring an increase of RS to decrease current to 12 A.
Zener diode: There are two approaches to modeling a zener diode:
set the BV parameter to the zener voltage in the model statement, or
model the zener with a subcircuit containing a diode clamper set to
the zener voltage. An example of the first approach sets the
breakdown voltage BV to 15 for the 1n4469 15 V zener diode model
(IBV optional):
.model D1N4469 D ( BV=15 IBV=17m )
The second approach models the zener with a subcircuit. Clamper D1
and VZ in Figure below models the 15 V reverse breakdown voltage of
a 1N4477A zener diode. Diode DR accounts for the forward
conduction of the zener in the subcircuit.
.SUBCKT DI-1N4744A 1 2
* Terminals AK
D1 1 2 DF
DZ 3 1 DR
VZ 2 3 13.7
.MODEL DF D ( IS=27.5p RS=0.620 N=1.10
+ CJ0O=78.3p VJ=1.00 M=0.330 TT=50.1n )
.MODEL DR D ( IS=5.49f RS=0.804 N=1.77 )
. ENDS
Zener diode subcircuit uses clamper (D1 and VZ) to model zener.
Tunnel diode: A tunnel diode may be modeled by a pair of field
effect transistors (JFET) in a SPICE subcircuit. [KHM] An oscillator
circuit is also shown in this reference.
Gunn diode: A Gunn diode may also be modeled by a pair of JFET's.
[ISG] This reference shows a microwave relaxation oscillator.
e REVIEW:
e Diodes are described in SPICE by a diode component statement
referring to .model statement. The .model statement contains
parameters describing the diode. If parameters are not provided,
the model takes on default values.
e Static DC parameters include N, IS, and RS. Reverse breakdown
parameters: BV, IBV.
e Accurate dynamic timing requires TT and CJO parameters
e Models provided by the manufacturer are highly recommended.
Contributors
Contributors to this chapter are listed in chronological order of their
contributions, from most recent to first. See Appendix 2 (Contributor
List) for dates and contact information.
Jered Wierzbicki (December 2002): Pointed out error in diode
equation -- Boltzmann's constant shown incorrectly.
Bibliography
1. [PAGM] Paolo Antognetti, Giuseppe Massobrio “Semiconductor
Device Modeling with SPICE,” ISBN 0-07-002107-4, 1988
2. [ATCOJATCO Newsletter, Volume 14 No. 1, January 1997 at
http://www.atco.tv/homepage/voll4_ 1.pdf
3. [ABR]D.A. Brunner, et al,, “A Cockcroft-Walton Base for the FEU84-
3 Photomultiplier Tube,” Department of Physics, Indiana
University, Bloomington, Indiana 47405 January 1998, at
http://dustbunny.physics.indiana.edu/~paul/cwbase/
4.[BRB]Brenton Burnet, “The Basic Physics and Design of III-V
Multijunction Solar,” NREL, at
photochemistry.epfl.ch/EDEY/NREL.pdf
5. [DIn] Diodes Incorporated
http://www.diodes.com/products/spicemodels/index.php
6. [DI4] Diodes Incorporated, “1N4001/L - 1N4007/I, 1.0A rectifier,”
at http://www.diodes.com/datasheets/ds28002.pdf
7. [EET] “Solar firm gains $30 million in funding,” EE Times,
07/12/2007 at
http://www.eetimes.com/news/latest/showArticle.jhtml?
articlelD=201001129
8. [CHS] Christiana Honsberg, Stuart Bowden, “Photovoltaics
CDROM,” at http://www.udel.edu/igert/pvcdrom/
9. [RRK]JR. R. King, et. al., “40% efficient metamorphic
GalnP/GalnAs/Ge multijunction solar cells”, Applied Physics
Letters, 90, 183516 (2007) , at
http://scitation.aip.org/getabs/servlet/GetabsServlet?
prog=normal&id=APPLAB000090000018183516000001&idtype
=cvips&gifs=yes
10. [KWM]Kim W Mitchell, “Method of making a thin film cadmium
telluride solar cell,” United States Patent
47 34381,http://www.freepatentsonline.com/47 34381.html
11. [KHM] Karl H. Muller “RF/Microwave Analysis” Intusoft Newsletter
#51, November 1997, at http://www.intusoft.com/nihtm/nI51.htm
12. [ISG] “A Gunn Diode Relaxation Oscillator,” Intusoft Newsletter
#52, February 1998, at http://www.intusoft.com/nihtm/ni52.htm
13. [OAK]JOAK Solar., “Technical LED's LED color chart,” at
http://www.oksolar.com/led/led_color chart.htm
14. [IAP]lan Poole, “Summary of the Gunn Diode,” at http://www.radio-
electronics.com/nfo/data/semicond/gunndiode/gunndiode.php
15. [MHR] Muhammad H. Rashid, “SPICE for Power Electronics and
Electric Power,” ISBN 0-13-030420-4, 1993
16. [smi] “SPICE model index,” V2.16 30-Nov-05, at
http://homepages.which.net/~ paul. hills/Circuits/Spice/Modellndex
-html
17. [NTH] Neil Thomas, “Advancing CIGS Solar Cell Manufacturing
Technology,” April 6, 2007 at
http://www.renewableenergyaccess.com/rea/news/story?
id=48033&src=rss
18. [VSW]P,J. Verlinden, Sinton, K. Wickham, R.M. Swanson Crane,
“BACKSIDE-CONTACT SILICON SOLAR CELLS WITH IMPROVED
EFFICIENCY.” at
http://www.sunpowercorp.com/techpapers/EPSEC97 .pdf
19. [CHW] Christian Wolff, “Radar Principles,” Radar components,
Gunn diodes at at
http://www.radartutorial.eu/17.bauteile/bt12.en.htm
20. [YMCW]L. Yuan, M. R. Melloch, J. A. Cooper, K. J. Webb,“Silicon
Carbide IMPATT Oscillators for High-Power Microwave and
Millimeter-Wave Generation,” IEEE/Cornell Conference on
Advanced Concepts in High Speed Semiconductor Devices and
Circuits, Ithaca, NY, August 7-9, 2000. at
http://www.ecn.purdue.edu/WBG/Device_Research/IMPATT Diodes/
Index.html
21. [SZL] Alan Seabaugh, Zhaoming HU, Qingmin LIU, David Rink,
Jinli Wang, “Silicon Based Tunnel Diodes and Integrated Circuits,”
at http://www.nd.edu/~nano/0al003QFDpaper v1l.pdf
22. [SGG] S. M. Sze, G. Gibbons, “Avalanche breakdown voltages of
abrupt and linearly graded p-n junctions in Ge, Si, GaAs, and Ga
P,” Appl. Phys. Lett., 8, 111 (1966).
23. [LZy] Lisa Zyga, “40% efficient solar cells to be used for solar
electricity”, PhysOrgForum, at
http://www.physorg.com/news99904887.html
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt,
under the terms and conditions of the Design Science License.
Previous Contents Next
— 4 —>
Lessons In Electric Circuits --
Volume Ill
Chapter 4
BIPOLAR JUNCTION TRANSISTORS
Introduction
The transistor as a switch
Meter check of a transistor
Active mode operation
The common-emitter amplifier
The common-collector amplifier
The common-base amplifier
The cascode amplifier
Biasing techniques
Biasing calculations
o Base Bias
o Collector-feedback bias
o Emitter-bias
o Voltage divider bias
Input and output coupling
Feedback
Amplifier impedances
Current mirrors
Transistor ratings and packages
BJT quirks
o Nonlinearity
o Temperature drift
o Thermal runaway
o Junction capacitance
o Noise
o Thermal mismatch (problem with paralleling transistors),
o High frequency effects
Bibliography.
Introduction
The invention of the bipolar transistor in 1948 ushered in a revolution in
electronics. Technical feats previously requiring relatively large, mechanically
fragile, power-hungry vacuum tubes were suddenly achievable with tiny,
mechanically rugged, power-thrifty specks of crystalline silicon. This revolution
made possible the design and manufacture of lightweight, inexpensive
electronic devices that we now take for granted. Understanding how transistors
function is of paramount importance to anyone interested in understanding
modern electronics.
My intent here is to focus as exclusively as possible on the practical function
and application of bipolar transistors, rather than to explore the quantum world
of semiconductor theory. Discussions of holes and electrons are better left to
another chapter in my opinion. Here | want to explore how to use these
components, not analyze their intimate internal details. | don't mean to
downplay the importance of understanding semiconductor physics, but
sometimes an intense focus on solid-state physics detracts from understanding
these devices' functions on a component level. In taking this approach,
however, | assume that the reader possesses a certain minimum knowledge of
semiconductors: the difference between “P” and “N” doped semiconductors, the
functional characteristics of a PN (diode) junction, and the meanings of the
terms “reverse biased” and “forward biased.” If these concepts are unclear to
you, it is best to refer to earlier chapters in this book before proceeding with
this one.
A bipolar transistor consists of a three-layer “sandwich” of doped (extrinsic)
semiconductor materials, either P-N-P in Figure below(b) or N-P-N at (d). Each
layer forming the transistor has a specific name, and each layer is provided with
a wire contact for connection to a circuit. The schematic symbols are shown in
Figure below(a) and (d).
collector collector
collector collector
=e =e
emitter emitter
(a) emitter (b) (c) emitter (d)
BJT transistor: (a) PNP schematic symbol, (b) physical layout (c) NPN symbol,
(d) layout.
The functional difference between a PNP transistor and an NPN transistor is the
proper biasing (polarity) of the junctions when operating. For any given state of
operation, the current directions and voltage polarities for each kind of
transistor are exactly opposite each other.
Bipolar transistors work as current-controlled current regu/ators. In other words,
transistors restrict the amount of current passed according to a smaller,
controlling current. The main current that is contro//led goes from collector to
emitter, or from emitter to collector, depending on the type of transistor it is
(PNP or NPN, respectively). The small current that contro/s the main current
goes from base to emitter, or from emitter to base, once again depending on
the kind of transistor it is (PNP or NPN, respectively). According to the standards
of semiconductor symbology, the arrow always points aga/nst the direction of
electron flow. (Figure below)
CO
o—>
B B
? E < E
——> = small, controlling current — = large, controlled current
Small Base-Emitter current controls large Collector-Emitter current flowing
against emitter arrow.
Bipolar transistors are called bipolar because the main flow of electrons through
them takes place in two types of semiconductor material: P and N, as the main
current goes from emitter to collector (or vice versa). In other words, two types
of charge carriers -- electrons and holes -- comprise this main current through
the transistor.
As you can see, the controlling current and the contro//ed current always mesh
together through the emitter wire, and their electrons always flow against the
direction of the transistor's arrow. This is the first and foremost rule in the use of
transistors: all currents must be going in the proper directions for the device to
work as a current regulator. The small, controlling current is usually referred to
simply as the base current because it is the only current that goes through the
base wire of the transistor. Conversely, the large, controlled current is referred
to as the collector current because it is the only current that goes through the
collector wire. The emitter current is the sum of the base and collector currents,
in compliance with Kirchhoff's Current Law.
No current through the base of the transistor, shuts it off like an open switch
and prevents current through the collector. A base current, turns the transistor
on like a closed switch and allows a proportional amount of current through the
collector. Collector current is primarily limited by the base current, regardless of
the amount of voltage available to push it. The next section will explore in more
detail the use of bipolar transistors as switching elements.
e REVIEW:
e Bipolar transistors are so named because the controlled current must go
through two types of semiconductor material: P and N. The current consists
of both electron and hole flow, in different parts of the transistor.
e Bipolar transistors consist of either a P-N-P or an N-P-N semiconductor
“sandwich” structure.
e The three leads of a bipolar transistor are called the Emitter, Base, and
Collector.
e Transistors function as current regulators by allowing a small current to
control a larger current. The amount of current allowed between collector
and emitter is primarily determined by the amount of current moving
between base and emitter.
e In order for a transistor to properly function as a current regulator, the
controlling (base) current and the controlled (collector) currents must be
going in the proper directions: meshing additively at the emitter and going
against the emitter arrow symbol.
The transistor as a switch
Because a transistor's collector current is proportionally limited by its base
current, it can be used as a Sort of current-controlled switch. A relatively small
flow of electrons sent through the base of the transistor has the ability to exert
control over a much larger flow of electrons through the collector.
Suppose we had a lamp that we wanted to turn on and off with a switch. Such a
circuit would be extremely simple as in Figure below(a).
For the sake of illustration, let's insert a transistor in place of the switch to show
how it can control the flow of electrons through the lamp. Remember that the
controlled current through a transistor must go between collector and emitter.
Since it is the current through the lamp that we want to control, we must
position the collector and emitter of our transistor where the two contacts of the
switch were. We must also make sure that the lamp's current will move against
the direction of the emitter arrow symbol to ensure that the transistor's junction
bias will be correct as in Figure below(b).
switch —
(a)
(a) mechanical switch, (b) NPN transistor switch, (c) PNP transistor switch.
A PNP transistor could also have been chosen for the job. Its application is
shown in Figure above(c).
The choice between NPN and PNP is really arbitrary. All that matters is that the
proper current directions are maintained for the sake of correct junction biasing
(electron flow going against the transistor symbol's arrow).
Going back to the NPN transistor in our example circuit, we are faced with the
need to add something more so that we can have base current. Without a
connection to the base wire of the transistor, base current will be zero, and the
transistor cannot turn on, resulting in a lamp that is always off. Remember that
for an NPN transistor, base current must consist of electrons flowing from
emitter to base (against the emitter arrow symbol, just like the lamp current).
Perhaps the simplest thing to do would be to connect a switch between the
base and collector wires of the transistor as in Figure below (a).
Transistor: (a) cutoff, lamp off; (b) saturated, lamp on.
If the switch is open as in Figure above (a), the base wire of the transistor will be
left “floating” (not connected to anything) and there will be no current through
it. In this state, the transistor is said to be cutoff. If the switch is closed as in
Figure above (b), electrons will be able to flow from the emitter through to the
base of the transistor, through the switch, up to the left side of the lamp, back
to the positive side of the battery. This base current will enable a much larger
flow of electrons from the emitter through to the collector, thus lighting up the
lamp. In this state of maximum circuit current, the transistor is said to be
saturated.
Of course, it may seem pointless to use a transistor in this capacity to control
the lamp. After all, we're still using a switch in the circuit, aren't we? If we're
still using a switch to control the lamp -- if only indirectly -- then what's the
point of having a transistor to control the current? Why not just go back to our
Original circuit and use the switch directly to control the lamp current?
Two points can be made here, actually. First is the fact that when used in this
manner, the switch contacts need only handle what little base current is
necessary to turn the transistor on; the transistor itself handles most of the
lamp's current. This may be an important advantage if the switch has a low
Current rating: a small switch may be used to control a relatively high-current
load. More importantly, the current-controlling behavior of the transistor
enables us to use something completely different to turn the lamp on or off.
Consider Figure below, where a pair of solar cells provides 1 V to overcome the
0.7 Vp of the transistor to cause base current flow, which in turn controls the
lamp.
Solar cell serves as light sensor.
Or, we could use a thermocouple (many connected in series) to provide the
necessary base current to turn the transistor on in Figure below.
thermocouple
source of
heat
A single thermocouple provides less than 40 mV. Many in series could produce
in excess of the 0.7 V transistor Vge to cause base current flow and consequent
collector current to the lamp.
Even a microphone (Figure below) with enough voltage and current (from an
amplifier) output could turn the transistor on, provided its output is rectified
from AC to DC so that the emitter-base PN junction within the transistor will
always be forward-biased:
microphone
>
source of
sound
Amplified microphone signal is rectified to DC to bias the base of the transistor
providing a larger collector current.
The point should be quite apparent by now: any sufficient source of DC current
may be used to turn the transistor on, and that source of current only need be a
fraction of the current needed to energize the lamp. Here we see the transistor
functioning not only as a switch, but as a true amplifier. using a relatively low-
power signal to contro/a relatively large amount of power. Please note that the
actual power for lighting up the lamp comes from the battery to the right of the
schematic. It is not as though the small signal current from the solar cell,
thermocouple, or microphone is being magically transformed into a greater
amount of power. Rather, those small power sources are simply controlling the
battery's power to light up the lamp.
¢ REVIEW:
e Transistors may be used as switching elements to control DC power to a
load. The switched (controlled) current goes between emitter and collector;
the controlling current goes between emitter and base.
e When a transistor has zero current through it, it is said to be in a state of
cutoff (fully nonconducting).
e When a transistor has maximum current through it, it is said to be in a state
of saturation (fully conducting).
Meter check of a transistor
Bipolar transistors are constructed of a three-layer semiconductor “sandwich,”
either PNP or NPN. As such, transistors register as two diodes connected back-
to-back when tested with a multimeter's “resistance” or “diode check” function
as illustrated in Figure below. Low resistance readings on the base with the
black negative (-) leads correspond to an N-type material in the base of a PNP
transistor. On the symbol, the N-type material is "pointed" to by the arrow of the
base-emitter junction, which is the base for this example. The P-type emitter
corresponds to the other end of the arrow of the base-emitter junction, the
emitter. The collector is very similar to the emitter, and is also a P-type material
of the PN junction.
O
lO*]
PNP transistor meter check: (a) forward B-E, B-C, resistance is low; (b) reverse
B-E, B-C, resistance is o,
Here I'm assuming the use of a multimeter with only a single continuity range
(resistance) function to check the PN junctions. Some multimeters are equipped
with two separate continuity check functions: resistance and “diode check,”
each with its own purpose. If your meter has a designated “diode check”
function, use that rather than the “resistance” range, and the meter will display
the actual forward voltage of the PN junction and not just whether or not it
conducts current.
Meter readings will be exactly opposite, of course, for an NPN transistor, with
both PN junctions facing the other way. Low resistance readings with the red (+)
lead on the base is the “opposite” condition for the NPN transistor.
If a multimeter with a “diode check” function is used in this test, it will be found
that the emitter-base junction possesses a slightly greater forward voltage drop
than the collector-base junction. This forward voltage difference is due to the
disparity in doping concentration between the emitter and collector regions of
the transistor: the emitter is a much more heavily doped piece of
semiconductor material than the collector, causing its junction with the base to
produce a higher forward voltage drop.
Knowing this, it becomes possible to determine which wire is which on an
unmarked transistor. This is important because transistor packaging,
unfortunately, is not standardized. All bipolar transistors have three wires, of
course, but the positions of the three wires on the actual physical package are
not arranged in any universal, standardized order.
Suppose a technician finds a bipolar transistor and proceeds to measure
continuity with a multimeter set in the “diode check” mode. Measuring between
pairs of wires and recording the values displayed by the meter, the technician
obtains the data in Figure below.
Meter touching wire 1 (+) and 2 (-): “OL’
Meter touching wire 1 (-) and 2 (+): “OL’
Meter touching wire 1 (+) and 3 (-): 0.655 V
Meter touching wire 1 (-) and 3 (+): “OL’
Meter touching wire 2 (+) and 3 (-): 0.621 V
Meter touching wire 2 (-) and 3 (+): “OL’
Unknown bipolar transistor. Which terminals are emitter, base, and collector?
Q-meter readings between terminals.
The only combinations of test points giving conducting meter readings are
wires 1 and 3 (red test lead on 1 and black test lead on 3), and wires 2 and 3
(red test lead on 2 and black test lead on 3). These two readings must indicate
forward biasing of the emitter-to-base junction (0.655 volts) and the collector-
to-base junction (0.621 volts).
Now we look for the one wire common to both sets of conductive readings. It
must be the base connection of the transistor, because the base is the only
layer of the three-layer device common to both sets of PN junctions (emitter-
base and collector-base). In this example, that wire is number 3, being common
to both the 1-3 and the 2-3 test point combinations. In both those sets of meter
readings, the black (-) meter test lead was touching wire 3, which tells us that
the base of this transistor is made of N-type semiconductor material (black =
negative). Thus, the transistor is a PNP with base on wire 3, emitter on wire 1
and collector on wire 2 as described in Figure below.
E and Chigh R: 1 (+) and 2 (-): “OL’
C and E high R: 1 (-) and 2 (+): “OL’
E and B forward: 1 (+) and 3 (-): 0.655 V
E and B reverse: 1 (-) and 3 (+): “OL’
C and B forward: 2 (+) and 3 (-): 0.621 V
C and B reverse: 2 (-) and 3 (+): “OL’
Emitter
Collector 3 Base
BJT terminals identified by Q-meter.
Please note that the base wire in this example is not the middle lead of the
transistor, as one might expect from the three-layer “sandwich” model of a
bipolar transistor. This is quite often the case, and tends to confuse new
students of electronics. The only way to be sure which lead is which is by a
meter check, or by referencing the manufacturer's “data sheet” documentation
on that particular part number of transistor.
Knowing that a bipolar transistor behaves as two back-to-back diodes when
tested with a conductivity meter is helpful for identifying an unknown transistor
purely by meter readings. It is also helpful for a quick functional check of the
transistor. If the technician were to measure continuity in any more than two or
any less than two of the six test lead combinations, he or she would
immediately know that the transistor was defective (or else that it wasn'ta
bipolar transistor but rather something else -- a distinct possibility if no part
numbers can be referenced for sure identification!). However, the “two diode”
model of the transistor fails to explain how or why it acts as an amplifying
device.
To better illustrate this paradox, let's examine one of the transistor switch
circuits using the physical diagram in Figure below rather than the schematic
symbol to represent the transistor. This way the two PN junctions will be easier
to see.
A small base current flowing in the forward biased base-emitter junction allows
a large current flow through the reverse biased base-collector junction.
A grey-colored diagonal arrow shows the direction of electron flow through the
emitter-base junction. This part makes sense, since the electrons are flowing
from the N-type emitter to the P-type base: the junction is obviously forward-
biased. However, the base-collector junction is another matter entirely. Notice
how the grey-colored thick arrow is pointing in the direction of electron flow
(up-wards) from base to collector. With the base made of P-type material and
the collector of N-type material, this direction of electron flow is clearly
backwards to the direction normally associated with a PN junction! A normal PN
junction wouldn't permit this “backward” direction of flow, at least not without
offering significant opposition. However, a saturated transistor shows very little
opposition to electrons, all the way from emitter to collector, as evidenced by
the lamp's illumination!
Clearly then, something is going on here that defies the simple “two-diode”
explanatory model of the bipolar transistor. When | was first learning about
transistor operation, | tried to construct my own transistor from two back-to-
back diodes, as in Figure below.
no light!
no current!
A pair of back-to-back diodes don't act like a transistor!
My circuit didn't work, and | was mystified. However useful the “two diode”
description of a transistor might be for testing purposes, it doesn't explain how
a transistor behaves as a controlled switch.
What happens in a transistor is this: the reverse bias of the base-collector
junction prevents collector current when the transistor is in cutoff mode (that is,
when there is no base current). If the base-emitter junction is forward biased by
the controlling signal, the normally-blocking action of the base-collector
junction is overridden and current is permitted through the collector, despite
the fact that electrons are going the “wrong way” through that PN junction. This
action is dependent on the quantum physics of semiconductor junctions, and
can only take place when the two junctions are properly spaced and the doping
concentrations of the three layers are properly proportioned. Two diodes wired
in series fail to meet these criteria; the top diode can never “turn on” when it is
reversed biased, no matter how much current goes through the bottom diode in
the base wire loop. See Bipolar junction transistors, Ch 2 for more details.
That doping concentrations play a crucial part in the special abilities of the
transistor is further evidenced by the fact that collector and emitter are not
interchangeable. If the transistor is merely viewed as two back-to-back PN
junctions, or merely as a plain N-P-N or P-N-P sandwich of materials, it may
seem as though either end of the transistor could serve as collector or emitter.
This, however, is not true. If connected “backwards” in a circuit, a base-collector
current will fail to control current between collector and emitter. Despite the
fact that both the emitter and collector layers of a bipolar transistor are of the
same doping type (either N or P), collector and emitter are definitely not
identical!
Current through the emitter-base junction allows current through the reverse-
biased base-collector junction. The action of base current can be thought of as
“opening a gate” for current through the collector. More specifically, any given
amount of emitter-to-base current permits a limited amount of base-to-collector
current. For every electron that passes through the emitter-base junction and
on through the base wire, a certain, number of electrons pass through the base-
collector junction and no more.
In the next section, this current-limiting of the transistor will be investigated in
more detail.
e REVIEW:
e Tested with a multimeter in the “resistance” or “diode check” modes, a
transistor behaves like two back-to-back PN (diode) junctions.
e The emitter-base PN junction has a slightly greater forward voltage drop
than the collector-base PN junction, because of heavier doping of the
emitter semiconductor layer.
e The reverse-biased base-collector junction normally blocks any current from
going through the transistor between emitter and collector. However, that
junction begins to conduct if current is drawn through the base wire. Base
current may be thought of as “opening a gate” for a certain, limited amount
of current through the collector.
Active mode operation
When a transistor is in the fully-off state (like an open switch), it is said to be
cutoff. Conversely, when it is fully conductive between emitter and collector
(passing as much current through the collector as the collector power supply
and load will allow), it is said to be saturated. These are the two modes of
operation explored thus far in using the transistor as a switch.
However, bipolar transistors don't have to be restricted to these two extreme
modes of operation. As we learned in the previous section, base current “opens
a gate” for a limited amount of current through the collector. If this limit for the
controlled current is greater than zero but less than the maximum allowed by
the power supply and load circuit, the transistor will “throttle” the collector
Current in a mode somewhere between cutoff and saturation. This mode of
operation is called the active mode.
An automotive analogy for transistor operation is as follows: cutoff is the
condition of no motive force generated by the mechanical parts of the car to
make it move. In cutoff mode, the brake is engaged (zero base current),
preventing motion (collector current). Active mode is the automobile cruising at
a constant, controlled speed (constant, controlled collector current) as dictated
by the driver. Saturation the automobile driving up a steep hill that prevents it
from going as fast as the driver wishes. In other words, a “saturated”
automobile is one with the accelerator pedal pushed all the way down (base
current calling for more collector current than can be provided by the power
supply/load circuit).
Let's set up a circuit for SPICE simulation to demonstrate what happens when a
transistor is in its active mode of operation. (Figure below)
V
ammeter
bipolar transistor simulation
| il 0 1 dc 20u
gl 2 1 0 modi
vammeter 3 2 dc 0
v1 3 0 dc
.model mod1 npn
-dc vl 0 2 0.05
.plot dc i(vammeter)
.end
Current I,
source
Circuit for “active mode” SPICE simulation, and netlist.
“Q” is the standard letter designation for a transistor in a schematic diagram,
just as “R” is for resistor and “C” is for capacitor. In this circuit, we have an NPN
transistor powered by a battery (V,) and controlled by current through a current
source (l,). A current source is a device that outputs a specific amount of
Current, generating as much or as little voltage across its terminals to ensure
that exact amount of current through it. Current sources are notoriously difficult
to find in nature (unlike voltage sources, which by contrast attempt to maintain
a constant voltage, outputting as much or as little current in the fulfillment of
that task), but can be simulated with a small collection of electronic
components. As we are about to see, transistors themselves tend to mimic the
constant-current behavior of a current source in their ability to regulate current
at a fixed value.
In the SPICE simulation, we'll set the current source at a constant value of 20
UA, then vary the voltage source (V,) over a range of 0 to 2 volts and monitor
how much current goes through it. The “dummy” battery (Vammeter) in Figure
above with its output of 0 volts serves merely to provide SPICE with a circuit
element for current measurement.
mA — mag(I(vammeter#branch) }
sePcsseccccccecesJececcccccseuscecesdeeceseuuevcccceeeJucueceeseeeesauaess
A Sweeping collector voltage 0 to 2 V with base current constant at 20 UA
yields constant 2 mA collector current in the saturation region.
The constant base current of 20 WA sets a collector current limit of 2 mA,
exactly 100 times as much. Notice how flat the curve is in (Figure above) for
collector current over the range of battery voltage from 0 to 2 volts. The only
exception to this featureless plot is at the very beginning, where the battery
increases from 0 volts to 0.25 volts. There, the collector current increases
rapidly from 0 amps to its limit of 2 mA.
Let's see what happens if we vary the battery voltage over a wider range, this
time from 0 to 50 volts. We'll keep the base current steady at 20 UA. (Figure
below)
bipolar transistor simulation
il @ 1 dc 20u
ql 2 1 0 modi
vammeter 3 2 dc 0
mA — mag(I(vammeter#branch) } vl 3 0 dc
.model mod1 npn
.dc vl 0 50 2
.plot dc i(vammeter)
.end
“0,0 20,0 40.0 60.0
Sweeping collector voltage 0 to 50 V with base current constant at 20 HA yields
constant 2 MA collector current.
Same result! The collector current in Figure above holds absolutely steady at 2
mA, although the battery (v1) voltage varies all the way from 0 to 50 volts. It
would appear from our simulation that collector-to-emitter voltage has little
effect over collector current, except at very low levels (just above 0 volts). The
transistor is acting as a current regulator, allowing exactly 2 mA through the
collector and no more.
Now let's see what happens if we increase the controlling (l,) current from 20
UA to 75 WA, once again sweeping the battery (V,) voltage from 0 to 50 volts
and graphing the collector current in Figure below.
bipolar transistor simulation
il @ 1 dc 75u
gl 2 1 0 modi
vammeter 3 2 dc 0
v1 3 0 dc
.model modi1 npn
-dc vl 0 50 2 il 15u 75u 15u
.plot dc i(vammeter)
.end
“0,0 20,0 40,0 60,0
sweep v Vee
Sweeping collector voltage 0 to 50 V (.dc v1 0 50 2) with base current constant
at 75 HA yields constant 7.5 mA collector current. Other curves are generated
by current sweep (i1 15u 75u 15u) in DC analysis statement (.dc v1 050 2 [1
15u 75u 15u).
Not surprisingly, SPICE gives us a similar plot: a flat line, holding steady this
time at 7.5 mA -- exactly 100 times the base current -- over the range of battery
voltages from just above 0 volts to 50 volts. It appears that the base current is
the deciding factor for collector current, the V, battery voltage being irrelevant
as long as it is above a certain minimum level.
This voltage/current relationship is entirely different from what we're used to
seeing across a resistor. With a resistor, current increases linearly as the voltage
across it increases. Here, with a transistor, current from emitter to collector
stays limited at a fixed, maximum value no matter how high the voltage across
emitter and collector increases.
Often it is useful to superimpose several collector current/voltage graphs for
different base currents on the same graph as in Figure below. A collection of
curves like this -- one curve plotted for each distinct level of base current -- fora
particular transistor is called the transistor's characteristic curves:
8 Thase = 79 PA
= : |
(mA) *7 |
| Inase = 40 LA
=20 pA
Lace
=5pA
Las
EF collector-to-emitter (V)
Voltage collector to emitter vs collector current for various base currents.
Each curve on the graph reflects the collector current of the transistor, plotted
over a range of collector-to-emitter voltages, for a given amount of base
current. Since a transistor tends to act as a current regulator, limiting collector
Current to a proportion set by the base current, it is useful to express this
proportion as a standard transistor performance measure. Specifically, the ratio
of collector current to base current is known as the Beta ratio (symbolized by
the Greek letter B):
I
collector
6 =
Tsase
B is also known as hy,
Sometimes the 8 ratio is designated as “hga,” a label used in a branch of
mathematical semiconductor analysis known as “hybrid parameters” which
strives to achieve precise predictions of transistor performance with detailed
equations. Hybrid parameter variables are many, but each is labeled with the
general letter “h” and a specific subscript. The variable “h;,” is just another
(standardized) way of expressing the ratio of collector current to base current,
and is interchangeable with “B.” The £ ratio is unitless.
8 for any transistor is determined by its design: it cannot be altered after
manufacture. It is rare to have two transistors of the same design exactly match
because of the physical variables afecting 8B . If a circuit design relies on equal B
ratios between multiple transistors, “matched sets” of transistors may be
purchased at extra cost. However, it is generally considered bad design practice
to engineer circuits with such dependencies.
The B of a transistor does not remain stable for all operating conditions. For an
actual transistor, the B ratio may vary by a factor of over 3 within its operating
current limits. For example, a transistor with advertised B of 50 may actually
test with I./l, ratios as low as 30 and as high as 100, depending on the amount
of collector current, the transistor's temperature, and frequency of amplified
signal, among other factors. For tutorial purposes it is adequate to assume a
constant B for any given transistor; realize that real life is not that simple!
Sometimes it is helpful for comprehension to “model” complex electronic
components with a collection of simpler, better-understood components. The
model in Figure below is used in many introductory electronics texts.
C
Cc
B B
E
NPN
diode-rheostat
model E
Elementary diode resistor transistor model.
This model casts the transistor as a combination of diode and rheostat (variable
resistor). Current through the base-emitter diode controls the resistance of the
collector-emitter rheostat (as implied by the dashed line connecting the two
components), thus controlling collector current. An NPN transistor is modeled in
the figure shown, but a PNP transistor would be only slightly different (only the
base-emitter diode would be reversed). This model succeeds in illustrating the
basic concept of transistor amplification: how the base current signal can exert
control over the collector current. However, | don't like this model because it
miscommunicates the notion of a set amount of collector-emitter resistance for
a given amount of base current. If this were true, the transistor wouldn't
regulate collector current at all like the characteristic curves show. Instead of
the collector current curves flattening out after their brief rise as the collector-
emitter voltage increases, the collector current would be directly proportional to
collector-emitter voltage, rising steadily in a straight line on the graph.
A better transistor model, often seen in more advanced textbooks, is shown in
Figure below.
Cc
Cc
B B
E
NPN
diode-current source
model E
Current source model of transistor.
It casts the transistor as a combination of diode and current source, the output
of the current source being set at a multiple (B ratio) of the base current. This
model is far more accurate in depicting the true input/output characteristics of
a transistor: base current establishes a certain amount of collector current,
rather than a certain amount of collector-emitter resistance as the first model
implies. Also, this model is favored when performing network analysis on
transistor circuits, the current source being a well-understood theoretical
component. Unfortunately, using a current source to model the transistor's
current-controlling behavior can be misleading: in no way will the transistor
ever act as a source of electrical energy. The current source does not model the
fact that its source of energy is a external power supply, similar to an amplifier.
e REVIEW:
e A transistor is said to be in its active mode if it is operating somewhere
between fully on (saturated) and fully off (cutoff).
e Base current regulates collector current. By regu/ate, we mean that no more
collector current can exist than what is allowed by the base current.
e The ratio between collector current and base current is called “Beta” (8) or
“hie”.
e B ratios are different for every transistor, and
e B changes for different operating conditions.
The common-emitter amplifier
At the beginning of this chapter we saw how transistors could be used as
switches, operating in either their “saturation” or “cutoff” modes. In the last
section we saw how transistors behave within their “active” modes, between
the far limits of saturation and cutoff. Because transistors are able to control
Current in an analog (infinitely divisible) fashion, they find use as amplifiers for
analog signals.
One of the simpler transistor amplifier circuits to study previously illustrated
the transistor's switching ability. (Figure below)
NPN transistor as a simple switch.
It is called the common-emitter configuration because (ignoring the power
supply battery) both the signal source and the load share the emitter lead asa
common connection point shown in Figure below. This is not the only way in
which a transistor may be used as an amplifier, as we will see in later sections
of this chapter.
Common-emitter amplifier: The input and output signals both share a
connection to the emitter.
Before, a small solar cell current saturated a transistor, illuminating a lamp.
Knowing now that transistors are able to “throttle” their collector currents
according to the amount of base current supplied by an input signal source, we
should see that the brightness of the lamp in this circuit is controllable by the
solar cell's light exposure. When there is just a little light shone on the solar
cell, the lamp will glow dimly. The lamp's brightness will steadily increase as
more light falls on the solar cell.
Suppose that we were interested in using the solar cell as a light intensity
instrument. We want to measure the intensity of incident light with the solar
cell by using its output current to drive a meter movement. It is possible to
directly connect a meter movement to a solar cell (Figure below) for this
purpose. In fact, the simplest light-exposure meters for photography work are
designed like this.
YX +I f
solar
cell
High intensity light directly drives light meter.
Although this approach might work for moderate light intensity measurements,
it would not work as well for low light intensity measurements. Because the
solar cell has to supply the meter movement's power needs, the system is
necessarily limited in its sensitivity. Supposing that our need here is to measure
very low-level light intensities, we are pressed to find another solution.
Perhaps the most direct solution to this measurement problem is to use a
transistor (Figure below) to amplify the solar cell's current so that more meter
deflection may be obtained for less incident light.
Cell current must be amplified for low intensity light.
Current through the meter movement in this circuit will be B times the solar cell
current. With a transistor B of 100, this represents a substantial increase in
measurement sensitivity. It is prudent to point out that the additional power to
move the meter needle comes from the battery on the far right of the circuit,
not the solar cell itself. All the solar cell's current does is contro/ battery current
to the meter to provide a greater meter reading than the solar cell could
provide unaided.
Because the transistor is a current-regulating device, and because meter
movement indications are based on the current through the movement coil,
meter indication in this circuit should depend only on the current from the solar
cell, not on the amount of voltage provided by the battery. This means the
accuracy of the circuit will be independent of battery condition, a significant
feature! All that is required of the battery is a certain minimum voltage and
current output ability to drive the meter full-scale.
Another way in which the common-emitter configuration may be used is to
produce an output vo/tage derived from the input signal, rather than a specific
output current. Let's replace the meter movement with a plain resistor and
measure voltage between collector and emitter in Figure below
V output
Common emitter amplifier develops voltage output due to current through load
resistor.
With the solar cell darkened (no current), the transistor will be in cutoff mode
and behave as an open switch between collector and emitter. This will produce
maximum voltage drop between collector and emitter for maximMUM Voutput:
equal to the full voltage of the battery.
At full power (maximum light exposure), the solar cell will drive the transistor
into saturation mode, making it behave like a closed switch between collector
and emitter. The result will be minimum voltage drop between collector and
emitter, or almost zero output voltage. In actuality, a saturated transistor can
never achieve zero voltage drop between collector and emitter because of the
two PN junctions through which collector current must travel. However, this
“collector-emitter saturation voltage” will be fairly low, around several tenths of
a volt, depending on the specific transistor used.
For light exposure levels somewhere between zero and maximum solar cell
output, the transistor will be in its active mode, and the output voltage will be
somewhere between zero and full battery voltage. An important quality to note
here about the common-emitter configuration is that the output voltage is
inverted with respect to the input signal. That is, the output voltage decreases
as the input signal increases. For this reason, the common-emitter amplifier
configuration is referred to as an /nverting amplifier.
A quick SPICE simulation (Figure below) of the circuit in Figure below will verify
our qualitative conclusions about this amplifier circuit.
Mconnon-enitter amplifier
R il 0 1 dc
gl 2 1 0 modl
5 kQ mn r 3 2 5000
V,—I15V v1 3 0 de 15
.model modi npn
.dc il 0 50u 2u
lA .plot dc v(2,0)
0 0 0 .end
Common emitter schematic with node numbers and corresponding SPICE
netlist.
“0.0 20,0 40.0 60,0
sweep uA
Common emitter: collector voltage output vs base current input.
At the beginning of the simulation in Figure above where the current source
(solar cell) is outputting zero current, the transistor is in cutoff mode and the
full 15 volts from the battery is shown at the amplifier output (between nodes 2
and 0). As the solar cell's current begins to increase, the output voltage
proportionally decreases, until the transistor reaches saturation at 30 YA of
base current (3 mA of collector current). Notice how the output voltage trace on
the graph is perfectly linear (1 volt steps from 15 volts to 1 volt) until the point
of saturation, where it never quite reaches zero. This is the effect mentioned
earlier, where a saturated transistor can never achieve exactly zero voltage
drop between collector and emitter due to internal junction effects. What we do
see is a Sharp output voltage decrease from 1 volt to 0.2261 volts as the input
Current increases from 28 YA to 30 HA, and then a continuing decrease in
output voltage from then on (albeit in progressively smaller steps). The lowest
the output voltage ever gets in this simulation is 0.1299 volts, asymptotically
approaching zero.
So far, we've seen the transistor used as an amplifier for DC signals. In the solar
cell light meter example, we were interested in amplifying the DC output of the
solar cell to drive a DC meter movement, or to produce a DC output voltage.
However, this is not the only way in which a transistor may be employed as an
amplifier. Often an AC amplifier for amplifying a/ternating current and voltage
signals is desired. One common application of this is in audio electronics
(radios, televisions, and public-address systems). Earlier, we saw an example of
the audio output of a tuning fork activating a transistor switch. (Figure below)
Let's see if we can modify that circuit to send power to a speaker rather than to
a lamp in Figure below.
microphone
c aon
source of
sound
Transistor switch activated by audio.
In the original circuit, a full-wave bridge rectifier was used to convert the
microphone's AC output signal into a DC voltage to drive the input of the
transistor. All we cared about here was turning the lamp on with a sound signal
from the microphone, and this arrangement sufficed for that purpose. But now
we want to actually reproduce the AC signal and drive a speaker. This means we
cannot rectify the microphone's output anymore, because we need undistorted
AC signal to drive the transistor! Let's remove the bridge rectifier and replace
the lamp with a speaker:
speaker
microphone
source of
sound
Common emitter amplifier drives speaker with audio frequency signal.
Since the microphone may produce voltages exceeding the forward voltage
drop of the base-emitter PN (diode) junction, I've placed a resistor in series with
the microphone. Let's simulate the circuit in Figure below with SPICE. The
netlist is included in (Figure below)
speaker
Vv, — 15V
V input
1.5 V
2 kHz W
0 0 0
SPICE version of common emitter audio amplifier.
Units v(1) — 10*I (v1#branch)
I(v(1)})
AUNNORDEROOOUOAAOADENOEEHNANOOOOSEOOELONOANODOERESSOOHOONOHFECOOSONOHOOOOES .
common-emitter amplifier
vinput 1 0 sin (0 1.5 2000 0 0)
.model mod1 npn
-tran 0.02m 0.74m
.plot tran v(1,0) i(v1)
.end
Signal clipped at collector due to lack of DC base bias.
The simulation plots (Figure above) both the input voltage (an AC signal of 1.5
volt peak amplitude and 2000 Hz frequency) and the current through the 15
volt battery, which is the same as the current through the speaker. What we see
here is a full AC sine wave alternating in both positive and negative directions,
and a half-wave output current waveform that only pulses in one direction. If we
were actually driving a speaker with this waveform, the sound produced would
be horribly distorted.
What's wrong with the circuit? Why won't it faithfully reproduce the entire AC
waveform from the microphone? The answer to this question is found by close
inspection of the transistor diode current source model in Figure below.
Cc
B B
E
NPN
diode-current source
model E
The model shows that base current flow in on direction.
Collector current is controlled, or regulated, through the constant-current
mechanism according to the pace set by the current through the base-emitter
diode. Note that both current paths through the transistor are monodirectional:
one way only! Despite our intent to use the transistor to amplify an AC signal, it
is essentially a DC device, capable of handling currents in a single direction. We
may apply an AC voltage input signal between the base and emitter, but
electrons cannot flow in that circuit during the part of the cycle that reverse-
biases the base-emitter diode junction. Therefore, the transistor will remain in
cutoff mode throughout that portion of the cycle. It will “turn on” in its active
mode only when the input voltage is of the correct polarity to forward-bias the
base-emitter diode, and only when that voltage is sufficiently high to overcome
the diode's forward voltage drop. Remember that bipolar transistors are current-
controlled devices: they regulate collector current based on the existence of
base-to-emitter current, not base-to-emitter vo/tage.
The only way we can get the transistor to reproduce the entire waveform as
current through the speaker is to keep the transistor in its active mode the
entire time. This means we must maintain current through the base during the
entire input waveform cycle. Consequently, the base-emitter diode junction
must be kept forward-biased at all times. Fortunately, this can be accomplished
with a DC bias voltage added to the input signal. By connecting a sufficient DC
voltage in series with the AC signal source, forward-bias can be maintained at
all points throughout the wave cycle. (Figure below)
speaker
Voias Keeps transistor in the active region.
Units v(1) — 10*I (v1#branch)
common-emitter amplifier
vinput 15 sin (0 1.5 2000 0 0)
vbias 5 0 dc 2.3
rl 12 1k
ql 3 2 0 modl
rspkr 3 4 8
vl 4 0 de 15
.model mod1 npn
tran 0.02m 0.78m
.plot tran v(1,0) i(v1)
.end
Undistorted output current I(v(1) due to Vbias
With the bias voltage source of 2.3 volts in place, the transistor remains in its
active mode throughout the entire cycle of the wave, faithfully reproducing the
waveform at the speaker. (Figure above) Notice that the input voltage
(measured between nodes 1 and 0) fluctuates between about 0.8 volts and 3.8
volts, a peak-to-peak voltage of 3 volts just as expected (Source voltage = 1.5
volts peak). The output (Speaker) current varies between zero and almost 300
mA, 180° out of phase with the input (microphone) signal.
The illustration in Figure below is another view of the same circuit, this time
with a few oscilloscopes (“scopemeters”) connected at crucial points to display
all the pertinent signals.
m a speaker
y, A
Vv, A ep ‘
V
input
Input is biased upward at base. Output is inverted.
The need for biasing a transistor amplifier circuit to obtain full waveform
reproduction is an important consideration. A separate section of this chapter
will be devoted entirely to the subject biasing and biasing techniques. For now,
it is enough to understand that biasing may be necessary for proper voltage
and current output from the amplifier.
Now that we have a functioning amplifier circuit, we can investigate its voltage,
current, and power gains. The generic transistor used in these SPICE analyses
has a B of 100, as indicated by the short transistor statistics printout included
in the text output in Table below (these statistics were cut from the last two
analyses for brevity's sake).
BJT SPICE model parameters.
type npn
is 1.00E-16
bf 100.000
nf 1.000
br 1.000
nr 1.000
B is listed under the abbreviation “bf,” which actually stands for “beta,
forward”. If we wanted to insert our own & ratio for an analysis, we could have
done so on the .model line of the SPICE netlist.
Since B is the ratio of collector current to base current, and we have our load
connected in series with the collector terminal of the transistor and our source
connected in series with the base, the ratio of output current to input current is
equal to beta. Thus, our current gain for this example amplifier is 100, or 40 dB.
Voltage gain is a little more complicated to figure than current gain for this
circuit. As always, voltage gain is defined as the ratio of output voltage divided
by input voltage. In order to experimentally determine this, we modify our last
SPICE analysis to plot output voltage rather than output current so we have two
voltage plots to compare in Figure below.
common-emitter amplifier
Vinput 15 sin (0 1.5 2000 0 0)
Vbias 5 0 dc 2.3
rl 12 1k
ql 3 2 0 modl
rspkr 3 4 8
v1 4 @ de 15
.model modi npn
.tran 0.02m 0.78m
.plot tran v(1,0) v(3)
.end
V(3), the output voltage across rox, compared to the input.
Plotted on the same scale (from 0 to 4 volts), we see that the output waveform
in Figure above has a smaller peak-to-peak amplitude than the input waveform
, in addition to being at a lower bias voltage, not elevated up from 0 volts like
the input. Since voltage gain for an AC amplifier is defined by the ratio of AC
amplitudes, we can ignore any DC bias separating the two waveforms. Even so,
the input waveform is still larger than the output, which tells us that the voltage
gain is less than 1 (a negative dB figure).
To be honest, this low voltage gain is not characteristic to a// common-emitter
amplifiers. It is a consequence of the great disparity between the input and load
resistances. Our input resistance (R;) here is 1000 Q, while the load (speaker) is
only 8 Q. Because the current gain of this amplifier is determined solely by the
8 of the transistor, and because that B figure is fixed, the current gain for this
amplifier won't change with variations in either of these resistances. However,
voltage gain /s dependent on these resistances. If we alter the load resistance,
making it a larger value, it will drop a proportionately greater voltage for its
range of load currents, resulting in a larger output waveform. Let's try another
simulation, only this time with a 30 Q in Figure below load instead of an 8 OQ
load.
y — v4) = ¥f3)
15.0 common-emitter amplifier
vVinput 15 sin (0 1.5 2000 0 0)
vbias 5 0 dc 2.3
10,0
a .model mod1 npn
-tran 0.02m 0.78m
.plot tran v(1,0) v(3)
.end
Increasing px, to 30 Q increases the output voltage.
This time the output voltage waveform in Figure above is significantly greater in
amplitude than the input waveform. Looking closely, we can see that the output
waveform crests between 0 and about 9 volts: approximately 3 times the
amplitude of the input voltage.
We can do another computer analysis of this circuit, this time instructing SPICE
to analyze it from an AC point of view, giving us peak voltage figures for input
and output instead of a time-based plot of the waveforms. (Table below)
SPICE netlist for printing AC input and output voltages.
common-emitter amplifier
vinput 15 ac 1.5
vbias 5 0 dc 2.3
rl 12 1k
ql 3 2 0 modl
rspkr 3 4 30
vl 4 @ de 15
.model modi npn
.ac Lin 1 2000 2000
.print ac v(1,0) v(4,3)
.end
freq v(1) v(4,3)
2.000E+03 1.500E+00 4.418E+00
Peak voltage measurements of input and output show an input of 1.5 volts and
an output of 4.418 volts. This gives us a voltage gain ratio of 2.9453 (4.418 V /
1.5 V), or 9.3827 dB.
f Vout
oy Vin
_ 4418 V
a 15V
Ay = 2.9453
Aviap) = 20 log Ay ratio)
Ayiapy = 20 log 2.9453
A\ apy = 9-3827 dB
Because the current gain of the common-emitter amplifier is fixed by B, and
since the input and output voltages will be equal to the input and output
currents multiplied by their respective resistors, we can derive an equation for
approximate voltage gain:
R
Av = ‘out
\ ss ay
ae _30Q
Ay = (100) = 9050
Ay =3
Ayiap) = 20 log Av(ratio)
Ayvap) = 20 log 3
Aap) = 9-5424 dB
As you can see, the predicted results for voltage gain are quite close to the
simulated results. With perfectly linear transistor behavior, the two sets of
figures would exactly match. SPICE does a reasonable job of accounting for the
many “quirks” of bipolar transistor function in its analysis, hence the slight
mismatch in voltage gain based on SPICE's output.
These voltage gains remain the same regardless of where we measure output
voltage in the circuit: across collector and emitter, or across the series load
resistor as we did in the last analysis. The amount of output voltage change for
any given amount of input voltage will remain the same. Consider the two
following SPICE analyses as proof of this. The first simulation in Figure below is
time-based, to provide a plot of input and output voltages. You will notice that
the two signals are 180° out of phase with each other. The second simulation in
Table below is an AC analysis, to provide simple, peak voltage readings for input
and output.
y — v(4) — ¥(3)
15.0 common-emitter amplifier
.model mod1 npn
.tran 0.02m 0.74m
.plot tran v(1,0) v(3,0)
.end
Common-emitter amplifier shows a voltage gain with Rep4-=30Q
SPICE netlist for AC analysis
common-emitter amplifier
vinput 15 ac 1.5
vbias 5 0 dc 2.3
rl 12 1k
ql 3 2 0 modl
rspkr 3 4 30
vl 4 @ de 15
.model modi npn
.ac Lin 1 2000 2000
.print ac v(1,0) v(3,0)
.end
freq v(1) v(3)
2.000E+03 1.500E+00 4.418E+00
We still have a peak output voltage of 4.418 volts with a peak input voltage of
1.5 volts. The only difference from the last set of simulations is the phase of the
output voltage.
So far, the example circuits shown in this section have all used NPN transistors.
PNP transistors are just as valid to use as NPN in any amplifier configuration, as
long as the proper polarity and current directions are maintained, and the
common-emitter amplifier is no exception. The output invertion and gain of a
PNP transistor amplifier are the same as its NPN counterpart, just the battery
polarities are different. (Figure below)
PNP version of common emitter amplifier.
e REVIEW:
¢ Common-emitter transistor amplifiers are so-called because the input and
output voltage points share the emitter lead of the transistor in common
with each other, not considering any power supplies.
e Transistors are essentially DC devices: they cannot directly handle voltages
or currents that reverse direction. To make them work for amplifying AC
signals, the input signal must be offset with a DC voltage to keep the
transistor in its active mode throughout the entire cycle of the wave. This is
called biasing.
e If the output voltage is measured between emitter and collector on a
common-emitter amplifier, it will be 180° out of phase with the input
voltage waveform. Thus, the common-emitter amplifier is called an
inverting amplifier circuit.
e The current gain of a common-emitter transistor amplifier with the load
connected in series with the collector is equal to B. The voltage gain of a
common-emitter transistor amplifier is approximately given here:
R
Ay - B out
e in
¢ Where “Roy” is the resistor connected in series with the collector and “R;,”
is the resistor connected in series with the base.
The common-collector amplifier
Our next transistor configuration to study is a bit simpler for gain calculations.
Called the common-collector configuration, its schematic diagram is shown in
Figure below.
Common collector amplifier has collector common to both input and output.
It is called the common-collector configuration because (ignoring the power
supply battery) both the signal source and the load share the collector lead asa
common connection point as in Figure below.
Common collector: Input is applied to base and collector. Output is from
emitter-collector circuit.
It should be apparent that the load resistor in the common-collector amplifier
circuit receives both the base and collector currents, being placed in series with
the emitter. Since the emitter lead of a transistor is the one handling the most
current (the sum of base and collector currents, since base and collector
currents always mesh together to form the emitter current), it would be
reasonable to presume that this amplifier will have a very large current gain.
This presumption is indeed correct: the current gain for a common-collector
amplifier is quite large, larger than any other transistor amplifier configuration.
However, this is not necessarily what sets it apart from other amplifier designs.
Let's proceed immediately to a SPICE analysis of this amplifier circuit, and you
will be able to immediately see what is unique about this amplifier. The circuit
is in Figure below. The netlist is in Figure below.
common-collector amplifier
.model mod1 npn
.dc vin 05 0.2
.plot dc v(3,0)
.end
“00 10 20 3.0 40 5,0
Common collector: Output equals input less a 0.7 V Vp- drop.
Unlike the common-emitter amplifier from the previous section, the common-
collector produces an output voltage in direct rather than /nverse proportion to
the rising input voltage. See Figure above. As the input voltage increases, so
does the output voltage. Moreover, a close examination reveals that the output
voltage is nearly /dentica/ to the input voltage, lagging behind by about 0.7
volts.
This is the unique quality of the common-collector amplifier: an output voltage
that is nearly equal to the input voltage. Examined from the perspective of
output voltage change for a given amount of input voltage change, this
amplifier has a voltage gain of almost exactly unity (1), or 0 dB. This holds true
for transistors of any B value, and for load resistors of any resistance value.
It is simple to understand why the output voltage of a common-collector
amplifier is always nearly equal to the input voltage. Referring to the diode
Current source transistor model in Figure below, we see that the base current
must go through the base-emitter PN junction, which is equivalent to a normal
rectifying diode. If this junction is forward-biased (the transistor conducting
current in either its active or saturated modes), it will have a voltage drop of
approximately 0.7 volts, assuming silicon construction. This 0.7 volt drop is
largely irrespective of the actual magnitude of base current; thus, we can
regard it as being constant:
Emitter follower: Emitter voltage follows base voltage (less a 0.7 V Vp- drop.)
Given the voltage polarities across the base-emitter PN junction and the load
resistor, we see that these must add together to equal the input voltage, in
accordance with Kirchhoff's Voltage Law. In other words, the load voltage will
always be about 0.7 volts less than the input voltage for all conditions where
the transistor is conducting. Cutoff occurs at input voltages below 0.7 volts, and
saturation at input voltages in excess of battery (Supply) voltage plus 0.7 volts.
Because of this behavior, the common-collector amplifier circuit is also Known
as the voltage-follower or emitter-follower amplifier, because the emitter load
voltages follow the input so closely.
Applying the common-collector circuit to the amplification of AC signals
requires the same input “biasing” used in the common-emitter circuit: a DC
voltage must be added to the AC input signal to keep the transistor in its active
mode during the entire cycle. When this is done, the result is the non-inverting
amplifier in Figure below.
common-collector amplifier
Vin 1 4 sin(0 1.5 2000 0 0)
vbias 4 0 dc 2.3
gl 2 1 3 modl
vl 2 0 de 15
rload 3 0 5k
.model modi npn
.tran .02m .78m
.plot tran v(1,0) v(3,0)
.end
| I |
Common collector (emitter-follower) amplifier.
The results of the SPICE simulation in Figure below show that the output follows
the input. The output is the same peak-to-peak amplitude as the input. Though,
the DC level is shifted downward by one Vp, diode drop.
Common collector (emitter-follower): Output V3 follows input V1 less a 0.7 V
VBE drop.
Here's another view of the circuit (Figure below) with oscilloscopes connected to
several points of interest.
Common collector non-inverting voltage gain is 1.
Since this amplifier configuration doesn't provide any voltage gain (in fact, in
practice it actually has a voltage gain of slightly /ess than 1), its only amplifying
factor is current. The common-emitter amplifier configuration examined in the
previous section had a current gain equal to the B of the transistor, being that
the input current went through the base and the output (load) current went
through the collector, and B by definition is the ratio between the collector and
base currents. In the common-collector configuration, though, the load is
situated in series with the emitter, and thus its current is equal to the emitter
current. With the emitter carrying collector current and base current, the load in
this type of amplifier has all the current of the collector running through it p/us
the input current of the base. This yields a current gain of B plus 1:
A = Lemitter
Lsase
A.= Tectlactor® Thase
|=—— =
Di
A = Tootlector 4 ]
base
A,=B+1
Once again, PNP transistors are just as valid to use in the common-collector
configuration as NPN transistors. The gain calculations are all the same, as is
the non-inverting of the amplified signal. The only difference is in voltage
polarities and current directions shown in Figure below.
PNP version of the common-collector amplifier.
A popular application of the common-collector amplifier is for regulated DC
power supplies, where an unregulated (varying) source of DC voltage is clipped
at a specified level to supply regulated (steady) voltage to a load. Of course,
zener diodes already provide this function of voltage regulation shown in Figure
below.
Unregulated
DC voltage —
source
—
Regulated voltage
across load
Zener diode voltage regulator.
However, when used in this direct fashion, the amount of current that may be
supplied to the load is usually quite limited. In essence, this circuit regulates
voltage across the load by keeping current through the series resistor at a high
enough level to drop all the excess power source voltage across it, the zener
diode drawing more or less current as necessary to keep the voltage across
itself steady. For high-current loads, a plain zener diode voltage regulator would
have to shunt a heavy current through the diode to be effective at regulating
load voltage in the event of large load resistance or voltage source changes.
One popular way to increase the current-handling ability of a regulator circuit
like this is to use a common-collector transistor to amplify current to the load,
so that the zener diode circuit only has to handle the amount of current
necessary to drive the base of the transistor. (Figure below)
Unregulated
DC voltage ——
source
Common collector application: voltage regulator.
There's really only one caveat to this approach: the load voltage will be
approximately 0.7 volts less than the zener diode voltage, due to the
transistor's 0.7 volt base-emitter drop. Since this 0.7 volt difference is fairly
constant over a wide range of load currents, a zener diode with a 0.7 volt higher
rating can be chosen for the application.
Sometimes the high current gain of a single-transistor, common-collector
configuration isn't enough for a particular application. If this is the case,
multiple transistors may be staged together in a popular configuration known
as a Darlington pair, just an extension of the common-collector concept shown
in Figure below.
Cc
E
An NPN darlington pair.
Darlington pairs essentially place one transistor as the common-collector load
for another transistor, thus multiplying their individual current gains. Base
current through the upper-left transistor is amplified through that transistor's
emitter, which is directly connected to the base of the lower-right transistor,
where the current is again amplified. The overall current gain is as follows:
Darlington pair current gain
A, = (B, + 1B, + 1)
Where,
6, = Beta of first transistor
B, = Beta of second transistor
Voltage gain is still nearly equal to 1 if the entire assembly is connected to a
load in common-collector fashion, although the load voltage will be a full 1.4
volts less than the input voltage shown in Figure below.
Vout = Vin - 1.4
Darlington pair based common-collector amplifier loses two Vgr diode drops.
Darlington pairs may be purchased as discrete units (two transistors in the
same package), or may be built up from a pair of individual transistors. Of
course, if even more current gain is desired than what may be obtained with a
pair, Darlington triplet or quadruplet assemblies may be constructed.
¢ REVIEW:
« Common-collector transistor amplifiers are so-called because the input and
output voltage points share the collector lead of the transistor in common
with each other, not considering any power supplies.
e The common-collector amplifier is also Known as an emitter-follower.
e The output voltage on a common-collector amplifier will be in phase with
the input voltage, making the common-collector a non-inverting amplifier
Circuit.
e The current gain of a common-collector amplifier is equal to B plus 1. The
voltage gain is approximately equal to 1 (in practice, just a little bit less).
¢ A Darlington pair is a pair of transistors “piggybacked” on one another so
that the emitter of one feeds current to the base of the other in common-
collector form. The result is an overall current gain equal to the product
(multiplication) of their individual common-collector current gains (8 plus
1).
The common-base amplifier
The final transistor amplifier configuration (Figure below) we need to study is
the common-base. This configuration is more complex than the other two, and
is less common due to its strange operating characteristics.
Common-base amplifier
It is called the common-base configuration because (DC power source aside),
the signal source and the load share the base of the transistor as a common
connection point shown in Figure below.
Common-base amplifier: Input between emitter and base, output between
collector and base.
Perhaps the most striking characteristic of this configuration is that the input
signal source must carry the full emitter current of the transistor, as indicated
by the heavy arrows in the first illustration. As we know, the emitter current is
greater than any other current in the transistor, being the sum of base and
collector currents. In the last two amplifier configurations, the signal source was
connected to the base lead of the transistor, thus handling the /east current
possible.
Because the input current exceeds all other currents in the circuit, including the
output current, the current gain of this amplifier is actually /ess than 1 (notice
how Rigag is connected to the collector, thus carrying slightly less current than
the signal source). In other words, it attenuates current rather than amplifying
it. With common-emitter and common-collector amplifier configurations, the
transistor parameter most closely associated with gain was B. In the common-
base circuit, we follow another basic transistor parameter: the ratio between
collector current and emitter current, which is a fraction always less than 1. This
fractional value for any transistor is called the a/pha ratio, or a ratio.
Since it obviously can't boost signal current, it only seems reasonable to expect
it to boost signal voltage. A SPICE simulation of the circuit in Figure below will
vindicate that assumption.
common-base amplifier
vin 01
rl 1 2 100
gl 4 0 2 modl
v1 3 0 de 15
rload 3 4 5k
.model mod1 npn
.dc vin 0.6 1.2 .02
.plot dc v(3,4)
.end
0,60 0,80 1,00 1.20
sweep Vv
Common-base amplifier DC transfer function.
Notice in Figure above that the output voltage goes from practically nothing
(cutoff) to 15.75 volts (saturation) with the input voltage being swept over a
range of 0.6 volts to 1.2 volts. In fact, the output voltage plot doesn't show a
rise until about 0.7 volts at the input, and cuts off (flattens) at about 1.12 volts
input. This represents a rather large voltage gain with an output voltage span of
15.75 volts and an input voltage span of only 0.42 volts: a gain ratio of 37.5, or
31.48 dB. Notice also how the output voltage (measured across Rjgaq) actually
exceeds the power supply (15 volts) at saturation, due to the series-aiding
effect of the input voltage source.
A second set of SPICE analyses (circuit in Figure below) with an AC signal source
(and DC bias voltage) tells the same story: a high voltage gain
Viias 0.95 V I5V
Common-base circuit for SPICE AC analysis.
As you can see, the input and output waveforms in Figure below are in phase
with each other. This tells us that the common-base amplifier is non-inverting.
Units — 10*¥(5,2- ¥(4)
15,0 common-base amplifier
vin 5 2 sin (0 0.12 2000 0 0)
vbias @ 1 dc 0.95
rl 2 1 100
ql 4 05 modl
vl 3 0 dc 15
rload 3 4 5k
.model mod1 npn
tran 0.02m 0.78m
.plot tran v(5,2) v(4)
.end
5,0
The AC SPICE analysis in Table below at a single frequency of 2 kHz provides
input and output voltages for gain calculation.
Common-base AC analysis at 2 kHz- netlist followed by output.
common-base amplifier
vin 5 2 ac @.1 sin
vbias 0 1 dc 0.95
rl 2 1 100
ql 4 05 modl1
vl 3 0 de 15
rload 3 4 5k
.model modi npn
.ac dec 1 2000 2000
.print ac vm(5,2) vm(4,3)
.end
frequency mag(v(5,2)) mag(v(4,3))
0.000000e+00 1.000000e-01 4.273864e+00
Voltage figures from the second analysis (Table above) show a voltage gain of
42.74 (4.274 V/0.1 V), or 32.617 GB:
Vv
A ,= out
, Vin
a — 4.274 V
a 0.10 V
Ay = 42.74
Ayap) = 20 log Ayiratio)
Aviap) a 20 log 42.74
Aviap) = 32.62 dB
Here's another view of the circuit in Figure below, summarizing the phase
relations and DC offsets of various signals in the circuit just simulated.
Phase relationships and offsets for NPN common base amplifier.
...and for a PNP transistor: Figure below.
Phase relationships and offsets for PNP common base amplifier.
Predicting voltage gain for the common-base amplifier configuration is quite
difficult, and involves approximations of transistor behavior that are difficult to
measure directly. Unlike the other amplifier configurations, where voltage gain
was either set by the ratio of two resistors (Ccommon-emitter), or fixed at an
unchangeable value (common-collector), the voltage gain of the common-base
amplifier depends largely on the amount of DC bias on the input signal. As it
turns out, the internal transistor resistance between emitter and base plays a
major role in determining voltage gain, and this resistance changes with
different levels of current through the emitter.
While this phenomenon is difficult to explain, it is rather easy to demonstrate
through the use of computer simulation. What I'm going to do here is run
several SPICE simulations on a common-base amplifier circuit (Figure previous),
changing the DC bias voltage slightly (vbias in Figure below ) while keeping the
AC signal amplitude and all other circuit parameters constant. As the voltage
gain changes from one simulation to another, different output voltage
amplitudes will be noted.
Although these analyses will all be conducted in the “transfer function” mode,
each was first “proofed” in the transient analysis mode (voltage plotted over
time) to ensure that the entire wave was being faithfully reproduced and not
“clipped” due to improper biasing. See "*.tran 0.02m 0.78m" in Figure below,
the “commented out” transient analysis statement. Gain calculations cannot be
based on waveforms that are distorted. SPICE can calculate the small signal DC
gain for us with the “.tf v(4) vin” statement. The output is v(4) and the input as
vin.
common-base amp current gain
Tin 55 5 0A
vin 55 2
vbias @ 1 dc 0.8753
common-base amp vbias=0.85V ONG Ue Ns)
.model modi npn
eK .tran 0.02m 0.78m
.tf v(4) vin
.end
vin 5 2 sin (0 0.12 2000 0 0)
: rl 211
vVbias @ 1 dc 0.85 ee
gl 4 05 modl
rl 2 1 100
v1 3 0 de 15
ql 4 05 modl
rload 3 4 5k
Ds tet model modil1 npn
rload 3 4 5k 5 P
*.tran 0.02m 0.78m
.tf I(vl1) Iin
.end
Transfer function information:
transfer function = 9.900990e-01
iin input impedance = 9.900923e+11
v1 output impedance 1.000000e+20
SPICE net list: Common-base, transfer function (voltage gain) for various DC
bias voltages. SPICE net list: Common-base amp current gain; Note .tf v(4) vin
statement. Transfer function for DC current gain I(vin)/lin; Note .tf I(vin) lin
statement.
At the command line, spice -b filename.cir produces a printed output due to
the .tf statement: transfer function, output_impedance, and input_impedance.
The abbreviated output listing is from runs with vbias at 0.85, 0.90, 0.95, 1.00
V as recorded in Table below.
SPICE output: Common-base transfer function.
Circuit: common-base amp vbias=0.85V
transfer function = 3.756565e+01
output _impedance at_v(4) = 5.000000e+03
vin#input_impedance = 1.317825e+02
Circuit: common-base amp vbias=0.8753V Ic=1 mA
Transfer function information:
transfer function = 3.942567e+01
output_impedance at_v(4) = 5.000000e+03
vin#input impedance = 1.255653e+02
Circuit: common-base amp vbias=0.9V
transfer function = 4.079542e+01
output _impedance at_v(4) = 5.000000e+03
vin#input_ impedance = 1.213493e+02
Circuit: common-base amp vbias=0.95V
transfer function = 4.273864e+01
output _impedance at_v(4) = 5.000000e+03
vin#input_ impedance = 1.158318e+02
Circuit: common-base amp vbias=1.00V
transfer function = 4.401137e+01
output _impedance at_v(4) = 5.000000e+03
vin#input_impedance = 1.124822e+02
A trend should be evident in Table above. With increases in DC bias voltage,
voltage gain (transfer function) increases as well. We can see that the voltage
gain is increasing because each subsequent simulation (vbias= 0.85, 0.8753,
0.90, 0.95, 1.00 V) produces greater gain (transfer function= 37.6, 39.4 40.8,
42.7, 44.0), respectively. The changes are largely due to minuscule variations in
bias voltage.
The last three lines of Table above(right) show the I(v1)/lin current gain of 0.99.
(The last two lines look invalid.) This makes sense for B=100; a= B/(B+1),
a=0.99=100/(100-1). The combination of low current gain (always less than 1)
and somewhat unpredictable voltage gain conspire against the common-base
design, relegating it to few practical applications.
Those few applications include radio frequency amplifiers. The grounded base
helps shield the input at the emitter from the collector output, preventing
instability in RF amplifiers. The common base configuration is usable at higher
frequencies than common emitter or common collector. See “Class C common-
base 750 mW RF power amplifier” Ch 9. For a more elaborate circuit see “Class
A common-base small-signal high gain amplifier”Ch 9 .
¢ REVIEW:
« Common-base transistor amplifiers are so-called because the input and
output voltage points share the base lead of the transistor in common with
each other, not considering any power supplies.
e The current gain of a common-base amplifier is always less than 1. The
voltage gain is a function of input and output resistances, and also the
internal resistance of the emitter-base junction, which is subject to change
with variations in DC bias voltage. Suffice to say that the voltage gain of a
common-base amplifier can be very high.
e The ratio of a transistor's collector current to emitter current is called a. The
a value for any transistor is always less than unity, or in other words, less
than 1.
The cascode amplifier
While the C-B (common-base) amplifier is known for wider bandwidth than the
C-E (common-emitter) configuration, the low input impedance (10s of Q) of C-B
is a limitation for many applications. The solution is to precede the C-B stage by
a low gain C-E stage which has moderately high input impedance (kQs). See
Figure below. The stages are in a cascode configuration, stacked in series, as
opposed to cascaded for a standard amplifier chain. See “Capacitor coupled
three stage common-emitter amplifier” Capacitor coupled for a cascade
example. The cascode amplifier configuration has both wide bandwidth and a
moderately high input impedance.
Vo
Vi Vo
Vi
Common
emitter
Common
base —— > Vo
Common-base Common-emitter Cascode
The cascode amplifier is combined common-emitter and common-base. This is
an AC circuit equivalent with batteries and capacitors replaced by short circuits.
The key to understanding the wide bandwidth of the cascode configuration is
the Miller effect. The Miller effect is the multiplication of the bandwidth robbing
collector-base capacitance by voltage gain A,. This C-B capacitance is smaller
than the E-B capacitance. Thus, one would think that the C-B capacitance
would have little effect. However, in the C-E configuration, the collector output
signal is out of phase with the input at the base. The collector signal
Capacitively coupled back opposes the base signal. Moreover, the collector
feedback is (1-A,) times larger than the base signal. Keep in mind that A, is a
negative number for the inverting C-E amplifier. Thus, the small C-B
capacitance appears (1+A|,|) times larger than its actual value. This capacitive
gain reducing feedback increases with frequency, reducing the high frequency
response of a C-E amplifier.
The approximate voltage gain of the C-E amplifier in Figure below is -R,/ree. The
emitter current is set to 1.0 mA by biasing. Ree= 26MV/I_ = 26MV/1.0ma = 26
Q. Thus, Ay = -Ri/Reg = -4700/26 = -181. The pn2222 datasheet list C.,, = 8 pF.
[FAR] The miller capacitance is C.p.(1-Ay). Gain Ay = -181, negative since it is
inverting gain. Crier = Cepo(l-Ay) = 8pF(1-(-181)=1456pF
A common-base configuration is not subject to the Miller effect because the
grounded base shields the collector signal from being fed back to the emitter
input. Thus, a C-B amplifier has better high frequency response. To have a
moderately high input impedance, the C-E stage is still desirable. The key is to
reduce the gain (to about 1) of the C-E stage which reduces the Miller effect C-B
feedback to 1-:Ccgo. The total C-B feedback is the feedback capacitance 1:Ccg
plus the actual capacitance Ccp for a total of 2-Ccego. This is a considerable
reduction from 181:Ccego. The miller capacitance for a gain of -2 C-E stage is
Critler = Ceboll-Ay)= Cmitier = Cebo(1-(-1)) = Copo'2-
The way to reduce the common-emitter gain is to reduce the load resistance.
The gain of a C-E amplifier is approximately R-/Re. The internal emitter
resistance ree at 1mMA emitter current is 26Q. For details on the 26Q, see
“Derivation of Ree”, see REE. The collector load R¢- is the resistance of the
emitter of the C-B stage loading the C-E stage, 260 again. CE gain amplifier
gain is approximately Ay = R¢/Re=26/26=1. This Miller capacitance is Crier =
Cepo(1-Ay) = 8pF(1-(-1)=16pF. We now have a moderately high input
impedance C-E stage without suffering the Miller effect, but no C-E dB voltage
gain. The C-B stage provides a high voltage gain, Ay = -181. Current gain of
cascode is B of the C-E stage, 1 for the C-B, B overall. Thus, the cascode has
moderately high input impedance of the C-E, good gain, and good bandwidth of
the C-B.
(a) Cascode (b) Common-emitter
SPICE: Cascode and common-emitter for comparison.
The SPICE version of both a cascode amplifier, and for comparison, a common-
emitter amplifier is shown in Figure above. The netlist is in Table below. The AC
source V3 drives both amplifiers via node 4. The bias resistors for this circuit are
calculated in an example problem cascode.
Units — vm(3} = = vm(13)
yo oo 10*vm(5)— vmfa)
20,0
15,0]
10,0
SPICE waveforms. Note that Input is multiplied by 10 for visibility.
SPICE netlist for printing AC input and output voltages.
*SPICE circuit <03502.eps> from XCircuit v3.20
v1 19 0 10
Q1 13 15 0 q2n2222
Q2 3 2 A q2n2222
R1 19 13 4.7k
v2 1601.5
Cl 4 15 10n
R2 15 16 80k
Q3 A 5 0 q2n2222
V3 4 6 SIN(O 0.1 1k) acl
R3 1 2 80k
R4 3 9 4.7k
C2 2 0 10n
C3 45 10n
R5 5 6 80k
v4.10 11.5
V5 9 0 20
V6 601.5
.model q2n2222 npn (is=19f bf=150
+ vaf=100 ikf=0.18 ise=50p ne=2.5 br=7.5
+ var=6.4 ikr=12m isc=8.7p nc=1.2 rb=50
+ re=0.4 rc=0.3 cje=26p tf=0.5n
+ cjc=llp tr=7n xtb=1.5 kf=0.032f af=1)
.tran lu 5m
.AC DEC 10 1k 100Meg
.end
The waveforms in Figure above show the operation of the cascode stage. The
input signal is displayed multiplied by 10 so that it may be shown with the
outputs. Note that both the Cascode, Common-emitter, and Va (intermediate
point) outputs are inverted from the input. Both the Cascode and Common
emitter have large amplitude outputs. The Va point has a DC level of about 10V,
about half way between 20V and ground. The signal is larger than can be
accounted for by a C-E gain of 1, It is three times larger than expected.
Yoo = db(vm(3)} -= db(vm(13))
dB Cascode Common-emitter
, 10°3 10%4 10° 10°6 10°7 10°8 10°9
frequency Hz
Cascode vs common-emitter banwidth.
Figure above shows the frequency response to both the cascode and common-
emitter amplifiers. The SPICE statements responsible for the AC analysis,
extracted from the listing:
V3 4 6 SIN(Q 0.1 1k) acl
.AC DEC 10 1k 100Meg
Note the “ac 1” is necessary at the end of the V3 statement. The cascode has
marginally better mid-band gain. However, we are primarily looking for the
bandwidth measured at the -3dB points, down from the midband gain for each
amplifier. This is shown by the vertical solid lines in Figure above. It is also
possible to print the data of interest from nutmeg to the screen, the SPICE
graphical viewer (command, first line):
nutmeg 6 -> print frequency db(vm(3)) db(vm(13) )
Index frequency db(vm(3)) db(vm(13))
22 0.158MHz 47.54 45.41
33 1.995MHz 46.95 42.06
37 5 .012MHz 44.63 36.17
Index 22 gives the midband dB gain for Cascode vm(3)=47 .5dB and Common-
emitter vm(13)=45.4dB. Out of many printed lines, Index 33 was the closest to
being 3dB down from 45.4dB at 42.0dB for the Common-emitter circuit. The
corresponding Index 33 frequency is approximately 2Mhz, the common-emitter
bandwidth. Index 37 vm(3)=44.6db is approximately 3db down from 47.5db.
The corresponding Index37 frequency is 5Mhz, the cascode bandwidth. Thus,
the cascode amplifier has a wider bandwidth. We are not concerned with the
low frequency degradation of gain. It is due to the capacitors, which could be
remedied with larger ones.
The 5MHz bandwith of our cascode example, while better than the common-
emitter example, is not exemplary for an RF (radio frequency) amplifier. A pair
of RF or microwave transistors with lower interelectrode capacitances should be
used for higher bandwidth. Before the invention of the RF dual gate MOSFET,
the BJT cascode amplifier could have been found in UHF (ultra high frequency)
TV tuners.
e REVIEW
e A cascode amplifier consists of a common-emitter stage loaded by the
emitter of a common-base stage.
e« The heavily loaded C-E stage has a low gain of 1, overcoming the Miller
effect
e A cascode amplifier has a high gain, moderately high input impedance, a
high output impedance, and a high bandwidth.
Biasing techniques
In the common-emitter section of this chapter, we saw a SPICE analysis where
the output waveform resembled a half-wave rectified shape: only half of the
input waveform was reproduced, with the other half being completely cut off.
Since our purpose at that time was to reproduce the entire waveshape, this
constituted a problem. The solution to this problem was to add a small bias
voltage to the amplifier input so that the transistor stayed in active mode
throughout the entire wave cycle. This addition was called a bias voltage.
A half-wave output is not problematic for some applications. In fact, some
applications may necessitate this very kind of amplification. Because it is
possible to operate an amplifier in modes other than full-wave reproduction and
specific applications require different ranges of reproduction, it is useful to
describe the degree to which an amplifier reproduces the input waveform by
designating it according to class. Amplifier class operation is categorized with
alphabetical letters: A, B, C, and AB.
For Class A operation, the entire input waveform is faithfully reproduced.
Although | didn't introduce this concept back in the common-emitter section,
this is what we were hoping to attain in our simulations. Class A operation can
only be obtained when the transistor spends its entire time in the active mode,
never reaching either cutoff or saturation. To achieve this, sufficient DC bias
voltage is usually set at the level necessary to drive the transistor exactly
halfway between cutoff and saturation. This way, the AC input signal will be
perfectly “centered” between the amplifier's high and low signal limit levels.
Class A
Amplifier
Class A: The amplifier output is a faithful reproduction of the input.
Class B operation is what we had the first time an AC signal was applied to the
common-emitter amplifier with no DC bias voltage. The transistor spent half its
time in active mode and the other half in cutoff with the input voltage too low
(or even of the wrong polarity!) to forward-bias its base-emitter junction.
Class B
Amplifier
Little orno DC bias voltage
Class B: Bias is such that half (180°) of the waveform is reproduced.
By itself, an amplifier operating in class B mode is not very useful. In most
circumstances, the severe distortion introduced into the waveshape by
eliminating half of it would be unacceptable. However, class B operation is a
useful mode of biasing if two amplifiers are operated as a push-pull pair, each
amplifier handling only half of the waveform at a time:
Input components
omitted for simplicity
Class B push pull amplifier: Each transistor reproduces half of the waveform.
Combining the halves produces a faithful reproduction of the whole wave.
Transistor Q; “pushes” (drives the output voltage in a positive direction with
respect to ground), while transistor Q> “pulls” the output voltage (in a negative
direction, toward 0 volts with respect to ground). Individually, each of these
transistors is operating in class B mode, active only for one-half of the input
waveform cycle. Together, however, both function as a team to produce an
output waveform identical in shape to the input waveform.
A decided advantage of the class B (push-pull) amplifier design over the class A
design is greater output power capability. With a class A design, the transistor
dissipates considerable energy in the form of heat because it never stops
conducting current. At all points in the wave cycle it is in the active
(conducting) mode, conducting substantial current and dropping substantial
voltage. There is substantial power dissipated by the transistor throughout the
cycle. In a class B design, each transistor spends half the time in cutoff mode,
where it dissipates zero power (zero current = zero power dissipation). This
gives each transistor a time to “rest” and cool while the other transistor carries
the burden of the load. Class A amplifiers are simpler in design, but tend to be
limited to low-power signal applications for the simple reason of transistor heat
dissipation.
Another class of amplifier operation known as class AB, is somewhere between
class A and class B: the transistor soends more than 50% but less than 100% of
the time conducting current.
If the input signal bias for an amplifier is slightly negative (opposite of the bias
polarity for class A operation), the output waveform will be further “clipped”
than it was with class B biasing, resulting in an operation where the transistor
spends most of the time in cutoff mode:
Class C
Amplifier
Class C: Conduction is for less than a half cycle (< 180°).
At first, this scheme may seem utterly pointless. After all, how useful could an
amplifier be if it clips the waveform as badly as this? If the output is used
directly with no conditioning of any kind, it would indeed be of questionable
utility. However, with the application of a tank circuit (parallel resonant
inductor-capacitor combination) to the output, the occasional output surge
produced by the amplifier can set in motion a higher-frequency oscillation
maintained by the tank circuit. This may be likened to a machine where a heavy
flywheel is given an occasional “kick” to keep it spinning:
LN
Class C
Amplifier
with resonant
Class C amplifier driving a resonant circuit.
Called class C operation, this scheme also enjoys high power efficiency due to
the fact that the transistor(s) spend the vast majority of time in the cutoff
mode, where they dissipate zero power. The rate of output waveform decay
(decreasing oscillation amplitude between “kicks” from the amplifier) is
exaggerated here for the benefit of illustration. Because of the tuned tank
circuit on the output, this circuit is usable only for amplifying signals of definite,
fixed amplitude. A class C amplifier may used in an FM (frequency modulation)
radio transmitter. However, the class C amplifier may not directly amplify an AM
(amplitude modulated) signal due to distortion.
Another kind of amplifier operation, significantly different from Class A, B, AB,
or C, is called Class D. It is not obtained by applying a specific measure of bias
voltage as are the other classes of operation, but requires a radical re-design of
the amplifier circuit itself. It is a little too early in this chapter to investigate
exactly how a class D amplifier is built, but not too early to discuss its basic
principle of operation.
A class D amplifier reproduces the profile of the input voltage waveform by
generating a rapidly-pulsing squarewave output. The duty cycle of this output
waveform (time “on” versus total cycle time) varies with the instantaneous
amplitude of the input signal. The plots in (Figure below demonstrate this
principle.
Input = / \ /
Output
Class D amplifier: Input signal and unfiltered output.
The greater the instantaneous voltage of the input signal, the greater the duty
cycle of the output squarewave pulse. If there can be any goal stated of the
class D design, it is to avoid active-mode transistor operation. Since the output
transistor of a class D amplifier is never in the active mode, only cutoff or
saturated, there will be little heat energy dissipated by it. This results in very
high power efficiency for the amplifier. Of course, the disadvantage of this
strategy is the overwhelming presence of harmonics on the output. Fortunately,
since these harmonic frequencies are typically much greater than the frequency
of the input signal, these can be filtered out by a low-pass filter with relative
ease, resulting in an output more closely resembling the original input signal
waveform. Class D technology is typically seen where extremely high power
levels and relatively low frequencies are encountered, such as in industrial
inverters (devices converting DC into AC power to run motors and other large
devices) and high-performance audio amplifiers.
A term you will likely come across in your studies of electronics is something
called quiescent, which is a modifier designating the zero input condition of a
circuit. Quiescent current, for example, is the amount of current in a circuit with
zero input signal voltage applied. Bias voltage in a transistor circuit forces the
transistor to operate at a different level of collector current with zero input
signal voltage than it would without that bias voltage. Therefore, the amount of
bias in an amplifier circuit determines its quiescent values.
In aclass A amplifier, the quiescent current should be exactly half of its
saturation value (halfway between saturation and cutoff, cutoff by definition
being zero). Class B and class C amplifiers have quiescent current values of
zero, since these are supposed to be cutoff with no signal applied. Class AB
amplifiers have very low quiescent current values, just above cutoff. To
illustrate this graphically, a “load line” is sometimes plotted over a transistor's
characteristic curves to illustrate its range of operation while connected to a
load resistance of specific value shown in Figure below.
saturation
= Thuse = 75 LA
~—"Load line"
Teotlector
Thuse = 40 WA
I/—_Nawse = 20 WA
LE hase = 5 WA
0 E cutoff
r
‘collector-to-emitter V supply
Example load line drawn over transistor characteristic curves from Veuppjyy to
saturation current.
A load line is a plot of collector-to-emitter voltage over a range of collector
currents. At the lower-right corner of the load line, voltage is at maximum and
Current is at zero, representing a condition of cutoff. At the upper-left corner of
the line, voltage is at zero while current is at a maximum, representing a
condition of saturation. Dots marking where the load line intersects the various
transistor curves represent realistic operating conditions for those base currents
given.
Quiescent operating conditions may be shown on this graph in the form of a
single dot along the load line. For a class A amplifier, the quiescent point will be
in the middle of the load line as in (Figure below.
Tyase =75 HA
| Quiescent point
Teotlector | for c ass
| Tyase = 40 HA _ if operation
—~e———
Vv
supply ‘
Esottector-t )-emilter
Quiescent point (dot) for class A.
In this illustration, the quiescent point happens to fall on the curve representing
a base current of 40 WA. If we were to change the load resistance in this circuit
to a greater value, it would affect the slope of the load line, since a greater load
resistance would limit the maximum collector current at saturation, but would
not change the collector-emitter voltage at cutoff. Graphically, the result is a
load line with a different upper-left point and the same lower-right point as in
(Figure below)
Thuse =75 LA
Teotiector
! a = 40 LA
| .
| /
The non- 7 ||/ = 5
horizontal ||/_ Thase = 20 HA
portion of /
e curve _
re resents hase = 5 WA 2
ransistor
saturation =
0 Fcottector-to-emitter supply 4
Load line resulting from increased load resistance.
Note how the new load line doesn't intercept the 75 WA curve along its flat
portion as before. This is very important to realize because the non-horizontal
portion of a characteristic curve represents a condition of saturation. Having the
load line intercept the 75 YA curve outside of the curve's horizontal range
means that the amplifier will be saturated at that amount of base current.
Increasing the load resistor value is what caused the load line to intercept the
75 WA curve at this new point, and it indicates that saturation will occur at a
lesser value of base current than before.
With the old, lower-value load resistor in the circuit, a base current of 75 WA
would yield a proportional collector current (base current multiplied by B). In
the first load line graph, a base current of 75 UWA gave a collector current almost
twice what was obtained at 40 UA, as the B ratio would predict. However,
collector current increases marginally between base currents 75 YA and 40 HA,
because the transistor begins to lose sufficient collector-emitter voltage to
continue to regulate collector current.
To maintain linear (no-distortion) operation, transistor amplifiers shouldn't be
operated at points where the transistor will saturate; that is, where the load line
will not potentially fall on the horizontal portion of a collector current curve.
We'd have to add a few more curves to the graph in Figure below before we
could tell just how far we could “push” this transistor with increased base
currents before it saturates.
base — 75 LA
base — 60 HA
base > 50 HA
base — 40 HA
Diets fr.
Teas = 20 pA
Tase = 5 WA
0 E,
‘collector-to-emitter V cupply 4
More base current curves shows saturation detail.
It appears in this graph that the highest-current point on the load line falling on
the straight portion of a curve is the point on the 50 WA curve. This new point
should be considered the maximum allowable input signal level for class A
operation. Also for class A operation, the bias should be set so that the
quiescent point is halfway between this new maximum point and cutoff shown
in Figure below.
Tyase= 75 PA
/ Thuse = 60 PA
4 I 50
I a the base — - LA
I/ Tse = 40 PA
New quiescent point
0 E,
‘collector-lo-emitter
V
supply 4
New quiescent point avoids saturation region.
Now that we know a little more about the consequences of different DC bias
voltage levels, it is time to investigate practical biasing techniques. So far, I've
shown a small DC voltage source (battery) connected in series with the AC
input signal to bias the amplifier for whatever desired class of operation. In real
life, the connection of a precisely-calibrated battery to the input of an amplifier
is Simply not practical. Even if it were possible to customize a battery to
produce just the right amount of voltage for any given bias requirement, that
battery would not remain at its manufactured voltage indefinitely. Once it
started to discharge and its output voltage drooped, the amplifier would begin
to drift toward class B operation.
Take this circuit, illustrated in the common-emitter section for a SPICE
simulation, for instance, in Figure below.
speaker
Vinput
15 V
2 kHz
5
Voias
2.3 V
Impractical base battery bias.
That 2.3 volt “Vpjs,” battery would not be practical to include in a real amplifier
circuit. A far more practical method of obtaining bias voltage for this amplifier
would be to develop the necessary 2.3 volts using a voltage divider network
connected across the 15 volt battery. After all, the 15 volt battery is already
there by necessity, and voltage divider circuits are easy to design and build.
Let's see how this might look in Figure below.
speaker
2 R, 2
; Q — 15V
, eee “1 kQ
2 kHz «Vis ]
0 0
Voltage divider bias.
If we choose a pair of resistor values for Rp and R3 that will produce 2.3 volts
across R3 from a total of 15 volts (such as 8466 © for Rz and 1533 Q for R3), we
should have our desired value of 2.3 volts between base and emitter for biasing
with no signal input. The only problem is, this circuit configuration places the
AC input signal source directly in parallel with R3 of our voltage divider. This is
not acceptable, as the AC source will tend to overpower any DC voltage
dropped across R3. Parallel components must have the same voltage, so if an
AC voltage source is directly connected across one resistor of a DC voltage
divider, the AC source will “win” and there will be no DC bias voltage added to
the signal.
One way to make this scheme work, although it may not be obvious why it will
work, is to place a coupling capacitor between the AC voltage source and the
voltage divider as in Figure below.
Coupling capacitor prevents voltage divider bias from flowing into signal
generator.
The capacitor forms a high-pass filter between the AC source and the DC
voltage divider, passing almost all of the AC signal voltage on to the transistor
while blocking all DC voltage from being shorted through the AC signal source.
This makes much more sense if you understand the superposition theorem and
how it works. According to superposition, any linear, bilateral circuit can be
analyzed in a piecemeal fashion by only considering one power source at a
time, then algebraically adding the effects of all power sources to find the final
result. If we were to separate the capacitor and R>--R3 voltage divider circuit
from the rest of the amplifier, it might be easier to understand how this
superposition of AC and DC would work.
With only the AC signal source in effect, and a capacitor with an arbitrarily low
impedance at signal frequency, almost all the AC voltage appears across R3:
Due to the coupling capacitor's very low impedance at the signal frequency, it
behaves much like a piece of wire, thus can be omitted for this step in
superposition analysis.
With only the DC source in effect, the capacitor appears to be an open circuit,
and thus neither it nor the shorted AC signal source will have any effect on the
operation of the R>--R3 voltage divider in Figure below.
The capacitor appears to be an open circuit as far at the DC analysis is
concerned
Combining these two separate analyses in Figure below, we get a superposition
of (almost) 1.5 volts AC and 2.3 volts DC, ready to be connected to the base of
the transistor.
Combined AC and DC circuit.
Enough talk -- its about time for a SPICE simulation of the whole amplifier
circuit in Figure below. We will use a capacitor value of 100 UF to obtain an
arbitrarily low (0.796 Q) impedance at 2000 Hz:
voltage divider biasing
vinput 1 0 sin (0 1.5 2000 0 0)
cl 15 100u
rl 5 2 1k
r2 4 5 8466
r3 5 0 1533
ql 3 2 0 modl
rspkr 3 4 8
vl 4 0 de 15
.model modi npn
.tran 0.02m 0.78m
.plot tran v(1,0) i(v1)
.end
v(1} ;
Units v(1) — 10*y1#branch Units
SPICE simulation of voltage divider bias.
Note the substantial distortion in the output waveform in Figure above. The sine
wave is being clipped during most of the input signal's negative half-cycle. This
tells us the transistor is entering into cutoff mode when it shouldn't (I'm
assuming a goal of class A operation as before). Why is this? This new biasing
technique should give us exactly the same amount of DC bias voltage as before,
right?
With the capacitor and R>--R3 resistor network unloaded, it will provide exactly
2.3 volts worth of DC bias. However, once we connect this network to the
transistor, it is no longer unloaded. Current drawn through the base of the
transistor will load the voltage divider, thus reducing the DC bias voltage
available for the transistor. Using the diode current source transistor model in
Figure below to illustrate, the bias problem becomes evident.
speaker
\
y
input
Diode transistor model shows loading of voltage divider.
A voltage divider's output depends not only on the size of its constituent
resistors, but also on how much current is being divided away from it through a
load. The base-emitter PN junction of the transistor is a load that decreases the
DC voltage dropped across R3, due to the fact that the bias current joins with
R3's current to go through R3, upsetting the divider ratio formerly set by the
resistance values of R> and R3. To obtain a DC bias voltage of 2.3 volts, the
values of Ry and/or R3 must be adjusted to compensate for the effect of base
current loading. To increase the DC voltage dropped across R3, lower the value
of Ro, raise the value of R3, or both.
v1) I(v())
Unite v(1) — 10*y1#branch Units
mA
voltage divider biasing
vinput 1 0 sin (0 1.5 2000 0 0)
cl 15 100u
rl 5 2 1k
r2 4 5 6k <--- R2 decreased to 6 k
r3 5 0 4k <--- R3 increased to 4 k
ql 3 2 0 modl
.model mod1 npn
.tran @.02m 0.78m
.plot tran v(1,0) i(v1)
end
No distortion of the output after adjusting R2 and R3.
The new resistor values of 6 kKQ and 4 kQ (R>2 and R3, respectively) in Figure
above results in class A waveform reproduction, just the way we wanted.
e REVIEW:
e Class A operation is an amplifier biased to be in the active mode throughout
the entire waveform cycle, thus faithfully reproducing the whole waveform.
e Class B operation is an amplifier biased so that only half of the input
waveform gets reproduced: either the positive half or the negative half. The
transistor spends half its time in the active mode and half its time cutoff.
Complementary pairs of transistors running in class B operation are often
used to deliver high power amplification in audio signal systems, each
transistor of the pair handling a separate half of the waveform cycle. Class B
operation delivers better power efficiency than a class A amplifier of similar
output power.
¢ Class AB operation is an amplifier is biased at a point somewhere between
class A and class B.
e Class Cis an amplifier biased to amplify only a small portion of the
waveform. Most of the transistor's time is spent in cutoff mode. In order for
there to be a complete waveform at the output, a resonant tank circuit is
often used as a “flywheel” to maintain oscillations for a few cycles after
each “kick” from the amplifier. Because the transistor is not conducting
most of the time, power efficiencies are high for a class C amplifier.
e Class D operation requires an advanced circuit design, and functions on the
principle of representing instantaneous input signal amplitude by the duty
cycle of a high-frequency squarewave. The output transistor(s) never
operate in active mode, only cutoff and saturation. Little heat energy
dissipated makes energy efficiency high.
¢« DC bias voltage on the input signal, necessary for certain classes of
operation (especially class A and class C), may be obtained through the use
of a voltage divider and coupling capacitor rather than a battery connected
in series with the AC signal source.
Biasing calculations
Although transistor switching circuits operate without bias, it is unusual for
analog circuits to operate without bias. One of the few examples is “TR One,
one transistor radio” TR One, Ch 9 with an amplified AM (amplitude modulation)
detector. Note the lack of a bias resistor at the base in that circuit. In this
section we look at a few basic bias circuits which can set a selected emitter
current Ir. Given a desired emitter current I-, what values of bias resistors are
required, Rg, Re, etc?
Base Bias
The simplest biasing applies a base-bias resistor between the base and a base
battery Vpp. It is convenient to use the existing Vcc supply instead of a new bias
supply. An example of an audio amplifier stage using base-biasing is “Crystal
radio with one transistor...” crystal radio, Ch 9 . Note the resistor from the
base to the battery terminal. A similar circuit is shown in Figure below.
Write a KVL (Krichhoff's voltage law) equation about the loop containing the
battery, Rg, and the Vp- diode drop on the transistor in Figure below. Note that
We USE Vpp for the base supply, even though it is actually Vcc. If B is large we
can make the approximation that Ic =l-. For silicon transistors Vp-=0.7 V.
Ves -1,Re, ~ Vee =0
Ves - Vee =1Re (KVL)
a Ves - Vze
BT Rs
1 = (B+1)I; = Bly
Vas - Ver i
= (IE base-bias)
Te R,/B
Base-bias
Silicon small signal transistors typically have a B in the range of 100-300.
Assuming that we have a B=100 transistor, what value of base-bias resistor is
required to yield an emitter current of 1mA?
Solving the IE base-bias equation for Rg and substituting 8, Veg, Veg, and I
yields 930kQ. The closest standard value is 910kQ.
B=100 Vp,=10V Ic~I,=I1ma
_ Vgp- Ver _ 10-0.7
What is the emitter current with a 910kQ resistor? What is the emitter current if
we randomly get a B=300 transistor?
B=100 Vgg=10V Rg=910k Vp =0.7V
r= Spe Ver _ 10-07 _ jgoma
=. Ran - 910k/ 100 Se
B = 300
, = 07 _ 7 3.07mA
7 910k/ 300
The emitter current is little changed in using the standard value 910kQ resistor.
However, with a change in B from 100 to 300, the emitter current has tripled.
This is not acceptable in a power amplifier if we expect the collector voltage to
swing from near Vcc to near ground. However, for low level signals from micro-
volts to a about a volt, the bias point can be centered for a B of square root of
(100-300)=173. The bias point will still drift by a considerable amount .
However, low level signals will not be clipped.
Base-bias by its self is not suitable for high emitter currents, as used in power
amplifiers. The base-biased emitter current is not temperature stable. Thermal!
run away is the result of high emitter current causing a temperature increase
which causes an increase in emitter current, which further increases
temperature.
Collector-feedback bias
Variations in bias due to temperature and beta may be reduced by moving the
Vpp end of the base-bias resistor to the collector as in Figure below. If the
emitter current were to increase, the voltage drop across Rc increases,
decreasing Vc¢, decreasing Ip fed back to the base. This, in turn, decreases the
emitter current, correcting the original increase.
Write a KVL equation about the loop containing the battery, Rc, Rg, and the
Vee drop. Substitute Ic=l_ and Ip=l_/B. Solving for I; yields the IE CFB-bias
equation. Solving for lp yields the IB CFB-bias equation.
I. = Bl, Io = Tp I, ~ Bly
Voc - IcRe - IpRg - Vgx = 0 (KVL)
Voc - TeRe - (Iy/B)Rp - Var=0
Vec- Var = TeRe + (Ie/B)Rp
Vee > Ver = 1:((Rg/B) + Re)
L= Vec - Vex (IE CFB-bias)
e R,/B + Re
R,= B eee -Ro (RB CFB-bias)
E
Collector-feedback bias.
Find the required collector feedback bias resistor for an emitter current of 1 mA,
a 4.7K collector load resistor, and a transistor with B=100 . Find the collector
voltage Vc. It should be approximately midway between Vcc and ground.
B=100 V,-=10V I[.=Iz=1ma R.-=4.7k
| a 7 | 100 | 2-27 ATk = 460k
ia B ImA
I;
Vo = Voc - eRe = 10 - (ImA)-(4.7k) = 5.3V
The closest standard value to the 460k collector feedback bias resistor is 47 Ok.
Find the emitter current I; with the 470 K resistor. Recalculate the emitter
current for a transistor with B=100 and B=300.
B=100 V~.-=10V R-=4.7k R,=470k
_ Voc - Var = 10 - 0.7 _ oy
ls -RB+Re ™ 470k/100+47k = °-759™mA
B = 300
_ Vcc - Ver _ 10 - 0.7 _
In= -ReiB+Ro ~ 470k/30004.7K ~ *8mA
We see that as beta changes from 100 to 300, the emitter current increases
from 0.989mA to 1.48mA. This is an improvement over the previous base-bias
circuit which had an increase from 1.02mA to 3.07mMmA. Collector feedback bias
is twice as stable as base-bias with respect to beta variation.
Emitter-bias
Inserting a resistor Re in the emitter circuit as in Figure below causes
degeneration, also known as negative feedback. This opposes a change in
emitter current I; due to temperature changes, resistor tolerances, beta
variation, or power supply tolerance. Typical tolerances are as follows: resistor—
5%, beta— 100-300, power supply— 5%. Why might the emitter resistor
stabilize a change in current? The polarity of the voltage drop across R-_ is due
to the collector battery Vcc. The end of the resistor closest to the (-) battery
terminal is (-), the end closest to the (+) terminal it (+). Note that the (-) end of
Re is connected via Vgp battery and Rg to the base. Any increase in current flow
through R¢ will increase the magnitude of negative voltage applied to the base
circuit, decreasing the base current, decreasing the emitter current. This
decreasing emitter current partially compensates the original increase.
Ves -IpRp - Vee - 1eRe = 0
I; = (B+) I, = BI,
Vpp -(e/B)Re - Vee- 1eRe = 0
Vos - Ver = Ip((Rp/B) + Re)
Vers - Vee
I= R,/BeRy (IE emitter-bias)
Vepp- V
R,/B + Re = _—BB BE
I,
Ves - Vee
R,= 8 —— = -R, (RB emitter-bias)
E
Emitter-bias
Note that base-bias battery Vpp is used instead of Vcc to bias the base in Figure
above. Later we will show that the emitter-bias is more effective with a lower
base bias battery. Meanwhile, we write the KVL equation for the loop through
the base-emitter circuit, paying attention to the polarities on the components.
We substitute Ip=I_-/B and solve for emitter current Ir. This equation can be
solved for Rg , equation: RB emitter-bias, Figure above.
Before applying the equations: RB emitter-bias and IE emitter-bias, Figure
above, we need to choose values for Rc and R_ . Rc is related to the collector
supply Vcc and the desired collector current Ic which we assume is
approximately the emitter current I-. Normally the bias point for Vc is set to half
of Vcc. Though, it could be set higher to compensate for the voltage drop across
the emitter resistor Re. The collector current is whatever we require or choose. It
could range from micro-Amps to Amps depending on the application and
transistor rating. We choose Ic = 1mA, typical of a small-signal transistor circuit.
We calculate a value for Rc and choose a close standard value. An emitter
resistor which is 10-50% of the collector load resistor usually works well.
Ve= Vec/2 = 10/2 =5V
Re = Ve/I, = 5/ImA = 5k (4.7k standard value)
Ry = 0.10R¢ = 0.10(4.7K) = 470Q
Our first example sets the base-bias supply to high at Veg = Vcc = 10V to show
why a lower voltage is desirable. Determine the required value of base-bias
resistor Rg. Choose a standard value resistor. Calculate the emitter current for
B=100 and B=300. Compare the stabilization of the current to prior bias
circuits.
B=100 I,=Ic=lma Vec=V,—=10V Ry=470Q
10-0.7
-V
Ra = BB VBE _ = 100 | ————- - 470 = 883k
Fr “eee Re 0.001
An 883k resistor was calculated for Rg, an 870k chosen. At B=100, I; is 1.01mA.
B=100 Rp =870k
Mise Ver 10 -0.7
. R,/B + Ry 870K/100 + 470
B=300
x. “Wane Vee 10 -0.7 ees
I, = eh
f Rp/B+Rp 870K/300 + 470
For B=300 the emitter currents are shown in Table below.
Emitter current comparison for B=100, B=300.
Bias circuit IC B=100)IC B=300
base-bias 1.02mA_|[3.07mA
collector feedback biasj0O.989MA |/1.48mA
emitter-bias, Vpg=1lOV |1.01mA 2.76mA
Table above shows that for Vag = 10V, emitter-bias does not do a very good job
of stabilizing the emitter current. The emitter-bias example is better than the
previous base-bias example, but, not by much. The key to effective emitter bias
is lowering the base supply Vgp nearer to the amount of emitter bias.
How much emitter bias do we Have? Rounding, that is emitter current times
emitter resistor: I-Re = (1mA)(470) = 0.47V. In addition, we need to overcome
the Vp = 0.7V. Thus, we need a Vag >(0.47 + 0.7)V or >1.17V. If emitter
current deviates, this number will change compared with the fixed base supply
Vpp,Causing a correction to base current Ip and emitter current I-. A good value
for Vg >1.17V is 2V.
B=100 I.=Ic=Ima Vec=10V. Vpg=2V Re = 4702
- . 9 =
R,= B| ever -p, =100| 2297 479) | = 83k
Le 0.001
The calculated base resistor of 83k is much lower than the previous 883k. We
choose 82k from the list of standard values. The emitter currents with the 82k
Rg for B=100 and B=300 are:
B=100 R, = 82k
I, = Vee ~ Ver = 2-07 = 1.0lmA
= R,/B + Ry 82K/100 + 470
B=300
=< 9a
= Nes~ Vee oO eee ak
io =
Z Ry/B + Ry 82K/300 + 470
Comparing the emitter currents for emitter-bias with Vpp = 2V at B=100 and
8=300 to the previous bias circuit examples in Table below, we see
considerable improvement at 1.75mA, though, not as good as the 1.48mA of
collector feedback.
Emitter current comparison for B=100, B=300.
Bias circuit IC B=100)\IC B=300
base-bias 1.02mA_ |3.07mA
collector feedback biasj0O.989MA ||1.48mA
emitter-bias, Vgg=1lOV |1.01mA_ |2.76mA
emitter-bias, Vgg=2V |1.01mA_ |/1.75mA
How can we improve the performance of emitter-bias? Either increase the
emitter resistor Re or decrease the base-bias supply Vpp or both. As an example,
we double the emitter resistor to the nearest standard value of 910Q.
B=100 I.~Ic=Ima Vec=10V Vpp=2V. Ry =910Q
y= Be oe, | =100} 2297 _o19 | =39k
I. 0.001
The calculated Rg = 39k is a standard value resistor. No need to recalculate I
for B = 100. For B = 300, it is:
B=300 Rx = 39k
ioe SO gg iam
: Rp/B + Ry 39K/300 + 910
The performance of the emitter-bias circuit with a 910 emitter resistor is much
improved. See Table below.
Emitter current comparison for B=100, B=300.
Bias circuit
base-bias
IC B=100|IC B=300
1.02mA_ |3.07mMA
0.989mA |1.48mA
collector feedback bias 0.989mA |1.48mA
emitter-bias, Va3=10V 1.01mA 2 76mA
emitter-bias, Vap=2V, RE=470//1.01mA [1.7 5mA
emitter-bias, Vpgp=2V, RE=910/1.00mA
1.25mA
As an exercise, rework the emitter-bias example with the emitter resistor
reverted back to 470Q, and the base-bias supply reduced to 1.5V.
B=100 I.=Ic=Ima Vec=10V Vpp=1.5V Ry=470Q
Ry | = 100 | 1:82.07
0.001
The 33k base resistor is a standard value, emitter current at B = 100 is OK. The
emitter current at B = 300 is:
Ves - Ver
R,= 6 -470 | = 33k
Ty
oe eee ge fain
R,/B + Ry 33K/300 + 470
Table below below compares the exercise results 1mA and 1.38mA to the
previous examples.
Emitter current comparison for B=100, B=300.
Bias circuit IC B=100)IC B=300
base-bias [1.02mA_ |3.07mA
collector feedback bias 0.989mA |1.48mA
emitter-bias, Van3=10V 1.01mA 2 76mA
emitter-bias, Vag=2V, Rg=470 |1.01mA [1.75mA
emitter-bias, Vgg=2V, Rg=910 |1.00mA [1.25mA
emitter-bias, Vgg=1.5V, Rg=470/1.00mA_ |1.38mA
The emitter-bias equations have been repeated in Figure below with the
internal emitter resistance included for better accuracy. The internal emitter
resistance is the resistance in the emitter circuit contained within the transistor
package. This internal resistance r¢r is significant when the (external) emitter
resistor Re is small, or even zero. The value of internal resistance Reg iS a
function of emitter current I-, Table below.
Derivation of rer
Vee = KT/Iem
where:
K=1.38x10°23 watt-sec/°C, Boltzman's constant
T= temperature in Kelvins =300.
Ir = emitter current
m = varies from 1 to 2 for Silicon
rep = 0.026V/I-_ = 26mV/I_
For reference the 26mV approximation is listed as equation rEE in Figure below.
Vep —pRep - Vee- lefee - IERE= 0 (KVL)
I; = (B+) I, = BI,
Vpp-e/ B)Rg- Vee-leter-1eRp=0
Vpp- Vpe=(p(Rp/B) + Teter + 1eRe)
I= _Vue- Vaz (IE EB)
. R,/B + rert Re
Vpp- V
R,/B + Ree + Re= — HE BE
Ip
Van - VBE
Ry = B) PEPE - ry -Ry (RB EB)
I;
Emitter-bias equations with internal emitter resistance rrr included...
The more accurate emitter-bias equations in Figure above may be derived by
writing a KVL equation. Alternatively, start with equations IE emitter-bias and
Rg emitter-bias in Figure previous, substituting Re with ree+Re_. The result is
equations IE EB and RB EB, respectively in Figure above.
Redo the Rg, calculation in the previous example emitter-bias with the inclusion
of ree and compare the results.
B=100 Ip=IQ=lma Vcc=10V Vpp=2V -Re=4702
tgp = 26mV/ImA = 260
2.0-0.7
-tep-Rp | = 100] ————— - 26-470 __| = 80.4k
0.001
Vec-V
Rz = B oe BE
Tp
The inclusion of reg in the calculation results in a lower value of the base resistor
Rg a Shown in Table below. It falls below the standard value 82k resistor instead
of above it.
Effect of inclusion of ree on calculated Rg
reg? ree Value
Without reei83k
With ree |80.4k
Bypass Capacitor for R_
One problem with emitter bias is that a considerable part of the output signal is
dropped across the emitter resistor Re (Figure below). This voltage drop across
the emitter resistor is in series with the base and of opposite polarity compared
with the input signal. (This is similar to a common collector configuration
having <1 gain.) This degeneration severely reduces the gain from base to
collector. The solution for AC signal amplifiers is to bypass the emitter resistor
with a capacitor. This restores the AC gain since the capacitor is a short for AC
signals. The DC emitter current still experiences degeneration in the emitter
resistor, thus, stabilizing the DC current.
Coupling Coupling
Vec +
_ Cc
a R, a -
V,
Cbypass Is required to prevent AC gain reduction.
in
i
What value should the bypass capacitor be? That depends on the lowest
frequency to be amplified. For radio frequencies Cbpass would be small. For an
audio amplifier extending down to 20Hz it will be large. A “rule of thumb” for
the bypass capacitor is that the reactance should be 1/10 of the emitter
resistance or less. The capacitor should be designed to accommodate the lowest
frequency being amplified. The capacitor for an audio amplifier covering 20HZ
to 20kHz would be:
l
Xo= sre
1
C=
2TfX¢
c= —! ___-169uF
2720(470/10)
Note that the internal emitter resistance ree is not bypassed by the bypass
Capacitor.
Voltage divider bias
Stable emitter bias requires a low voltage base bias supply, Figure below. The
alternative to a base supply Vpp Is a voltage divider based on the collector
supply Vcc.
Emitter-bias Voltage divider bias
Voltage Divider bias replaces base battery with voltage divider.
The design technique is to first work out an emitter-bias design, Then convert it
to the voltage divider bias configuration by using Thevenin's Theorem. [TK1]
The steps are shown graphically in Figure below. Draw the voltage divider
without assigning values. Break the divider loose from the base. (The base of
the transistor is the load.) Apply Thevenin's Theorem to yield a single Thevenin
equivalent resistance Rth and voltage source Vth.
Rth
Thevenin's Theorem converts voltage divider to single supply Vth and
resistance Rth.
The Thevenin equivalent resistance is the resistance from load point (arrow)
with the battery (Vcc) reduced to 0 (ground). In other words, R1||R2.The
Thevenin equivalent voltage is the open circuit voltage (load removed). This
calculation is by the voltage divider ratio method. R1 is obtained by eliminating
R2 from the pair of equations for Rth and Vth. The equation of R1 is in terms of
known quantities Rth, Vth, Vcc. Note that Rth is Rg, the bias resistor from the
emitter-bias design. The equation for R2 is in terms of R1 and Rth.
= R2
Rth = R1 II R2 Vth = Ve
R1 +R2
I
—-—_—_— +e
Rth RI R2 pa Nth _ R2
Voc R1 +R2
J R24R1) © so 1[R24R1]_ 1
Rth RI-R2 ~ RI[R2 |] RI
Rth Vee I I
Rl= *— =Rth —_— = —- —
f Vth R2 Rth_ RI
Emitter-bias example converted to voltage divider bias.
These values were previously selected or calculated for an emitter-bias example
B=100 1:=Ic=Ima Vec=10V_ Vgg=1-5V Ryp=4702
Vin - V =
R,= B| Be-‘se -R, = 100 | 1: OF gy | ease
ik 0.001
Substituting Vcc, Veg, Rg yields Rl and R2 for the voltage divider bias
configuration.
Vep= Vth=1.5V | 1 1
R,, = Rth = 33k R2 Rth RI
Vec 1 1 l
RI = Rth oe == asp
RI = 33k if = 220k R2 = 38.8k
salt DE,
R1 is a standard value of 220K. The closest standard value for R2 corresponding
to 38.8k is 39k. This does not change I|_ enough for us to calculate it.
Problem: Calculate the bias resistors for the cascode amplifier in Figure below.
Vp is the bias voltage for the common emitter stage. Vp, is a fairly high voltage
at 11.5 because we want the common-base stage to hold the emitter at 11.5-
0.7=10.8V, about 11V. (It will be LOV after accounting for the voltage drop
across Rg, .) That is, the common-base stage is the load, substitute for a
resistor, for the common-emitter stage's collector. We desire a 1mA emitter
current.
Voc=20V. Ip=ImA B=100 V,y=10V— R,=4.7k
Vapi =11.5V Vago =1.5V
Ves 5 Var (IE base-bias
_ -bias)
lg Rz/p
R — Ves - Ver _ (Vppi- Va) - Ver _ (11.5-10) - 0.7 = 80k
ce I./p 1/8 ~ |mA/100——
Var - V (1.5) - 0.7
R,, = —BB2 BE _ = x0)
Be 1,./B ImA/100 :
Cascode
Bias for a cascode amplifier.
Problem: Convert the base bias resistors for the cascode amplifier to voltage
divider bias resistors driven by the V¢c of 20V.
Ryy = 80k Vec= Vth = 20V
Von = 11.5V
Van = Vth = 11.5V
R,, = Rth = 80k
Voc
Rl = Rth Vib
R1 = 80k 2% = 139.1k
7 Ns
] l
Rth
I
R2 RI
Itoi 1
R2 80k 139.1k
R2 = 210k
Ryp2 = 80k
Voan2 = 1.5V
Vay = Vth = 1.5V
R,, = Rth = 80k
Voc
R3 = Rth Vih
20
R3 = 80k is = 1.067Meg
l 1 1
R4.—s Rth”~_ R3
tot tL
R4.——s- 80k_—swi1067k
R4 = 86.5k
The final circuit diagram is shown in the “Practical Analog Circuits” chapter,
“Class A cascode amplifier. .
e REVIEW:
e See Figure below.
.” cascode, Ch9.
e Select bias circuit configuration
¢ Select Rc and I; for the intended application. The values for Rc and I
should normally set collector voltage Vc to 1/2 of Vee.
¢ Calculate base resistor Rp to achieve desired emitter current.
¢ Recalculate emitter current I; for standard value resistors if necessary.
¢ For voltage divider bias, perform emitter-bias calculations first, then
determine R1 and R2.
¢ For AC amplifiers, a bypass capacitor in parallel with Re improves AC gain.
Set X-S0.10R; for lowest frequency.
V~.< ¥; Ven- V
Va - V Ip= —SC_—BE = Vp = Vil
b= Op E~ Ry/Pt Re E~ R,/Pt+Rp alias
Voc - Vee Re = Re + lee R= Rth
Rg = A Se Me Ry= B LE “Re to include ree 7 Vv
E cc
Tpp = 26mv/1, Ri= Rth Gr
Vern - Vee 1 1 1
R= —BB__BE _R a
. 6| ia - R2 Rth_ RI
Base-bias Collector feedback bias Emitter-bias Voltage divider bias
Biasing equations summary.
Input and output coupling
To overcome the challenge of creating necessary DC bias voltage for an
amplifier's input signal without resorting to the insertion of a battery in series
with the AC signal source, we used a voltage divider connected across the DC
power source. To make this work in conjunction with an AC input signal, we
“coupled” the signal source to the divider through a capacitor, which acted asa
high-pass filter. With that filtering in place, the low impedance of the AC signal
source couldn't “short out” the DC voltage dropped across the bottom resistor
of the voltage divider. A simple solution, but not without any disadvantages.
Most obvious is the fact that using a high-pass filter capacitor to couple the
signal source to the amplifier means that the amplifier can only amplify AC
signals. A steady, DC voltage applied to the input would be blocked by the
coupling capacitor just as much as the voltage divider bias voltage is blocked
from the input source. Furthermore, since capacitive reactance is frequency-
dependent, lower-frequency AC signals will not be amplified as much as higher-
frequency signals. Non-sinusoidal signals will tend to be distorted, as the
Capacitor responds differently to each of the signal's constituent harmonics. An
extreme example of this would be a low-frequency square-wave signal in Figure
below.
V input
Capacitively coupled low frequency square-wave shows distortion.
Incidentally, this same problem occurs when oscilloscope inputs are set to the
“AC coupling” mode as in Figure below. In this mode, a coupling capacitor is
inserted in series with the measured voltage signal to eliminate any vertical
offset of the displayed waveform due to DC voltage combined with the signal.
This works fine when the AC component of the measured signal is of a fairly
high frequency, and the capacitor offers little impedance to the signal.
However, if the signal is of a low frequency, or contains considerable levels of
harmonics over a wide frequency range, the oscilloscope's display of the
waveform will not be accurate. (Figure below) Low frequency signals may be
viewed by setting the oscilloscope to “DC coupling” in Figure below.
FUNCTION GENERATOR
40.00 Hz OOmgoaodoadaq
1 10 100 1k 10k 100k 1M
® @® soo og
SS 00 c0@® ©
DC output
OSCILLOSCOPE
vertical
— DC _GND AC
Vidiv ao
trigger ©
ecjfjNj373Omnnmnmr
timebase
X
DC_GND AC
Cc
With DC coupling, the oscilloscope properly indicates the shape of the square
wave coming from the signal generator.
FUNCTION GENERATOR
40.00#) DOmoooa
1 10 100 1k 10k100k 1M
@©@ @ soo co
fu
coarse fine N % pc output
OSCILLOSCOPE
vertical
rl DC_GND AC
Vidiv —o
trigger ©
(el
timebase
X
1.)
DC_GND AC
Cc
sidiv
Low frequency: With AC coupling, the high-pass filtering of the coupling
capacitor distorts the square wave's shape so that what is seen is not an
accurate representation of the real signal.
In applications where the limitations of capacitive coupling (Figure above)
would be intolerable, another solution may be used: direct coupling. Direct
coupling avoids the use of capacitors or any other frequency-dependent
coupling component in favor of resistors. A direct-coupled amplifier circuit is
shown in Figure below.
Vv
input
Direct coupled amplifier: direct coupling to speaker.
With no capacitor to filter the input signal, this form of coupling exhibits no
frequency dependence. DC and AC signals alike will be amplified by the
transistor with the same gain (the transistor itself may tend to amplify some
frequencies better than others, but that is another subject entirely!).
If direct coupling works for DC as well as for AC signals, then why use capacitive
coupling for any application? One reason might be to avoid any unwanted DC
bias voltage naturally present in the signal to be amplified. Some AC signals
may be superimposed on an uncontrolled DC voltage right from the source, and
an uncontrolled DC voltage would make reliable transistor biasing impossible.
The high-pass filtering offered by a coupling capacitor would work well here to
avoid biasing problems.
Another reason to use capacitive coupling rather than direct is its relative lack
of signal attenuation. Direct coupling through a resistor has the disadvantage of
diminishing, or attenuating, the input signal so that only a fraction of it reaches
the base of the transistor. In many applications, some attenuation is necessary
anyway to prevent signal levels from “overdriving” the transistor into cutoff and
saturation, so any attenuation inherent to the coupling network is useful
anyway. However, some applications require that there be no signal loss from
the input connection to the transistor's base for maximum voltage gain, anda
direct coupling scheme with a voltage divider for bias simply won't suffice.
So far, we've discussed a couple of methods for coupling an /nput signal to an
amplifier, but haven't addressed the issue of coupling an amplifier's output to a
load. The example circuit used to illustrate input coupling will serve well to
illustrate the issues involved with output coupling.
In our example circuit, the load is a speaker. Most speakers are electromagnetic
in design: that is, they use the force generated by an lightweight electromagnet
coil suspended within a strong permanent-magnet field to move a thin paper or
plastic cone, producing vibrations in the air which our ears interpret as sound.
An applied voltage of one polarity moves the cone outward, while a voltage of
the opposite polarity will move the cone inward. To exploit cone's full freedom
of motion, the speaker must receive true (unbiased) AC voltage. DC bias applied
to the speaker coil offsets the cone from its natural center position, and this
limits the back-and-forth motion it can sustain from the applied AC voltage
without overtraveling. However, our example circuit (Figure above) applies a
varying voltage of only one polarity across the speaker, because the speaker is
connected in series with the transistor which can only conduct current one way.
This would be unacceptable for any high-power audio amplifier.
Somehow we need to isolate the speaker from the DC bias of the collector
current so that it only receives AC voltage. One way to achieve this goal is to
couple the transistor collector circuit to the speaker through a transformer in
Figure below)
Transformer coupling isolates DC from the load (speaker).
Voltage induced in the secondary (speaker-side) of the transformer will be
strictly due to variations in collector current, because the mutual inductance of
a transformer only works on Changes in winding current. In other words, only
the AC portion of the collector current signal will be coupled to the secondary
side for powering the speaker. The speaker will “see” true alternating current at
its terminals, without any DC bias.
Transformer output coupling works, and has the added benefit of being able to
provide impedance matching between the transistor circuit and the speaker coil
with custom winding ratios. However, transformers tend to be large and heavy,
especially for high-power applications. Also, it is difficult to engineer a
transformer to handle signals over a wide range of frequencies, which is almost
always required for audio applications. To make matters worse, DC current
through the primary winding adds to the magnetization of the core in one
polarity only, which tends to make the transformer core saturate more easily in
one AC polarity cycle than the other. This problem is reminiscent of having the
speaker directly connected in series with the transistor: a DC bias current tends
to limit how much output signal amplitude the system can handle without
distortion. Generally, though, a transformer can be designed to handle a lot
more DC bias current than a speaker without running into trouble, so
transformer coupling is still a viable solution in most cases. See the coupling
transformer between Q4 and the speaker, Regency TR1,Ch 9 as an example of
transformer coupling.
Another method to isolate the speaker from DC bias in the output signal is to
alter the circuit a bit and use a coupling capacitor in a manner similar to
coupling the input signal (Figure below) to the amplifier.
mq te
Capacitor coupling isolates DC from the load.
This circuit in Figure above resembles the more conventional form of common-
emitter amplifier, with the transistor collector connected to the battery through
a resistor. The capacitor acts as a high-pass filter, passing most of the AC
voltage to the speaker while blocking all DC voltage. Again, the value of this
coupling capacitor is chosen so that its impedance at the expected signal
frequency will be arbitrarily low.
The blocking of DC voltage from an amplifier's output, be it via a transformer or
a capacitor, is useful not only in coupling an amplifier to a load, but also in
coupling one amplifier to another amplifier. “Staged” amplifiers are often used
to achieve higher power gains than what would be possible using a single
transistor as in Figure below.
\
Vv
output
r
V input
Firststage Secondstage Third stage
Capacitor coupled three stage common-emitter amplifier.
While it is possible to directly couple each stage to the next (via a resistor
rather than a capacitor), this makes the whole amplifier very sensitive to
variations in the DC bias voltage of the first stage, since that DC voltage will be
amplified along with the AC signal until the last stage. In other words, the
biasing of the first stage will affect the biasing of the second stage, and so on.
However, if the stages are capacitively coupled shown in the above illustration,
the biasing of one stage has no effect on the biasing of the next, because DC
voltage is blocked from passing on to the next stage.
Transformer coupling between amplifier stages is also a possibility, but less
often seen due to some of the problems inherent to transformers mentioned
previously. One notable exception to this rule is in radio-frequency amplifiers
(Figure below) with small coupling transformers, having air cores (making them
immune to saturation effects), that are part of a resonant circuit to block
unwanted harmonic frequencies from passing on to subsequent stages. The use
of resonant circuits assumes that the signal frequency remains constant, which
is typical of radio circuitry. Also, the “flywheel” effect of LC tank circuits allows
for class C operation for high efficiency.
\ =
y
V output
\
y
input
First stage Second stage Third stage
Three stage tuned RF amplifier illustrates transformer coupling.
Note the transformer coupling between transistors Q1, Q2, Q3, and Q4,
Regency TR1, Ch 9 . The three intermediate frequency (IF) transformers within
the dashed boxes couple the IF signal from collector to base of following
transistor IF amplifiers. The intermediate freqency ampliers are RF amplifiers,
though, at a different frequency than the antenna RPF input.
Having said all this, it must be mentioned that it /s possible to use direct
coupling within a multi-stage transistor amplifier circuit. In cases where the
amplifier is expected to handle DC signals, this is the only alternative.
The trend of electronics to more widespread use of integrated circuits has
encouraged the use of direct coupling over transformer or capacitor coupling.
The only easily manufactured integrated circuit component is the transistor.
Moderate quality resistors can also be produced. Though, transistors are
favored. Integrated capacitors to only a few 10's of pF are possible. Large
Capacitors are not integrable. If necessary, these can be external components.
The same is true of transformers. Since integrated transistors are inexpensive,
as many transistors as possible are substituted for the offending capacitors and
transformers. As much direct coupled gain as possible is designed into ICs
between the external coupling components. While external capacitors and
transformers are used, these are even being designed out if possible. The result
is that a modern IC radio (See “IC radio”, Ch 9 _) looks nothing like the original 4-
transistor radio Regency TR1, Ch 9.
Even discrete transistors are inexpensive compared with transformers. Bulky
audio transformers can be replaced by transistors. For example, a common-
collector (emitter follower) configuration can impedance match a low output
impedance like a speaker. It is also possible to replace large coupling capacitors
with transistor circuitry.
We still like to illustrate texts with transformer coupled audio amplifiers. The
circuits are simple. The component count is low. And, these are good
introductory circuits— easy to understand.
The circuit in Figure below (a) is a simplified transformer coupled push-pull
audio amplifier. In push-pull, pair of transistors alternately amplify the positive
and negative portions of the input signal. Neither transistor nor the other
conducts for no signal input. A positive input signal will be positive at the top of
the transformer secondary causing the top transistor to conduct. A negative
input will yield a positive signal at the bottom of the secondary, driving the
bottom transistor into conduction. Thus the transistors amplify alternate halves
of a signal. As drawn, neither transistor in Figure below (a) will conduct for an
input below 0.7 Vpeak. A practical circuit connects the secondary center tap to
a 0.7 V (or greater) resistor divider instead of ground to bias both transistor for
true class B.
(a) Transformer coupled push-pull amplifier. (b) Direct coupled complementary-
pair amplifier replaces transformers with transistors.
The circuit in Figure above (b) is the modern version which replaces the
transformer functions with transistors. Transistors Q,; and Q, are common
emitter amplifiers, inverting the signal with gain from base to collector.
Transistors Q3 and Qy are known as a complementary pair because these NPN
and PNP transistors amplify alternate halves (positive and negative,
respectively) of the waveform. The parallel connection the bases allows phase
splitting without an input transformer at (a). The speaker is the emitter load for
Q3 and Q,. Parallel connection of the emitters of the NPN and PNP transistors
eliminates the center-tapped output transformer at (a) The low output
impedance of the emitter follower serves to match the low 8 Q impedance of
the speaker to the preceding common emitter stage. Thus, inexpensive
transistors replace transformers. For the complete circuit see“ Direct coupled
complementary symmetry 3 w audio amplifier,”Ch 9
e REVIEW:
e Capacitive coupling acts like a high-pass filter on the input of an amplifier.
This tends to make the amplifier's voltage gain decrease at lower signal
frequencies. Capacitive-coupled amplifiers are all but unresponsive to DC
input signals.
e Direct coupling with a series resistor instead of a series capacitor avoids the
problem of frequency-dependent gain, but has the disadvantage of
reducing amplifier gain for all signal frequencies by attenuating the input
signal.
e Transformers and capacitors may be used to couple the output of an
amplifier to a load, to eliminate DC voltage from getting to the load.
e Multi-stage amplifiers often make use of capacitive coupling between
stages to eliminate problems with the bias from one stage affecting the bias
of another.
Feedback
If some percentage of an amplifier's output signal is connected to the input, so
that the amplifier amplifies part of its own output signal, we have what is
known as feedback. Feedback comes in two varieties: positive (also called
regenerative), and negative (also called degenerative). Positive feedback
reinforces the direction of an amplifier's output voltage change, while negative
feedback does just the opposite.
A familiar example of feedback happens in public-address (“PA”) systems where
someone holds the microphone too close to a speaker: a high-pitched “whine”
or “howl” ensues, because the audio amplifier system is detecting and
amplifying its own noise. Specifically, this is an example of positive or
regenerative feedback, as any sound detected by the microphone is amplified
and turned into a louder sound by the speaker, which is then detected by the
microphone again, and soon...the result being a noise of steadily increasing
volume until the system becomes “saturated” and cannot produce any more
volume.
One might wonder what possible benefit feedback is to an amplifier circuit,
given such an annoying example as PA system “howl.” If we introduce positive,
or regenerative, feedback into an amplifier circuit, it has the tendency of
creating and sustaining oscillations, the frequency of which determined by the
values of components handling the feedback signal from output to input. This is
one way to make an oscillator circuit to produce AC from a DC power supply.
Oscillators are very useful circuits, and so feedback has a definite, practical
application for us. See “Phase shift oscillator” , Ch 9 for a practical application
of positive feedback.
Negative feedback, on the other hand, has a “dampening” effect on an
amplifier: if the output signal happens to increase in magnitude, the feedback
signal introduces a decreasing influence into the input of the amplifier, thus
opposing the change in output signal. While positive feedback drives an
amplifier circuit toward a point of instability (oscillations), negative feedback
drives it the opposite direction: toward a point of stability.
An amplifier circuit equipped with some amount of negative feedback is not
only more stable, but it distorts the input waveform less and is generally
capable of amplifying a wider range of frequencies. The tradeoff for these
advantages (there just has to be a disadvantage to negative feedback, right?)
is decreased gain. If a portion of an amplifier's output signal is “fed back” to the
input to oppose any changes in the output, it will require a greater input signal
amplitude to drive the amplifier's output to the same amplitude as before. This
constitutes a decreased gain. However, the advantages of stability, lower
distortion, and greater bandwidth are worth the tradeoff in reduced gain for
many applications.
Let's examine a simple amplifier circuit and see how we might introduce
negative feedback into it, starting with Figure below.
Vv
input
Common-emitter amplifier without feedback.
The amplifier configuration shown here is a common-emitter, with a resistor
bias network formed by R, and R92. The capacitor couples Vi,py; to the amplifier
so that the signal source doesn't have a DC voltage imposed on it by the R,/R>
divider network. Resistor R3 serves the purpose of controlling voltage gain. We
could omit it for maximum voltage gain, but since base resistors like this are
common in common-emitter amplifier circuits, we'll keep it in this schematic.
Like all common-emitter amplifiers, this one inverts the input signal as it is
amplified. In other words, a positive-going input voltage causes the output
voltage to decrease, or move toward negative, and vice versa. The oscilloscope
waveforms are shown in Figure below.
Common-emitter amplifier, no feedback, with reference waveforms for
comparison.
Because the output is an inverted, or mirror-image, reproduction of the input
signal, any connection between the output (collector) wire and the input (base)
wire of the transistor in Figure below will result in negative feedback.
Negative feedback, collector feedback, decreases the output signal.
The resistances of Ry, Ro, R3, and Rreegback function together as a signal-mixing
network so that the voltage seen at the base of the transistor (with respect to
ground) is a weighted average of the input voltage and the feedback voltage,
resulting in signal of reduced amplitude going into the transistor. So, the
amplifier circuit in Figure above will have reduced voltage gain, but improved
linearity (reduced distortion) and increased bandwidth.
A resistor connecting collector to base is not the only way to introduce negative
feedback into this amplifier circuit, though. Another method, although more
difficult to understand at first, involves the placement of a resistor between the
transistor's emitter terminal and circuit ground in Figure below.
Emitter feedback: A different method of introducing negative feedback into a
circuit.
This new feedback resistor drops voltage proportional to the emitter current
through the transistor, and it does so in such a way as to oppose the input
signal's influence on the base-emitter junction of the transistor. Let's take a
closer look at the emitter-base junction and see what difference this new
resistor makes in Figure below.
With no feedback resistor connecting the emitter to ground in Figure below (a) ,
whatever level of input signal (Vinout) Makes it through the coupling capacitor
and R,/R>/R3 resistor network will be impressed directly across the base-emitter
junction as the transistor's input voltage (Vp). In other words, with no
feedback resistor, Vg.¢ equals Vinout- Therefore, if Vinout increases by 100 mV,
then Vp.- increases by 100 mV: a change in one is the same as a change in the
other, since the two voltages are equal to each other.
Now let's consider the effects of inserting a resistor (Rreegpack) between the
transistor's emitter lead and ground in Figure below (b).
7 Lectlector | Teottsctor
| re
+
(a) No feedback vs (b) emitter feedback. A waveform at the collector is inverted
with respect to the base. At (b) the emitter waveform is in-phase (emitter
Lemitter
follower) with base, out of phase with collector. Therefore, the emitter signal
subtracts from the collector output signal.
Note how the voltage dropped across Ryeedback Adds with Vg.e to equal Vinput-
With Rreedback iN the Vinput -- Ve-e loop, Vg-g will no longer be equal to Vinour. We
know that Reeegback Will drop a voltage proportional to emitter current, which is
in turn controlled by the base current, which is in turn controlled by the voltage
dropped across the base-emitter junction of the transistor (Vp.-). Thus, if Vinput
were to increase in a positive direction, it would increase Vp.¢, causing more
base current, causing more collector (load) current, causing more emitter
current, and causing more feedback voltage to be dropped across Rfeegback: This
increase of voltage drop across the feedback resistor, though, subtracts from
Vinput to reduce the Vgc, so that the actual voltage increase for Vg_¢ will be less
than the voltage increase of Vinput- No longer will a 100 mV increase in Vinpout
result in a full 100 mV increase for Vg_¢, because the two voltages are not equal
to each other.
Consequently, the input voltage has less control over the transistor than before,
and the voltage gain for the amplifier is reduced: just what we expected from
negative feedback.
In practical common-emitter circuits, negative feedback isn't just a luxury; its a
necessity for stable operation. In a perfect world, we could build and operate a
common-emitter transistor amplifier with no negative feedback, and have the
full amplitude of Vinout impressed across the transistor's base-emitter junction.
This would give us a large voltage gain. Unfortunately, though, the relationship
between base-emitter voltage and base-emitter current changes with
temperature, as predicted by the “diode equation.” As the transistor heats up,
there will be less of a forward voltage drop across the base-emitter junction for
any given current. This causes a problem for us, as the R;/R> voltage divider
network is designed to provide the correct quiescent current through the base
of the transistor so that it will operate in whatever class of operation we desire
(in this example, I've shown the amplifier working in class-A mode). If the
transistor's voltage/current relationship changes with temperature, the amount
of DC bias voltage necessary for the desired class of operation will change. A
hot transistor will draw more bias current for the same amount of bias voltage,
making it heat up even more, drawing even more bias current. The result, if
unchecked, is called thermal runaway.
Common-collector amplifiers, (Figure below) however, do not suffer from
thermal runaway. Why is this? The answer has everything to do with negative
feedback.
Common collector (emitter follower) amplifier.
Note that the common-collector amplifier (Figure above) has its load resistor
placed in exactly the same spot as we had the Ryeegback resistor in the last
circuit in Figure above (b): between emitter and ground. This means that the
only voltage impressed across the transistor's base-emitter junction is the
difference between Vinout ANd Voutput resulting in a very low voltage gain
(usually close to 1 for a common-collector amplifier). Thermal runaway is
impossible for this amplifier: if base current happens to increase due to
transistor heating, emitter current will likewise increase, dropping more voltage
across the load, which in turn subtracts from Vinput to reduce the amount of
voltage dropped between base and emitter. In other words, the negative
feedback afforded by placement of the load resistor makes the problem of
thermal runaway se/f-correcting. In exchange for a greatly reduced voltage
gain, we get superb stability and immunity from thermal runaway.
By adding a “feedback” resistor between emitter and ground in a common-
emitter amplifier, we make the amplifier behave a little less like an “ideal”
common-emitter and a little more like a common-collector. The feedback
resistor value is typically quite a bit less than the load, minimizing the amount
of negative feedback and keeping the voltage gain fairly high.
Another benefit of negative feedback, seen clearly in the common-collector
circuit, is that it tends to make the voltage gain of the amplifier less dependent
on the characteristics of the transistor. Note that in a common-collector
amplifier, voltage gain is nearly equal to unity (1), regardless of the transistor's
8B. This means, among other things, that we could replace the transistor ina
common-collector amplifier with one having a different B and not see any
significant changes in voltage gain. In a common-emitter circuit, the voltage
gain is highly dependent on 8B. If we were to replace the transistor in a common-
emitter circuit with another of differing B, the voltage gain for the amplifier
would change significantly. In a common-emitter amplifier equipped with
negative feedback, the voltage gain will still be dependent upon transistor B to
some degree, but not as much as before, making the circuit more predictable
despite variations in transistor B.
The fact that we have to introduce negative feedback into a common-emitter
amplifier to avoid thermal runaway is an unsatisfying solution. Is it possibe to
avoid thermal runaway without having to suppress the amplifier's inherently
high voltage gain? A best-of-both-worlds solution to this dilemma is available to
us if we closely examine the problem: the voltage gain that we have to
minimize in order to avoid thermal runaway is the DC voltage gain, not the AC
voltage gain. After all, it isn't the AC input signal that fuels thermal runaway: its
the DC bias voltage required for a certain class of operation: that quiescent DC
signal that we use to “trick” the transistor (fundamentally a DC device) into
amplifying an AC signal. We can suppress DC voltage gain in a common-emitter
amplifier circuit without suppressing AC voltage gain if we figure out a way to
make the negative feedback only function with DC. That is, if we only feed back
an inverted DC signal from output to input, but not an inverted AC signal.
The Reeedback Emitter resistor provides negative feedback by dropping a voltage
proportional to load current. In other words, negative feedback is accomplished
by inserting an impedance into the emitter current path. If we want to feed
back DC but not AC, we need an impedance that is high for DC but low for AC.
What kind of circuit presents a high impedance to DC but a low impedance to
AC? A high-pass filter, of course!
By connecting a Capacitor in parallel with the feedback resistor in Figure below,
we create the very situation we need: a path from emitter to ground that is
easier for AC than it is for DC.
RY] ok,
Vv A
@
Vinput (v)
High AC voltage gain reestablished by adding Cpypass in parallel with Rreedback
The new capacitor “bypasses” AC from the transistor's emitter to ground, so
that no appreciable AC voltage will be dropped from emitter to ground to “feed
back” to the input and suppress voltage gain. Direct current, on the other hand,
cannot go through the bypass capacitor, and so must travel through the
feedback resistor, dropping a DC voltage between emitter and ground which
lowers the DC voltage gain and stabilizes the amplifier's DC response,
preventing thermal runaway. Because we want the reactance of this capacitor
(Xc¢) to be as low as possible, Cpyy5ac5 should be sized relatively large. Because
the polarity across this capacitor will never change, it is safe to use a polarized
(electrolytic) capacitor for the task.
Another approach to the problem of negative feedback reducing voltage gain is
to use multi-stage amplifiers rather than single-transistor amplifiers. If the
attenuated gain of a single transistor is insufficient for the task at hand, we can
use more than one transistor to make up for the reduction caused by feedback.
An example circuit showing negative feedback in a three-stage common-
emitter amplifier is Figure below.
R feedback
Feedback around an “odd” number of direct coupled stages produce negative
feedback.
The feedback path from the final output to the input is through a single resistor,
Rreedback: SINCe each stage is a common-emitter amplifier (thus inverting), the
odd number of stages from input to output will invert the output signal; the
feedback will be negative (degenerative). Relatively large amounts of feedback
may be used without sacrificing voltage gain, because the three amplifier
stages provide much gain to begin with.
At first, this design philosophy may seem inelegant and perhaps even counter-
productive. Isn't this a rather crude way to overcome the loss in gain incurred
through the use of negative feedback, to simply recover gain by adding stage
after stage? What is the point of creating a huge voltage gain using three
transistor stages if we're just going to attenuate all that gain anyway with
negative feedback? The point, though perhaps not apparent at first, is
increased predictability and stability from the circuit as a whole. If the three
transistor stages are designed to provide an arbitrarily high voltage gain (in the
tens of thousands, or greater) with no feedback, it will be found that the
addition of negative feedback causes the overall voltage gain to become less
dependent of the individual stage gains, and approximately equal to the simple
ratio Rreegback/Rin. The more voltage gain the circuit has (without feedback), the
more closely the voltage gain will approximate Reeegpack/Rin once feedback is
established. In other words, voltage gain in this circuit is fixed by the values of
two resistors, and nothing more.
This is an advantage for mass-production of electronic circuitry: if amplifiers of
predictable gain may be constructed using transistors of widely varied B values,
it eases the selection and replacement of components. It also means the
amplifier's gain varies little with changes in temperature. This principle of
stable gain control through a high-gain amplifier “tamed” by negative feedback
is elevated almost to an art form in electronic circuits called operational
amplifiers, or op-amps. You may read much more about these circuits in a later
chapter of this book!
REVIEW:
Feedback is the coupling of an amplifier's output to its input.
Positive, or regenerative feedback has the tendency of making an amplifier
circuit unstable, so that it produces oscillations (AC). The frequency of these
oscillations is largely determined by the components in the feedback
network.
Negative, or degenerative feedback has the tendency of making an
amplifier circuit more stable, so that its output changes /ess for a given
input signal than without feedback. This reduces the gain of the amplifier,
but has the advantage of decreasing distortion and increasing bandwidth
(the range of frequencies the amplifier can handle).
Negative feedback may be introduced into a common-emitter circuit by
coupling collector to base, or by inserting a resistor between emitter and
ground.
An emitter-to-ground “feedback” resistor is usually found in common-
emitter circuits as a preventative measure against thermal runaway.
Negative feedback also has the advantage of making amplifier voltage gain
more dependent on resistor values and less dependent on the transistor's
characteristics.
Common-collector amplifiers have much negative feedback, due to the
placement of the load resistor between emitter and ground. This feedback
accounts for the extremely stable voltage gain of the amplifier, as well as its
immunity against thermal runaway.
Voltage gain for a common-emitter circuit may be re-established without
sacrificing immunity to thermal runaway, by connecting a bypass capacitor
in parallel with the emitter “feedback resistor.”
If the voltage gain of an amplifier is arbitrarily high (tens of thousands, or
greater), and negative feedback is used to reduce the gain to reasonable
levels, it will be found that the gain will approximately equal Rreegpack/Rin-
Changes in transistor B or other internal component values will have little
effect on voltage gain with feedback in operation, resulting in an amplifier
that is stable and easy to design.
Amplifier impedances
Input impedance varies considerably with the circuit configuration shown in
Figure below. It also varies with biasing. Not considered here, the input
impedance is complex and varies with frequency. For the common-emitter and
common-collector it is base resistance times B. The base resistance can be both
internal and external to the transistor. For the common-collector:
Rin = BRE
It is a bit more complicated for the common-emitter circuit. We need to know
the internal emitter resistance reg. This is given by:
Vee = KT/Iem
where:
K=1.38x10°23 watt-sec/°C, Boltzman's constant
T= temperature in Kelvins =300.
Ir = emitter current
m = varies from 1 to 2 for Silicon
Re = 0.026V/I_ = 26mV/I;-
Thus, for the common-emitter circuit Rin is
Rin = Breer
As an example the input resistance of a, 8 = 100, CE configuration biased at 1
MA is:
rep = 26mV/1mA = 260
Rin = Bree = 100(26) = 26000
Moreover, a more accurate Rin for the common-collector should have included
TEE
Rin = B(Re + Teg)
This equation (above) is also applicable to a common-emitter configuration with
an emitter resistor.
Input impedance for the common-base configuration is Rin = rer.
The high input impedance of the common-collector configuration matches high
impedance sources. A crystal or ceramic microphone is one such high
impedance source. The common-base arrangement is sometimes used in RF
(radio frequency) circuits to match a low impedance source, for example, a 50 QO
coaxial cable feed. For moderate impedance sources, the common-emitter is a
good match. An example is a dynamic microphone.
The output impedances of the three basic configurations are listed in Figure
below. The moderate output impedance of the common-emitter configuration
helps make it a popular choice for general use. The Low output impedance of
the common-collector is put to good use in impedance matching, for example,
tranformerless matching to a 4 Ohm speaker. There do not appear to be any
simple formulas for the output impedances. However, R. Victor Jones develops
expressions for output resistance. [RVJ]
Basic circuit Common emitter Common collector Common base Cascode
Voltage gain high less than unity high, same as CE | high, same as CB
Current gain high high less than unity high, same as CE
Power gain high moderate moderate highest
Phase inversion| yes no no yes
Input moderate = 1k highest = 300k low = 502 same asCE, =1k
impedance
Output moderate = 50k low = 300 Q highest = 1Meg same as CB,=1Meg
impedance
Amplifier characteristics, adapted from GE Transistor Manual, Figure 1.21.[GET]
« REVIEW:
e See Figure above.
Current mirrors
An often-used circuit applying the bipolar junction transistor is the so-called
current mirror, which serves as a simple current regulator, supplying nearly
constant current to a load over a wide range of load resistances.
We know that in a transistor operating in its active mode, collector current is
equal to base current multiplied by the ratio 8. We also know that the ratio
between collector current and emitter current is called a. Because collector
current is equal to base current multiplied by B, and emitter current is the sum
of the base and collector currents, a should be mathematically derivable from B.
If you do the algebra, you'll find that a = B/(8+1) for any transistor.
We've seen already how maintaining a constant base current through an active
transistor results in the regulation of collector current, according to the B ratio.
Well, the a ratio works similarly: if emitter current is held constant, collector
current will remain at a stable, regulated value so long as the transistor has
enough collector-to-emitter voltage drop to maintain it in its active mode.
Therefore, if we have a way of holding emitter current constant through a
transistor, the transistor will work to regulate collector current at a constant
value.
Remember that the base-emitter junction of a BJT is nothing more than a PN
junction, just like a diode, and that the “diode equation” specifies how much
current will go through a PN junction given forward voltage drop and junction
temperature:
Ip =I, (et VONET _ 1)
Where,
I, = Diode current in amps
I, = Saturation current.in amps
(typically 1 x 10°'? amps)
e = Euler's constant (~ 2.718281828)
q = charge of electron (1.6 x 10°'? coulombs)
Vp = Voltage applied across diode in volts
N = "Nonideality" or "emission" coefficient
(typically between 1 and 2)
k = Boltzmann’s constant (1.38 x 10°)
T = Junction temperature in Kelvins
If both junction voltage and temperature are held constant, then the PN
junction current will be constant. Following this rationale, if we were to hold the
base-emitter voltage of a transistor constant, then its emitter current will be
constant, given a constant temperature. (Figure below)
I eollector Rioad
(constant)
(constant)
Thase —
L emitter
(constant)
—>
(constant)
a (constant)
Voase
(constant)
Constant Vp- gives constant Ig, constant I-, and constant Ic.
This constant emitter current, multiplied by a constant a ratio, gives a constant
collector current through Rjgag, if enough battery voltage is available to keep
the transistor in its active mode for any change in Rjgag's resistance.
To maintain a constant voltage across the transistor's base-emitter junction use
a forward-biased diode to establish a constant voltage of approximately 0.7
volts, and connect it in parallel with the base-emitter junction as in Figure
below.
Roias I
collector
(constant)
(constant)
Lease oo
Lemmitter
(constant)
(constant)
Diode junction 0.7 V maintains constant base voltage, and constant base
current.
The voltage dropped across the diode probably won't be 0.7 volts exactly. The
exact amount of forward voltage dropped across it depends on the current
through the diode, and the diode's temperature, all in accordance with the
diode equation. If diode current is increased (say, by reducing the resistance of
Rpias), its voltage drop will increase slightly, increasing the voltage drop across
the transistor's base-emitter junction, which will increase the emitter current by
the same proportion, assuming the diode's PN junction and the transistor's
base-emitter junction are well-matched to each other. In other words, transistor
emitter current will closely equal diode current at any given time. If you change
the diode current by changing the resistance value of Rpja,, then the transistor's
emitter current will follow suit, because the emitter current is described by the
same equation as the diode's, and both PN junctions experience the same
voltage drop.
Remember, the transistor's collector current is almost equal to its emitter
current, as the a ratio of a typical transistor is almost unity (1). If we have
control over the transistor's emitter current by setting diode current with a
simple resistor adjustment, then we likewise have control over the transistor's
collector current. In other words, collector current mimics, or mirrors, diode
current.
Current through resistor Rjgaqg is therefore a function of current set by the bias
resistor, the two being nearly equal. This is the function of the current mirror
circuit: to regulate current through the load resistor by conveniently adjusting
the value of Rpia,- Current through the diode is described by a simple equation:
power supply voltage minus diode voltage (almost a constant value), divided
by the resistance of Rpjas-
To better match the characteristics of the two PN junctions (the diode junction
and the transistor base-emitter junction), a transistor may be used in place of a
regular diode, as in Figure below (a).
|
(a) current sinking (b) current-sourcing
Current mirror circuits.
Because temperature is a factor in the “diode equation,” and we want the two
PN junctions to behave identically under all operating conditions, we should
maintain the two transistors at exactly the same temperature. This is easily
done using discrete components by gluing the two transistor cases back-to-
back. If the transistors are manufactured together on a single chip of silicon (as
a so-called integrated circuit, or IC), the designers should locate the two
transistors close to one another to facilitate heat transfer between them.
The current mirror circuit shown with two NPN transistors in Figure above (a) is
sometimes called a current-sinking type, because the regulating transistor
conducts current to the load from ground (“sinking” current), rather than from
the positive side of the battery (“sourcing” current). If we wish to have a
grounded load, and a current sourcing mirror circuit, we may use PNP
transistors like Figure above (b).
While resistors can be manufactured in ICs, it is easier to fabricate transistors.
IC designers avoid some resistors by replacing load resistors with current
sources. A circuit like an operational amplifier built from discrete components
will have a few transistors and many resistors. An integrated circuit version will
have many transistors and a few resistors. In Figure below One voltage
reference, Q1, drives multiple current sources: Q2, Q3, and Q4. If Q2 and Q3 are
equal area transistors the load currents lj,3q will be equal. If we need a 2'ligag,
parallel Q2 and Q3. Better yet fabricate one transistor, say Q3 with twice the
area of Q2. Current I3 will then be twice I2. In other words, load current scales
with transistor area.
Multiple current mirrors may be slaved from a single (Q1 - Ryjz;) voltage source.
Note that it is customary to draw the base voltage line right through the
transistor symbols for multiple current mirrors! Or in the case of Q4 in Figure
above, two current sources are associated with a single transistor symbol. The
load resistors are drawn almost invisible to emphasize the fact that these do not
exist in most cases. The load is often another (multiple) transistor circuit, say a
pair of emitters of a differential amplifier, for example Q3 and Q4 in "A simple
operational amplifier", Ch 8 . Often, the collector load of a transistor is nota
resistor but a current mirror. For example the collector load of Q4 collector , Ch
8 is a current mirror (Q2).
For an example of a current mirror with multiple collector outputs see Q13 in
the model 741 op-amp ,_Ch 8 .. The Q13 current mirror outputs substitute for
resistors as collector loads for Q15 and Q17. We see from these examples that
Current mirrors are preferred as loads over resistors in integrated circuitry.
e REVIEW:
e A current mirror is a transistor circuit that regulates current through a load
resistance, the regulation point being set by a simple resistor adjustment.
e Transistors in a current mirror circuit must be maintained at the same
temperature for precise operation. When using discrete transistors, you may
glue their cases together to do this.
e Current mirror circuits may be found in two basic varieties: the current
sinking configuration, where the regulating transistor connects the load to
ground; and the current sourcing configuration, where the regulating
transistor connects the load to the positive terminal of the DC power supply.
Transistor ratings and packages
Like all electrical and electronic components, transistors are limited in the
amounts of voltage and current each one can handle without sustaining
damage. Since transistors are more complex than some of the other
components you're used to seeing at this point, these tend to have more kinds
of ratings. What follows is an itemized description of some typical transistor
ratings.
Power dissipation: When a transistor conducts current between collector and
emitter, it also drops voltage between those two points. At any given time, the
power dissipated by a transistor is equal to the product (multiplication) of
collector current and collector-emitter voltage. Just like resistors, transistors are
rated for how many watts each can safely dissipate without sustaining damage.
High temperature is the mortal enemy of all semiconductor devices, and bipolar
transistors tend to be more susceptible to thermal damage than most. Power
ratings are always referenced to the temperature of ambient (Surrounding) air.
When transistors are to be used in hotter environments (>25,, their power
ratings must be derated to avoid a shortened service life.
Reverse voltages: As with diodes, bipolar transistors are rated for maximum
allowable reverse-bias voltage across their PN junctions. This includes voltage
ratings for the emitter-base junction Veg , collector-base junction Veg, and also
from collector to emitter Vc, .
Veg , the maximum reverse voltage from emitter to base is approximately 7 V
for some small signal transistors. Some circuit designers use discrete BJTs as 7 V
zener diodes with a series current limiting resistor. Transistor inputs to analog
integrated circuits also have a V_ep rating, which if exceeded will cause damage,
no zenering of the inputs is allowed.
The rating for maximum collector-emitter voltage Vce can be thought of as the
maximum voltage it can withstand while in full-cutoff mode (no base current).
This rating is of particular importance when using a bipolar transistor as a
switch. A typical value for a small signal transistor is 60 to 80 V. In power
transistors, this could range to 1000 V, for example, a horizontal deflection
transistor in a cathode ray tube display.
Collector current: A maximum value for collector current Ic will be given by the
manufacturer in amps. Typical values for small signal transistors are 10s to 100s
of mA, 10s of A for power transistors. Understand that this maximum figure
assumes a Saturated state (minimum collector-emitter voltage drop). If the
transistor is not saturated, and in fact is dropping substantial voltage between
collector and emitter, the maximum power dissipation rating will probably be
exceeded before the maximum collector current rating. Just something to keep
in mind when designing a transistor circuit!
Saturation voltages: |deally, a saturated transistor acts as a closed switch
contact between collector and emitter, dropping zero voltage at full collector
current. In reality this is nevertrue. Manufacturers will specify the maximum
voltage drop of a transistor at saturation, both between the collector and
emitter, and also between base and emitter (forward voltage drop of that PN
junction). Collector-emitter voltage drop at saturation is generally expected to
be 0.3 volts or less, but this figure is of course dependent on the specific type of
transistor. Low voltage transistors, low Vce , show lower saturation voltages. The
saturation voltage is also lower for higher base drive current.
Base-emitter forward voltage drop, kVp- , is similar to that of an equivalent
diode, =0.7 V, which should come as no Surprise.
Beta: The ratio of collector current to base current, B is the fundamental
parameter characterizing the amplifying ability of a bipolar transistor. B is
usually assumed to be a constant figure in circuit calculations, but
unfortunately this is far from true in practice. As such, manufacturers provide a
set of B (or “hge”) figures for a given transistor over a wide range of operating
conditions, usually in the form of maximum/minimum/typical ratings. It may
surprise you to see just how widely B can be expected to vary within normal
operating limits. One popular small-signal transistor, the 2N3903, is advertised
as having a B ranging from 15 to 150 depending on the amount of collector
current. Generally, B is highest for medium collector currents, decreasing for
very low and very high collector currents. hg. is small signal AC gain; N¢¢ is large
AC signal gain or DC gain.
Alpha: the ratio of collector current to emitter current, a=I-/I-. a may be
derived from B, being a=B/(B+1) .
Bipolar transistors come in a wide variety of physical packages. Package type is
primarily dependent upon the required power dissipation of the transistor,
much like resistors: the greater the maximum power dissipation, the larger the
device has to be to stay cool. Figure below shows several standardized package
types for three-terminal semiconductor devices, any of which may be used to
house a bipolar transistor. There are many other semiconductor devices other
than bipolar transistors which have three connection points. Note that the pin-
outs of plastic transistors can vary within a single package type, e.g. TO-92 in
Figure below. It is impossibl/e to positively identify a three-terminal
semiconductor device without referencing the part number printed on it, or
subjecting it to a set of electrical tests.
158
hs Lo
EBC
TO-39
TO-3 case, Collector
Fo4 4 10.7 4 |s2
ok 66 oi [ys
O O LL. 15.5
oB EBC al Y
E
|- 16.89 - 710-09
— 30.15
39.37 TO-18 BCE BCE
TO-3 (300 w) TO-220 (150 w) (TO-247 250 w)
Transistor packages, dimensions in mm.
Small plastic transistor packages like the TO-92 can dissipate a few hundred
milliwatts. The metal cans, TO-18 and TO-39 can dissipate more power, several
hundred milliwatts. Plastic power transistor packages like the TO-220 and TO-
247 dissipate well over 100 watts, approaching the dissipation of the all metal
TO-3. The dissipation ratings listed in Figure above are the maximum ever
encountered by the author for high powered devices. Most power transistors are
rated at half or less than the listed wattage. Consult specific device datasheets
for actual ratings. The semiconductor die in the TO-220 and TO-247 plastic
packages is mounted to a heat conductive metal slug which transfers heat from
the back of the package to a metal heatsink, not shown. A thin coating of
thermally conductive grease is applied to the metal before mounting the
transistor to the heatsink. Since the TO-220 and TO-247 slugs, and the TO-3
case are connected to the collector, it is sometimes necessary to electrically
isolate these from a grounded heatsink by an interposed mica or polymer
washer. The datasheet ratings for the power packages are only valid when
mounted to a heatsink. Without a heatsink, a TO-220 dissipates approximately
1 watt safely in free air.
Datasheet maximum power disipation ratings are difficult to acheive in practice.
The maximum power dissipation is based on a heatsink maintaining the
transistor case at no more than 25°C. This is difficult with an air cooled
heatsink. The allowable power dissipation decreases with increasing
temperature. This is known as derating. Many power device datasheets include
a dissipation versus case termperaure graph.
e REVIEW:
e Power dissipation: maximum allowable power dissipation on a sustained
basis.
¢ Reverse voltages: maximum allowable Vce, Vcg, Veg -
¢ Collector current. the maximum allowable collector current.
¢ Saturation voltage is the Vcg¢ voltage drop in a saturated (fully conducting)
transistor.
¢ Beta: B=Ic/lp
¢ Alpha: a=Ic/Ie a= B/(B+1)
¢ TransistorPackages are a major factor in power dissipation. Larger packages
dissipate more power.
BJT quirks
An ideal transistor would show 0% distortion in amplifying a signal. Its gain
would extend to all frequencies. It would control hundreds of amperes of
current, at hundreds of degrees C. In practice, available devices show distortion.
Amplification is limited at the high frequency end of the spectrum. Real parts
only handle tens of amperes with precautions. Care must be taken when
paralleling transistors for higher current. Operation at elevated temperatures
can destroy transistors if precautions are not taken.
Nonlinearity
The class A common-emitter amplifier (similar to Figure previous)is driven
almost to clipping in Figure below . Note that the positive peak is flatter than
the negative peaks. This distortion is unacceptable in many applications like
high-fidelity audio.
Distortion in large signal common-emitter amplifier.
Small signal amplifiers are relatively linear because they use a small linear
section of the transistor characteristics. Large signal amplifiers are not 100%
linear because transistor characteristics like B are not constant, but vary with
collector current. B is high at low collector current, and low at very low current
or high current. Though, we primarily encounter decreasing B with increasing
collector current.
. cs spice -b ce.cir
common-emitter amplifier j : .
Vbias 4 0 0.74 ee
Vsig 5 4 sin (0 125m 2000 0 0) : ;
rbias 6 5 2k
qi 2 6 © q2n2222 fe ter Sota
r 3 2 1000
v1 3 0 dc 10 : ee ;
.model q2n2222 npn (is=19f bf=150 5 4000 0.0979929
+ vaf=100 ikf=0.18 ise=50p ne=2.5 br=7.5 3 6000 0.0365461
+ var=6.4 ikr=12m isc=8.7p nc=1.2 rb=50 °
: 4 8000 0.00438709
+ re=0.4 rc=0.3 cje=26p tf=0.5n 5 10000 0.00115878
+ cjc=llp tr=7n xtb=1.5 kf=0.032f af=1) 6 12000 0. 00089388
ee ae 7 14000 6.00021169
end 7 : 8 16000 3.8158e-05
j 9 18000 3.3726e-05
SPICE net list: for transient and fourier analyses. Fourier analysis shows 10%
total harmonic distortion (THD).
The SPICE listing in Table above illustrates how to quantify the amount of
distortion. The ".fourier 2000 v(2)" command tells SPICE to perm a fourier
analysis at 2000 Hz on the output v(2). At the command line "spice -b
circuitname.cir" produces the Fourier analysis output in Table above. It shows
THD (total harmonic distortion) of over 10%, and the contribution of the
individual harmonics.
A partial solution to this distortion is to decrease the collector current or
operate the amplifier over a smaller portion of the load line. The ultimate
solution is to apply negative feedback. See Feedback.
Temperature drift
Temperature affects the AC and DC characteristics of transistors. The two
aspects to this problem are environmental temperature variation and self-
heating. Some applications, like military and automotive, require operation over
an extended temperature range. Circuits in a benign environment are subject to
self-heating, in particular high power circuits.
Leakage current Icg and B increase with temperature. The DC B he¢ increases
exponentially. The AC B hy, increases, but not as rapidly. It doubles over the
range of -55° to 85° C. As temperature increases, the increase in h¢. will yield a
larger common-emitter output, which could be clipped in extreme cases. The
increase in hre shifts the bias point, possibly clipping one peak. The shift in bias
point is amplified in multi-stage direct-coupled amplifiers. The solution is some
form of negative feedback to stabilize the bias point. This also stabilizes AC
gain.
Increasing temperature in Figure below (a) will decrease Vp- from the nominal
0.7V for silicon transistors. Decreasing Vp- increases collector current in a
common-emitter amplifier, further shifting the bias point. The cure for shifting
Vee iS a pair of transistors configured as a differential amplifier. If both
transistors in Figure below (b) are at the same temperature, the Vpe- will track
with changing temperature and cancel.
+Vcc
(a) single ended CE amplifier vs (b) differential amplifier with Vg cancellation.
The maximum recommended junction temperature for silicon devices is
frequently 125° C. Though, this should be derated for higher reliability.
Transistor action ceases beyond 150° C. Silicon carbide and diamond transistors
will operate considerably higher.
Thermal runaway
The problem with increasing temperature causing increasing collector current is
that more current increase the power dissipated by the transistor which, in turn,
increases its temperature. This self-reinforcing cycle is known as thermal run
away, which may destroy the transistor. Again, the solution is a bias scheme
with some form of negative feedback to stabilize the bias point.
Junction capacitance
Capacitance exists between the terminals of a transistor. The collector-base
capacitance Ccg and emitter-base capacitance Crp decrease the gain of a
common emitter circuit at higher frequencies.
In acommon emitter amplifier, the capacitive feedback from collector to base
effectively multiplies Ccg by B. The amount of negative gain-reducing feedback
is related to both current gain, and amount of collector-base capacitance. This is
known as the Miller effect, Miller effect.
Noise
The ultimate sensitivity of small signal amplifiers is limited by noise due to
random variations in current flow. The two major sources of noise in transistors
are shot noise due to current flow of carriers in the base and thermal! noise. The
source of thermal noise is device resistance and increases with temperature:
V,, = V4kTRB,
where
k = boltzman’s conatant (1.38¢10~>° watt-sec/K)
T = resistor tempeature in kelvins
R = resistance in Ohms
B, = noise bandwidth in Hz
Noise in a transistor amplifier is defined in terms of excess no/se generated by
the amplifier, not that noise amplified from input to output, but that generated
within the amplifier. This is determined by measuring the signal to noise ratio
(S/N) at the amplifier input and output. The AC voltage output of an amplifier
with a small signal input corresponds to S+N, signal plus noise. The AC voltage
with no signal in corresponds to noise N. The no/se figure F is defined in terms
of S/N of amplifier input and output:
__ (S/N);
~ (S/N),
Fag = 10 log F
The noise figure F for RF (radio frequency) transistors is usually listed on
transistor data sheets in decibels, Fyg. A good VHF (very high frequency, 30
MHz to 300 Mhz) noise figure is < 1 dB. The noise figure above VHF increases
considerable, 20 dB per decade as shown in Figure below.
Oy. shot noise and
Se
thermal noise
NS
{ &
Pp
Noise figure F (decibels)
S)
Log Frequency
Small signal transistor noise figure vs Frequency. After Thiele, Figure 11.147
[AGT]
Figure above also shows that noise at low frequencies increases at 10 dB per
decade with decreasing frequency. This noise is known as 1/f noise.
Noise figure varies with the transistor type (part number). Small signal RF
transistors used at the antenna input of a radio receiver are specifically
designed for low noise figure. Noise figure varies with bias current and
impedance matching. The best noise figure for a transistor is achieved at lower
bias current, and possibly with an impedance mismatch.
Thermal mismatch (problem with paralleling transistors)
If two identical power transistors were paralleled for higher current, one would
expect them to share current equally. Because of differences in
characteristerics, transistors do not share current equally.
+V
Incorrect Correct
Transistors paralleled for increased power require emitter ballast resistors
It is not practical to select identical transistors. The B for small signal transistors
typically has a range of 100-300, power transistors: 20-50. If each one could be
matched, one still might run hotter than the other due to environmental
conditions. The hotter transistor draws more current resulting in thermal
runaway. The solution when paralleling bipolar transistors is to insert emitter
resistors known as ballast resistors of less than an ohm. If the hotter transistor
draws more current, the voltage drop across the ballast resistor increases—
negative feedback. This decreases the current. Mounting all transistors on the
same heatsink helps equalize current too.
High frequency effects
The performance of a transistor amplifier is relatively constant, up to a point, as
shown by the small signal common-emitter current gain with increasing
frequency in Figure below. Beyond that point the performance of a transistor
degrades as frequency increases.
Beta cutoff frequency, f; is the frequency at which common-emitter small
signal current gain (hq) falls to unity. (Figure below) A practical amplifier must
have a gain >1. Thus, a transistor cannot be used in a practical amplifier at f;. A
more usable limit for a transistor is 0.1-f;.
100
log f
Common-emitter small signal current gain (hyo) vs frequency.
Some RF silicon bipolar transistors are usable as amplifers up to a few GHz.
Silicon-germanium devices extend the upper range to 10 GHz.
Alpha cutoff frequency, fajpnq is the frequency at which the a falls to 0.707 of
low frequency ,9 A=0.707 >. Alpha cutoff and beta cutoff are nearly equal:
falpha=fr Beta cutoff f; is the preferred figure of merit of high frequency
performance.
fmax IS the highest frequency of oscillation possible under the most favorable
conditions of bias and impedance matching. It is the frequency at which the
power gain is unity. All of the output is fed back to the input to sustain
oscillations. f,,4, is an upper limit for frequency of operation of a transistor as an
active device. Though, a practical amplifier would not be usable at f,,3..
Miller effect: The high frequency limit for a transistor is related to the junction
capacitances. For example a PN2222A has an input capacitance C,,,=9pF and
an output capacitance C;,,=25pF from C-B and E-B respectively. [FAR] Although
the C-E capacitance of 25 pF seems large, it is less of a factor than the C-B
(9pF) capacitance. because of the Miller effect, the C-B capacitance has an
effect on the base equivalent to beta times the capacitance in the common-
emitter amplifier. Why might this be? A common-emitter amplifier inverts the
signal from base to collector. The inverted collector signal fed back to the base
opposes the input on the base. The collector signal is beta times larger than the
input. For the PN2222A, B=50-300. Thus, the 9pF C-E capacitance looks like
9:50=450pF to 9:300=27 OOpF.
The solution to the junction capacitance problem is to select a high frequency
transistor for wide bandwidth applications— RF (radio frequency) or microwave
transistor. The bandwidth can be extended further by using the common-base
instead of the common-emitter configuration. The grounded base shields the
emitter input from capacitive collector feedback. A two-transistor cascode
arrangement will yield the same bandwidth as the common-base, with the
higher input impedance of the common-emitter.
« REVIEW:
e Transistor amplifiers exhibit distortion because of B variation with collector
current.
° |, Vee, B and junction capacitance vary with temperature.
¢ An increase in temperature can cause an increase in Ic, causing an increase
in temperature, a vicious cycle known as thermal runaway.
e Junction capacitance limits high frequency gain of a transistor. The Miller
effect makes C,, look B times larger at the base of a CE amplifier.
e Transistor noise limits the ability to amplify small signals. No/se figure is a
figure of merit concerning transistor noise.
e When paralleling power transistors for increased current, insert ba/last
resistors in series with the emitters to equalize current.
¢ Fr is the absolute upper frequency limit for a CE amplifier, small signal
current gain falls to unity, hy.=1.
© Fimax is the upper frequency limit for an oscillator under the most ideal
conditions.
Bibliography
1. [AGT] A. G. Thiele in Loyd P. Hunter, “Handbook of Semiconductor
Electronics,” Low Frequency Amplifiers, ISBN -07-031305-9, 1970
. [GET] “GE Transistor Manual”, General Electric, 1964.
. [RVJ] R. Victor Jones, “Basic BJT Amplifier Configurations”, November 7,
2001. at
http://people.seas.harvard.edu/~jones/es154/lectures/lecture_3/bjt_amps/b
jt_amps.html
4. [TK1] Tony Kuphaldt,“Lessons in Electric Circuits”, Vol. 1, DC, DC Network
Analysis, Thevenin's Theorem, at
http://www.openbookproject.net/electricCircuits/DC/DC_10.html# xtocid102
679
5. [FAR] “PN2222 Datasheet”, Fairchild Semiconductor Corporation, 2007 at
http://www.fairchildsemi.com/ds/PN/PN2222A.pdf
WN
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt, under
the terms and conditions of the Design Science License.
Previous Contents Next
— 4 — >
Previous Contents Next
— 4 —>
Lessons In Electric Circuits --
Volume Ill
Chapter 5
JUNCTION FIELD-EFFECT
TRANSISTORS
e Introduction
e The transistor as a switch
e Meter check of a transistor
e« Active-mode operation
¢ The common-source amplifier -- PENDING
e The common-drain amplifier -- PENDING
e The common-gate amplifier -- PENDING
e Biasing techniques -- PENDING
¢ Transistor ratings and packages -- PENDING
e JFET quirks -- PENDING
4 INCOMPLETE ***
Introduction
A transistor is a linear semiconductor device that controls current with the
application of a lower-power electrical signal. Transistors may be roughly
grouped into two major divisions: bipolar and field-effect. In the last chapter we
studied bipolar transistors, which utilize a small current to control a large
current. In this chapter, we'll introduce the general concept of the field-effect
transistor -- a device utilizing a small vo/tage to control current -- and then
focus on one particular type: the junction field-effect transistor. In the next
chapter we'll explore another type of field-effect transistor, the /nsulated gate
variety.
All field-effect transistors are unipo/ar rather than bipolar devices. That is, the
main current through them is comprised either of electrons through an N-type
semiconductor or holes through a P-type semiconductor. This becomes more
evident when a physical diagram of the device is seen:
N-channel JFET
drain
drain
gate gate P
source
source
schematic symbol physical diagram
In a junction field-effect transistor, or JFET, the controlled current passes from
source to drain, or from drain to source as the case may be. The controlling
voltage is applied between the gate and source. Note how the current does not
have to cross through a PN junction on its way between source and drain: the
path (called a channe/) is an uninterrupted block of semiconductor material. In
the image just shown, this channel is an N-type semiconductor. P-type channel
JFETs are also manufactured:
P-channel JFET
drain
drain
gate gate
source
source
schematic symbol physical diagram
Generally, N-channel JFETs are more commonly used than P-channel. The
reasons for this have to do with obscure details of semiconductor theory, which
I'd rather not discuss in this chapter. As with bipolar transistors, | believe the
best way to introduce field-effect transistor usage is to avoid theory whenever
possible and concentrate instead on operational characteristics. The only
practical difference between N- and P-channel JFETs you need to concern
yourself with now is biasing of the PN junction formed between the gate
material and the channel.
With no voltage applied between gate and source, the channel is a wide-open
path for electrons to flow. However, if a voltage is applied between gate and
source of such polarity that it reverse-biases the PN junction, the flow between
source and drain connections becomes limited, or regulated, just as it was for
bipolar transistors with a set amount of base current. Maximum gate-source
voltage "pinches off" all current through source and drain, thus forcing the JFET
into cutoff mode. This behavior is due to the depletion region of the PN junction
expanding under the influence of a reverse-bias voltage, eventually occupying
the entire width of the channel if the voltage is great enough. This action may
be likened to reducing the flow of a liquid through a flexible hose by squeezing
it: with enough force, the hose will be constricted enough to completely block
the flow.
water
hose n
* een ERR ay,
fe)
N
N
a)
Mt
water
=
h
Hose constricted by squeezing,
water flow reduced or stoppe
Note how this operational behavior is exactly opposite of the bipolar junction
transistor. Bipolar transistors are normally-off devices: no current through the
base, no current through the collector or the emitter. JFETs, on the other hand,
are normally-on devices: no voltage applied to the gate allows maximum
current through the source and drain. Also take note that the amount of current
allowed through a JFET is determined by a voltage signal rather than a current
signal as with bipolar transistors. In fact, with the gate-source PN junction
reverse-biased, there should be nearly zero current through the gate
connection. For this reason, we classify the JFET as a vo/tage-controlled device,
and the bipolar transistor as a current-controlled device.
If the gate-source PN junction is forward-biased with a small voltage, the JFET
channel will "open" a little more to allow greater currents through. However, the
PN junction of a JFET is not built to handle any substantial current itself, and
thus it is not recommended to forward-bias the junction under any
circumstances.
This is a very condensed overview of JFET operation. In the next section, we'll
explore the use of the JFET as a switching device.
The transistor as a switch
Like its bipolar cousin, the field-effect transistor may be used as an on/off
switch controlling electrical power to a load. Let's begin our investigation of the
JFET as a switch with our familiar switch/lamp circuit:
ro
switch —
Remembering that the contro/led current in a JFET flows between source and
drain, we substitute the source and drain connections of a JFET for the two ends
of the switch in the above circuit:
If you haven't noticed by now, the source and drain connections on a JFET look
identical on the schematic symbol. Unlike the bipolar junction transistor where
the emitter is clearly distinguished from the collector by the arrowhead, a JFET's
source and drain lines both run perpendicular into the bar representing the
semiconductor channel. This is no accident, as the source and drain lines of a
JFET are often interchangeable in practice! In other words, JFETs are usually able
to handle channel current in either direction, from source to drain or from drain
to source.
Now all we need in the circuit is a way to control the JFET's conduction. With
zero applied voltage between gate and source, the JFET's channel will be
"open," allowing full current to the lamp. In order to turn the lamp off, we will
need to connect another source of DC voltage between the gate and source
connections of the JFET like this:
ben
Closing this switch will "pinch off" the JFET's channel, thus forcing it into cutoff
and turning the lamp off:
switch
Note that there is no current going through the gate. As a reverse-biased PN
junction, it firmly opposes the flow of any electrons through it. As a voltage-
controlled device, the JFET requires negligible input current. This is an
advantageous trait of the JFET over the bipolar transistor: there is virtually zero
power required of the controlling signal.
Opening the control switch again should disconnect the reverse-biasing DC
voltage from the gate, thus allowing the transistor to turn back on. Ideally,
anyway, this is how it works. In practice this may not work at all:
switch
No lamp current after the switch opens!
Why is this? Why doesn't the JFET's channel open up again and allow lamp
current through like it did before with no voltage applied between gate and
source? The answer lies in the operation of the reverse-biased gate-source
junction. The depletion region within that junction acts as an insulating barrier
separating gate from source. As such, it possesses a certain amount of
capacitance capable of storing an electric charge potential. After this junction
has been forcibly reverse-biased by the application of an external voltage, it will
tend to hold that reverse-biasing voltage as a stored charge even after the
source of that voltage has been disconnected. What is needed to turn the JFET
on again is to bleed off that stored charge between the gate and source through
a resistor:
Resistor bleeds off stored charge in
PN junction to allow transistor to
turn on once again.
This resistor's value is not very important. The capacitance of the JFET's gate-
source junction is very small, and so even a rather high-value bleed resistor
creates a fast RC time constant, allowing the transistor to resume conduction
with little delay once the switch is opened.
Like the bipolar transistor, it matters little where or what the controlling voltage
comes from. We could use a solar cell, thermocouple, or any other sort of
voltage-generating device to supply the voltage controlling the JFET's
conduction. All that is required of a voltage source for JFET switch operation is
sufficient voltage to achieve pinch-off of the JFET channel. This level is usually
in the realm of a few volts DC, and is termed the pinch-off or cutoff voltage. The
exact pinch-off voltage for any given JFET is a function of its unique design, and
is not a universal figure like 0.7 volts is for a silicon BJT's base-emitter junction
voltage.
REVIEW:
Field-effect transistors control the current between source and drain
connections by a voltage applied between the gate and source. In a
junction field-effect transistor (JFET), there is a PN junction between the
gate and source which is normally reverse-biased for control of source-drain
current.
JFETs are normally-on (normally-saturated) devices. The application of a
reverse-biasing voltage between gate and source causes the depletion
region of that junction to expand, thereby "pinching off" the channel
between source and drain through which the controlled current travels.
It may be necessary to attach a "bleed-off" resistor between gate and
source to discharge the stored charge built up across the junction's natural
Capacitance when the controlling voltage is removed. Otherwise, a charge
may remain to keep the JFET in cutoff mode even after the voltage source
has been disconnected.
Meter check of a transistor
Testing a JFET with a multimeter might seem to be a relatively easy task, seeing
as how it has only one PN junction to test: either measured between gate and
source, or between gate and drain.
N-channel transistor
drain
source
source
physical diagram
Both meters show non-continuity
(high resistance) through gate-
channel junction.
N-channel transistor
<
ca
Oa com drain
drain :
| |
gate \—+ gate +r] N
7 +
0 source
source
| physical diagram
vo Both meters show continuity (low
resistance) through gate-channel
O* | junction.
Testing continuity through the drain-source channel is another matter, though.
Remember from the last section how a stored charge across the capacitance of
the gate-channel PN junction could hold the JFET in a pinched-off state without
any external voltage being applied across it? This can occur even when you're
holding the JFET in your hand to test it! Consequently, any meter reading of
continuity through that channel will be unpredictable, since you don't
necessarily know if a charge is being stored by the gate-channel junction. Of
course, if you Know beforehand which terminals on the device are the gate,
source, and drain, you may connect a jumper wire between gate and source to
eliminate any stored charge and then proceed to test source-drain continuity
with no problem. However, if you don't know which terminals are which, the
unpredictability of the source-drain connection may confuse your determination
of terminal identity.
A good strategy to follow when testing a JFET is to insert the pins of the
transistor into anti-static foam (the material used to ship and store static-
sensitive electronic components) just prior to testing. The conductivity of the
foam will make a resistive connection between all terminals of the transistor
when it is inserted. This connection will ensure that all residual voltage built up
across the gate-channel PN junction will be neutralized, thus "opening up" the
channel for an accurate meter test of source-to-drain continuity.
Since the JFET channel is a single, uninterrupted piece of semiconductor
material, there is usually no difference between the source and drain terminals.
A resistance check from source to drain should yield the same value as a check
from drain to source. This resistance should be relatively low (a few hundred
ohms at most) when the gate-source PN junction voltage is zero. By applying a
reverse-bias voltage between gate and source, pinch-off of the channel should
be apparent by an increased resistance reading on the meter.
Active-mode operation
JFETs, like bipolar transistors, are able to "throttle" current in a mode between
cutoff and saturation called the active mode. To better understand JFET
operation, let's set up a SPICE simulation similar to the one used to explore
basic bipolar transistor function:
V
ammeter
jfet simulation
vin 01dcl1
jl 2 1 0 modl
vammeter 3 2 dc 0
v1 3 0 dc
.model modi njf
.dc v1 0 2 0.05
.plot dc i(vammeter)
.end
Note that the transistor labeled "Q," in the schematic is represented in the
SPICE netlist as j1. Although all transistor types are commonly referred to as "Q"
devices in circuit schematics -- just as resistors are referred to by "R"
designations, and capacitors by "C" -- SPICE needs to be told what type of
transistor this is by means of a different letter designation: q for bipolar junction
transistors, and j for junction field-effect transistors.
uA = yeaeterse anch
vanmeter
Here, the controlling signal is a steady voltage of 1 volt, applied with negative
towards the JFET gate and positive toward the JFET source, to reverse-bias the
PN junction. In the first BJT simulation of chapter 4, a constant-current source of
20 yA was used for the controlling signal, but remember that a JFET isa
voltage-controlled device, not a current-controlled device like the bipolar
junction transistor.
Like the BJT, the JFET tends to regulate the controlled current at a fixed level
above a certain power supply voltage, no matter how high that voltage may
climb. Of course, this current regulation has limits in real life -- no transistor can
withstand infinite voltage from a power source -- and with enough drain-to-
source voltage the transistor will "break down" and drain current will Surge. But
within normal operating limits the JFET keeps the drain current at a steady level
independent of power supply voltage. To verify this, we'll run another computer
simulation, this time sweeping the power supply voltage (Vj) all the way to 50
volts:
jfet simulation
vin 01dcl1
jl 2 1 © modl
vammeter 3 2 dc 0
v1 3 @ dc
.model modi njf
.dc v1 0 50 2
.plot dc i(vammeter)
.end
uA — vanmeter#branch
I(vammeter}
100.0 ¢°
50,0
0,0 20,0 40,0 60,0
sweep Y
Sure enough, the drain current remains steady at a value of 100 HA (1.000E-04
amps) no matter how high the power supply voltage is adjusted.
Because the input voltage has control over the constriction of the JFET's
channel, it makes sense that changing this voltage should be the only action
capable of altering the current regulation point for the JFET, just like changing
the base current on a BJT is the only action capable of altering collector current
regulation. Let's decrease the input voltage from 1 volt to 0.5 volts and see
what happens:
jfet simulation
vin @ 1 dc 0.5
jl 2 1 © modl
vammeter 3 2 dc 0
vl 3 0 dc
.model modi njf
.dc vl 0 50 2
.plot dc i(vammeter)
.end
uA — yvanmeter#branch
I(vammeter )
300,0
200.0 J
100,0
0,0 20,0 40,0 60,0
sweep Y
As expected, the drain current is greater now than it was in the previous
simulation. With less reverse-bias voltage impressed across the gate-source
junction, the depletion region is not as wide as it was before, thus "opening" the
channel for charge carriers and increasing the drain current figure.
Please note, however, the actual value of this new current figure: 225 pA
(2.250E-04 amps). The last simulation showed a drain current of 100 yA, and
that was with a gate-source voltage of 1 volt. Now that we've reduced the
controlling voltage by a factor of 2 (from 1 volt down to 0.5 volts), the drain
Current increased, but not by the same 2:1 proportion! Let's reduce our gate-
source voltage once more by another factor of 2 (down to 0.25 volts) and see
what happens:
jfet simulation
vin 0 1 dc 0.25
jl 2 1 0 modl
vammeter 3 2 dc 0
v1 3 @ dc
.model modi njf
.dc v1 0 50 2
.plot dc i(vammeter)
.end
uA — vanmeter#branch
I(vammeter)
400,0
300.0]
200,0
100,0
0,0 20,0 40,0 60,0
sweep Vv
With the gate-source voltage set to 0.25 volts, one-half what it was before, the
drain current is 306.3 YA. Although this is still an increase over the 225 yA from
the prior simulation, it isn't proportional to the change of the controlling
voltage.
To obtain a better understanding of what is going on here, we should run a
different kind of simulation: one that keeps the power supply voltage constant
and instead varies the controlling (voltage) signal. When this kind of simulation
was run on a BJT, the result was a straight-line graph, showing how the input
current / output current relationship of a BJT is linear. Let's see what kind of
relationship a JFET exhibits:
jfet simulation
vin 0 1 dc
jl 2 1 © modl
vammeter 3 2 dc 0
v1 3 0 de 25
.model modi njf
.dc vin 0 2 0.1
.plot dc i(vammeter)
.end
uA = fers anch
I(vammeter
This simulation directly reveals an important characteristic of the junction field-
effect transistor: the control effect of gate voltage over drain current is
nonlinear. Notice how the drain current does not decrease linearly as the gate-
source voltage is increased. With the bipolar junction transistor, collector
Current was directly proportional to base current: output signal proportionately
followed input signal. Not so with the JFET! The controlling signal (gate-source
voltage) has less and less effect over the drain current as it approaches cutoff.
In this simulation, most of the controlling action (75 percent of drain current
decrease -- from 400 HA to 100 WA) takes place within the first volt of gate-
source voltage (from 0 to 1 volt), while the remaining 25 percent of drain
Current reduction takes another whole volt worth of input signal. Cutoff occurs
at 2 volts input.
Linearity is generally important for a transistor because it allows it to faithfully
amplify a waveform without distorting it. If a transistor is nonlinear in its
input/output amplification, the shape of the input waveform will become
corrupted in some way, leading to the production of harmonics in the output
signal. The only time linearity is not important in a transistor circuit is when its
being operated at the extreme limits of cutoff and saturation (off and on,
respectively, like a switch).
A JFET's characteristic curves display the same current-regulating behavior as
for a BJT, and the nonlinearity between gate-to-source voltage and drain current
is evident in the disproportionate vertical spacings between the curves:
AV ps! = IVpl - IVs!
Below pinch-off | Above pinch-off
Ul
Tricde region ¢# Saturation region
3 ; J
Vv = OV
gate-to-source
Tacain 0.5 V
‘ —
‘ gate-to-source ~~
lV
V pate-to-source oe
(pinch-off)
= 9 _
V pate-to-source =2V= Vp
Earai n-to-source
To better comprehend the current-regulating behavior of the JFET, it might be
helpful to draw a model made up of simpler, more common components, just as
we did for the BJT:
S
N-channel JFET diode-regulating diode model
D
S
In the case of the JFET, it is the vo/tage across the reverse-biased gate-source
diode which sets the current regulation point for the pair of constant-current
diodes. A pair of opposing constant-current diodes is included in the model to
facilitate current in either direction between source and drain, a trait made
possible by the unipolar nature of the channel. With no PN junctions for the
source-drain current to traverse, there is no polarity sensitivity in the controlled
Current. For this reason, JFETs are often referred to as bilateral devices.
A contrast of the JFET's characteristic curves against the curves for a bipolar
transistor reveals a notable difference: the linear (straight) portion of each
curve's non-horizontal area is surprisingly long compared to the respective
portions of a BJT's characteristic curves:
pate-to-source — OV
Train & in teseurd =O5V
Vv 1V
gate-to-source ~
= 2V (pinch-off)
gate-to-source
Fy rain-to-source
"Ohmic regions"
Tbase = 79 PA
/
|
Tealleetnts |
| Inase = 40 LA
‘g
Thase = 20 PA
Taint =5 HA
Foot lector-to-emitter
A JFET transistor operated in the triode region tends to act very much like a
plain resistor as measured from drain to source. Like all simple resistances, its
current/voltage graph is a straight line. For this reason, the triode region (non-
horizontal) portion of a JFET's characteristic curve is sometimes referred to as
the ohmic region. |In this mode of operation where there isn't enough drain-to-
source voltage to bring drain current up to the regulated point, the drain
current is directly proportional to the drain-to-source voltage. In a carefully
designed circuit, this phenomenon can be used to an advantage. Operated in
this region of the curve, the JFET acts like a voltage-controlled resistance rather
than a voltage-controlled current regu/ator, and the appropriate model for the
transistor is different:
D
Ss
N-channel JFET diode-rheostat model
(for saturation, or "ohmic," mode only!)
D
S
Here and here alone the rheostat (variable resistor) model of a transistor is
accurate. It must be remembered, however, that this model of the transistor
holds true only for a narrow range of its operation: when it is extremely
saturated (far less voltage applied between drain and source than what is
needed to achieve full regulated current through the drain). The amount of
resistance (measured in ohms) between drain and source in this mode is
controlled by how much reverse-bias voltage is applied between gate and
source. The less gate-to-source voltage, the less resistance (steeper line on
graph).
Because JFETs are vo/tage-controlled current regulators (at least when they're
allowed to operate in their active), their inherent amplification factor cannot be
expressed as a unitless ratio as with BJTs. In other words, there is no B ratio fora
JFET. This is true for all voltage-controlled active devices, including other types
of field-effect transistors and even electron tubes. There is, however, an
expression of controlled (drain) current to controlling (gate-source) voltage, and
itis called transconductance. Its unit is Siemens, the same unit for conductance
(formerly known as the mho).
Why this choice of units? Because the equation takes on the general form of
current (output signal) divided by voltage (input signal).
Ofs AVos
Where,
g,, = Transconductance in Siemens
AI, = Change in drain current
AV, = Change in gate-source voltage
Unfortunately, the transconductance value for any JFET is not a stable quantity:
it varies significantly with the amount of gate-to-source control voltage applied
to the transistor. AS we saw in the SPICE simulations, the drain current does not
change proportionally with changes in gate-source voltage. To calculate drain
current for any given gate-source voltage, there is another equation that may
be used. It is obviously nonlinear upon inspection (note the power of 2),
reflecting the nonlinear behavior we've already experienced in simulation:
V 2
Ip = Ipss (1 - —S—)
Ves(cutotf)
Where,
I, = Drain current
Iss = Drain current with gate shorted to source
Vs = Gate-to-source voltage
Vosvcutorr) = Pinch-off gate-to-source voltage
e REVIEW:
e In their active modes, JFETs regulate drain current according to the amount
of reverse-bias voltage applied between gate and source, much like a BJT
regulates collector current according to base current. The mathematical
ratio between drain current (output) and gate-to-source voltage (input) is
called transconductance, and it is measured in units of Siemens.
e« The relationship between gate-source (control) voltage and drain
(controlled) current is nonlinear: as gate-source voltage is decreased, drain
current increases exponentially. That is to say, the transconductance of a
JFET is not constant over its range of operation.
e In their triode region, JFETs regulate drain-to-source resistance according to
the amount of reverse-bias voltage applied between gate and source. In
other words, they act like voltage-controlled resistances.
The common-source amplifier -- PENDING
#4 PENDING ***
¢ REVIEW:
The common-drain amplifier -- PENDING
*e PENDING ***
¢ REVIEW:
The common-gate amplifier -- PENDING
** PENDING ***
¢ REVIEW:
Biasing techniques -- PENDING
** PENDING ***
¢ REVIEW:
Transistor ratings and packages -- PENDING
#4 PENDING ***
¢ REVIEW:
JFET quirks -- PENDING
+ PENDING ***
¢ REVIEW:
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt, under
the terms and conditions of the Design Science License.
—||+4/|—
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— 4 —>
Lessons In Electric Circuits --
Volume Ill
Chapter 6
INSULATED-GATE FIELD-EFFECT
TRANSISTORS
Introduction
Depletion-type IGFETs
Enhancement-type IGFETs -- PENDING
Active-mode operation -- PENDING
The common-source amplifier -- PENDING
The common-drain amplifier -- PENDING
The common-gate amplifier -- PENDING
Biasing techniques -- PENDING
Transistor ratings and packages -- PENDING
IGFET quirks -- PENDING
MESFETs -- PENDING
IGBTs
4 INCOMPLETE ***
Introduction
As was Stated in the last chapter, there is more than one type of field-effect
transistor. The junction field-effect transistor, or JFET, uses voltage applied
across a reverse-biased PN junction to control the width of that junction's
depletion region, which then controls the conductivity of a semiconductor
channel through which the controlled current moves. Another type of field-
effect device -- the insulated gate field-effect transistor, or IGFET -- exploits a
similar principle of a depletion region controlling conductivity through a
semiconductor channel, but it differs primarily from the JFET in that there is no
direct connection between the gate lead and the semiconductor material itself.
Rather, the gate lead is insulated from the transistor body by a thin barrier,
hence the term insulated gate. This insulating barrier acts like the dielectric
layer of a capacitor, and allows gate-to-source voltage to influence the
depletion region electrostatically rather than by direct connection.
In addition to a choice of N-channel versus P-channel design, IGFETs come in
two major types: enhancement and depletion. The depletion type is more
closely related to the JFET, so we will begin our study of IGFETs with it.
Depletion-type IGFETs
Insulated gate field-effect transistors are unipolar devices just like JFETs: that is,
the controlled current does not have to cross a PN junction. There is a PN
junction inside the transistor, but its only purpose is to provide that
nonconducting depletion region which is used to restrict current through the
channel.
Here is a diagram of an N-channel IGFET of the "depletion" type:
N-channel, D-type IGFET
drain
drain
gate IE substrate gate Jv substrate
oie "Sarier®
source
schematic symbol physical diagram
Notice how the source and drain leads connect to either end of the N channel,
and how the gate lead attaches to a metal plate separated from the channel by
a thin insulating barrier. That barrier is sometimes made from silicon dioxide
(the primary chemical compound found in sand), which is a very good insulator.
Due to this Metal (gate) - Oxide (barrier) - Semiconductor (channel)
construction, the IGFET is sometimes referred to as a MOSFET. There are other
types of IGFET construction, though, and so "IGFET" is the better descriptor for
this general class of transistors.
Notice also how there are four connections to the IGFET. In practice, the
substrate lead is directly connected to the source lead to make the two
electrically common. Usually, this connection is made internally to the IGFET,
eliminating the separate substrate connection, resulting in a three-terminal
device with a slightly different schematic symbol:
N-channel, D-type IGFET
drain
drain
ate
wo IF vie :
insulatin
source barrier?
source
substrate
schematic symbol physical diagram
With source and substrate common to each other, the N and P layers of the
IGFET end up being directly connected to each other through the outside wire.
This connection prevents any voltage from being impressed across the PN
junction. As a result, a depletion region exists between the two materials, but it
can never be expanded or collapsed. JFET operation is based on the expansion
of the PN junction's depletion region, but here in the IGFET that cannot happen,
so IGFET operation must be based on a different effect.
Indeed it is, for when a controlling voltage is applied between gate and source,
the conductivity of the channel is changed as a result of the depletion region
moving closer to or further away from the gate. In other words, the channel's
effective width changes just as with the JFET, but this change in channel width
is due to depletion region displacement rather than depletion region expansion.
In an N-channel IGFET, a controlling voltage applied positive (+) to the gate
and negative (-) to the source has the effect of repelling the PN junction's
depletion region, expanding the N-type channel and increasing conductivity:
R load
controlling
voltage
Channel expands for greater conductivity
Reversing the controlling voltage's polarity has the opposite effect, attracting
the depletion region and narrowing the channel, consequently reducing
channel conductivity:
R load
controlling
voltage
Channel narrows for less conductivity
The insulated gate allows for controlling voltages of any polarity without danger
of forward-biasing a junction, as was the concern with JFETs. This type of IGFET,
although its called a "depletion-type," actually has the capability of having its
channel e/ther depleted (channel narrowed) or enhanced (channel expanded).
Input voltage polarity determines which way the channel will be influenced.
Understanding which polarity has which effect is not as difficult as it may seem.
The key is to consider the type of semiconductor doping used in the channel (N-
channel or P-channel?), then relate that doping type to the side of the input
voltage source connected to the channel by means of the source lead. If the
IGFET is an N-channel and the input voltage is connected so that the positive
(+) side is on the gate while the negative (-) side is on the source, the channel
will be enhanced as extra electrons build up on the channel side of the
dielectric barrier. Think, "negative (-) correlates with N-type, thus enhancing the
channel with the right type of charge carrier (electrons) and making it more
conductive." Conversely, if the input voltage is connected to an N-channel
IGFET the other way, so that negative (-) connects to the gate while positive (+)
connects to the source, free electrons will be "robbed" from the channel as the
gate-channel capacitor charges, thus depleting the channel of majority charge
carriers and making it less conductive.
For P-channel IGFETs, the input voltage polarity and channel effects follow the
same rule. That is to say, it takes just the opposite polarity as an N-channel
IGFET to either deplete or enhance:
controlling
voltage
Channel expands for greater conductivity
Ryoaa
controlling
voltage
Channel narrows for less conductivity
Illustrating the proper biasing polarities with standard IGFET symbols:
N-channel P-channel
Enhanced
(more drain
current)
Depleted
(less drain
current)
When there is zero voltage applied between gate and source, the IGFET will
conduct current between source and drain, but not as much current as it would
if it were enhanced by the proper gate voltage. This places the depletion-type,
or simply D-type, IGFET in a category of its own in the transistor world. Bipolar
junction transistors are normally-off devices: with no base current, they block
any current from going through the collector. Junction field-effect transistors are
normally-on devices: with zero applied gate-to-source voltage, they allow
maximum drain current (actually, you can coax a JFET into greater drain
currents by applying a very small forward-bias voltage between gate and
source, but this should never be done in practice for risk of damaging its fragile
PN junction). D-type IGFETs, however, are normally half-on devices: with no
gate-to-source voltage, their conduction level is somewhere between cutoff and
full saturation. Also, they will tolerate applied gate-source voltages of any
polarity, the PN junction being immune from damage due to the insulating
barrier and especially the direct connection between source and substrate
preventing any voltage differential across the junction.
Ironically, the conduction behavior of a D-type IGFET is strikingly similar to that
of an electron tube of the triode/tetrode/pentode variety. These devices were
voltage-controlled current regulators that likewise allowed current through
them with zero controlling voltage applied. A controlling voltage of one polarity
(grid negative and cathode positive) would diminish conductivity through the
tube while a voltage of the other polarity (grid positive and cathode negative)
would enhance conductivity. | find it curious that one of the later transistor
designs invented exhibits the same basic properties of the very first active
(electronic) device.
A few SPICE analyses will demonstrate the current-regulating behavior of D-
type IGFETs. First, a test with zero input voltage (gate shorted to source) and
the power supply swept from 0 to 50 volts. The graph shows drain current:
Vv
ammeter
n-channel igfet characteristic curve
m1 100 0 modl1
vammeter 2 1 dc 0
vl 2 0
.model mod1 nmos vto=-1
.dc vl 0 50 2
.plot dc i(vammeter)
.end
uA = erie lead
I(vammeter )
“0,0 20,0 40,0 60,0
sweep Vv
As expected for any transistor, the controlled current holds steady at a
regulated value over a wide range of power supply voltages. In this case, that
regulated point is 10 UA (1.000E-05). Now let's see what happens when we
apply a negative voltage to the gate (with reference to the source) and sweep
the power supply over the same range of 0 to 50 volts:
Vv
ammeter
n-channel igfet characteristic curve
m1 13 0 0 modl
vin 0 3 de 0.5
vammeter 2 1 dc 0
vl 20
.model modl nmos vto=-1
.dc vl 0 50 2
.plot dc i(vammeter)
.end
uA — vanmeter#branch
I(vammeter)
Not surprisingly, the drain current is now regulated at a lower value of 2.5 WA
(down from 10 YA with zero input voltage). Now let's apply an input voltage of
the other polarity, to enhance the IGFET:
Vv
ammeter
n-channel igfet characteristic curve
m1 13 0 0 modl1
vin 3 0 dc 0.5
vammeter 2 1 dc 0
vl 20
.model modl nmos vto=-1
.dc vl 0 50 2
.plot dc i(vammeter)
.end
uA — vanmeter#branch
I(vammeter)
“0,0 20,0 40,0 60.0
sweep Y
With the transistor enhanced by the small controlling voltage, the drain current
is now at an increased value of 22.5 WA (2.250E-05). It should be apparent from
these three sets of voltage and current figures that the relationship of drain
current to gate-source voltage is nonlinear just as it was with the JFET. With 1/2
volt of depleting voltage, the drain current is 2.5 WA; with 0 volts input the
drain current goes up to 10 yA; and with 1/2 volt of enhancing voltage, the
current is at 22.5 YA. To obtain a better understanding of this nonlinearity, we
can use SPICE to plot the drain current over a range of input voltage values,
sweeping from a negative (depleting) figure to a positive (enhancing) figure,
maintaining the power supply voltage of V; at a constant value:
n-channel igfet
ml 13 0 0 modl
vin 3 0
vammeter 2 1 dc 0
vl 2 0 dc 24
.model modl nmos vto=-1
.dc vin -1 1 0.1
.plot dc i(vammeter)
.end
uA — vanmeter#branch
I(vammeter)
Just as it was with JFETs, this inherent nonlinearity of the IGFET has the
potential to cause distortion in an amplifier circuit, as the input signal will not
be reproduced with 100 percent accuracy at the output. Also notice that a gate-
source voltage of about 1 volt in the depleting direction is able to pinch off the
channel so that there is virtually no drain current. D-type IGFETs, like JFETs,
have a certain pinch-off voltage rating. This rating varies with the precise
unique of the transistor, and may not be the same as in our simulation here.
Plotting a set of characteristic curves for the IGFET, we see a pattern not unlike
that of the JFET:
V gate-to-source = +0.5V
Tirain /
|
}
/ —_—
| V pate-to-source = OV
| =
| a
I/
/ _
I/ V gate-to-source = -05V
Ey Tain-to-source
¢ REVIEW:
Enhancement-type IGFETs -- PENDING
¢ REVIEW:
Active-mode operation -- PENDING
¢ REVIEW:
The common-source amplifier -- PENDING
¢ REVIEW:
The common-drain amplifier -- PENDING
¢ REVIEW:
The common-gate amplifier -- PENDING
¢ REVIEW:
Biasing techniques -- PENDING
¢ REVIEW:
Transistor ratings and packages -- PENDING
¢ REVIEW:
IGFET quirks -- PENDING
¢ REVIEW:
MESFETs -- PENDING
¢ REVIEW:
IGBTs
Because of their insulated gates, IGFETs of all types have extremely high
Current gain: there can be no sustained gate current if there is no continuous
gate circuit in which electrons may continually flow. The only current we see
through the gate terminal of an IGFET, then, is whatever transient (brief surge)
may be required to charge the gate-channel capacitance and displace the
depletion region as the transistor switches from an "on" state to an "off" state,
or vice versa.
This high current gain would at first seem to place IGFET technology ata
decided advantage over bipolar transistors for the control of very large currents.
If a bipolar junction transistor is used to control a large collector current, there
must be a substantial base current sourced or sunk by some control circuitry, in
accordance with the £ ratio. To give an example, in order for a power BJT with a
B of 20 to conduct a collector current of 100 amps, there must be at least 5
amps of base current, a substantial amount of current in itself for miniature
discrete or integrated control circuitry to handle:
Control
circuitry
It would be nice from the standpoint of control circuitry to have power
transistors with high current gain, so that far less current is needed for control
of load current. Of course, we can use Darlington pair transistors to increase the
current gain, but this kind of arrangement still requires far more controlling
current than an equivalent power IGFET:
Control
circuitry
Control
circuitry
Unfortunately, though, IGFETs have problems of their own controlling high
current: they typically exhibit greater drain-to-source voltage drop while
saturated than the collector-to-emitter voltage drop of a saturated BJT. This
greater voltage drop equates to higher power dissipation for the same amount
of load current, limiting the usefulness of IGFETs as high-power devices.
Although some specialized designs such as the so-called VMOS transistor have
been designed to minimize this inherent disadvantage, the bipolar junction
transistor is still Superior in its ability to switch high currents.
An interesting solution to this dilemma leverages the best features of IGFETs
with the best of features of BJTs, in one device called an /nsulated-Gate Bipolar
Transistor, or IGBT. Also known as an Bipolar-mode MOSFET, a Conductivity-
Modulated Field-Effect Transistor (COMFET), or simply as an /nsulated-Gate
Transistor (IGT), it is equivalent to a Darlington pair of IGFET and BJT:
Insulated-Gate Bipolar Transistor (IGBT)
(N-channel)
Schematic symbols Equivalent circuit
Collector Collector Collector
Gate : q | a
Emitter ica Jae Gate _
Emitter
In essence, the IGFET controls the base current of a BJT, which handles the main
load current between collector and emitter. This way, there is extremely high
Current gain (since the insulated gate of the IGFET draws practically no current
from the control circuitry), but the collector-to-emitter voltage drop during full
conduction is as low as that of an ordinary BJT.
One disadvantage of the IGBT over a standard BJT is its slower turn-off time. For
fast switching and high current-handling capacity, its difficult to beat the
bipolar junction transistor. Faster turn-off times for the IGBT may be achieved
by certain changes in design, but only at the expense of a higher saturated
voltage drop between collector and emitter. However, the IGBT provides a good
alternative to IGFETs and BJTs for high-power control applications.
¢ REVIEW:
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt, under
the terms and conditions of the Design Science License.
Previous Contents Next
— + —$
—/ | 4]
Lessons In Electric Circuits
-- Volume lll
Chapter 7
THYRISTORS
Hysteresis
Gas discharge tubes
The Shockley Diode
The DIAC
The Silicon-Controlled Rectifier (SCR)
The TRIAC
Optothyristors
The Unijunction Transistor (UJT)
The Silicon-Controlled Switch (SCS)
Field-effect-controlled thyristors
Bibliography
Hysteresis
Thyristors are a class of semiconductor components
exhibiting hysteresis, that property whereby a system fails
to return to its original state after some cause of state
change has been removed. A very simple example of
hysteresis is the mechanical action of a toggle switch: when
the lever is pushed, it flips to one of two extreme states
(positions) and will remain there even after the source of
motion is removed (after you remove your hand from the
switch lever). To illustrate the absence of hysteresis,
consider the action of a "momentary" pushbutton switch,
which returns to its original state after the button is no
longer pressed: when the stimulus is removed (your hand),
the system (switch) immediately and fully returns to its prior
state with no "latching" behavior.
Bipolar, junction field-effect, and insulated gate field-effect
transistors are all non-hysteric devices. That is, these do not
inherently "latch" into a state after being stimulated by a
voltage or current signal. For any given input signal at any
given time, a transistor will exhibit a predictable output
response as defined by its characteristic curve. Thyristors,
on the other hand, are semiconductor devices that tend to
stay "on" once turned on, and tend to stay "off" once turned
off. A momentary event is able to flip these devices into
either their on or off states where these will remain that way
on their own, even after the cause of the state change is
taken away. As such, these are useful only as on/off
switching devices -- much like a toggle switch -- and cannot
be used as analog signal amplifiers.
Thyristors are constructed using the same technology as
bipolar junction transistors, and in fact may be analyzed as
circuits comprised of transistor pairs. How then, cana
hysteric device (a thyristor) be made from non-hysteric
devices (transistors)? The answer to this question is positive
feedback, also known as regenerative feedback. As you
should recall, feedback is the condition where a percentage
of the output signal is "fed back" to the input of an
amplifying device. Negative, or degenerative, feedback
results in a diminishing of voltage gain with increases in
stability, linearity, and bandwidth. Positive feedback, on the
other hand, results in a kind of instability where the
amplifier's output tends to "saturate." In the case of
thyristors, this saturating tendency equates to the device
“wanting" to stay on once turned on, and off once turned off.
In this chapter we will explore several different kinds of
thyristors, most of which stem from a single, basic two-
transistor core circuit. Before we do that, though, it would be
beneficial to study the technological predecessor to
thyristors: gas discharge tubes.
Gas discharge tubes
If you've ever witnessed a lightning storm, you've seen
electrical hysteresis in action (and probably didn't realize
what you were seeing). The action of strong wind and rain
accumulates tremendous static electric charges between
cloud and earth, and between clouds as well. Electric charge
imbalances manifest themselves as high voltages, and when
the electrical resistance of air can no longer hold these high
voltages at bay, huge surges of current travel between
opposing poles of electrical charge which we call "lightning."
The buildup of high voltages by wind and rain is a fairly
continuous process, the rate of charge accumulation
increasing under the proper atmospheric conditions.
However, lightning bolts are anything but continuous: they
exist as relatively brief surges rather than continuous
discharges. Why is this? Why don't we see soft, glowing
lightning arcs instead of violently brief lightning bo/ts? The
answer lies in the nonlinear (and hysteric) resistance of air.
Under ordinary conditions, air has an extremely high
amount of resistance. It is so high, in fact, that we typically
treat its resistance as infinite and electrical conduction
through the air as negligible. The presence of water and
dust in air lowers its resistance some, but it is still an
insulator for most practical purposes. When enough high
voltage is applied across a distance of air, though, its
electrical properties change: electrons become "stripped"
from their normal positions around their respective atoms
and are liberated to constitute a current. In this state, air is
considered to be /onized and is called a plasma rather than a
gas. This usage of the word "plasma" is not to be confused
with the medical term (meaning the fluid portion of blood),
but is a fourth state of matter, the other three being solid,
liquid, and vapor (gas). Plasma is a relatively good
conductor of electricity, its specific resistance being much
lower than that of the same substance in its gaseous state.
As an electric current moves through the plasma, there is
energy dissipated in the plasma in the form of heat, just as
current through a solid resistor dissipates energy in the form
of heat. In the case of lightning, the temperatures involved
are extremely high. High temperatures are also sufficient to
convert gaseous air into a plasma or maintain plasma in that
state without the presence of high voltage. As the voltage
between cloud and earth, or between cloud and cloud,
decreases as the charge imbalance is neutralized by the
current of the lightning bolt, the heat dissipated by the bolt
maintains the air path in a plasma state, keeping its
resistance low. The lightning bolt remains a plasma until the
voltage decreases to too low a level to sustain enough
current to dissipate enough heat. Finally, the air returns to a
gaseous state and stops conducting current, thus allowing
voltage to build up once more.
Note how throughout this cycle, the air exhibits hysteresis.
When not conducting electricity, it tends to remain an
insulator until voltage builds up past a critical threshold
point. Then, once it changes state and becomes a plasma, it
tends to remain a conductor until voltage falls below a lower
critical threshold point. Once "turned on" it tends to stay
"on," and once "turned off" it tends to stay "off." This
hysteresis, combined with a steady buildup of voltage due to
the electrostatic effects of wind and rain, explains the action
of lightning as brief bursts.
In electronic terms, what we have here in the action of
lightning is a simple relaxation oscillator. Oscillators are
electronic circuits that produce an oscillating (AC) voltage
from a steady supply of DC power. A relaxation oscillator is
one that works on the principle of a charging capacitor that
is suddenly discharged every time its voltage reaches a
critical threshold value. One of the simplest relaxation
oscillators in existence is comprised of three components
(not counting the DC power supply): a resistor, capacitor,
and neon lamp in Figure below.
Neon lamp
Simple relaxation oscillator
Neon lamps are nothing more than two metal electrodes
inside a sealed glass bulb, separated by the neon gas inside.
At room temperatures and with no applied voltage, the lamp
has nearly infinite resistance. However, once a certain
threshold voltage is exceeded (this voltage depends on the
gas pressure and geometry of the lamp), the neon gas will
become ionized (turned into a plasma) and its resistance
dramatically reduced. In effect, the neon lamp exhibits the
Same characteristics as air in a lightning storm, complete
with the emission of light as a result of the discharge, albeit
on a much smaller scale.
The capacitor in the relaxation oscillator circuit shown above
charges at an inverse exponential rate determined by the
size of the resistor. When its voltage reaches the threshold
voltage of the lamp, the lamp suddenly "turns on" and
quickly discharges the capacitor to a low voltage value.
Once discharged, the lamp "turns off" and allows the
Capacitor to build up a charge once more. The result is a
series of brief flashes of light from the lamp, the rate of
which is dictated by battery voltage, resistor resistance,
Capacitor capacitance, and lamp threshold voltage.
While gas-discharge lamps are more commonly used as
sources of illumination, their hysteric properties were
leveraged in slightly more sophisticated variants known as
thyratron tubes. Essentially a gas-filled triode tube (a triode
being a three-element vacuum electron tube performing
much a similar function to the N-channel, D-type IGFET), the
thyratron tube could be turned on with a small control
voltage applied between grid and cathode, and turned off by
reducing the plate-to-cathode voltage.
high voltage
AC source
control
voltage
Simple thyratron contro! circuit
In essence, thyratron tubes were controlled versions of neon
lamps built specifically for switching current to a load. The
dot inside the circle of the schematic symbol indicates a gas
fill, as opposed to the hard vacuum normally seen in other
electron tube designs. In the circuit shown above in Figure
above. the thyratron tube allows current through the load in
one direction (note the polarity across the load resistor)
when triggered by the small DC control voltage connected
between grid and cathode. Note that the load's power
source is AC, which provides a clue about how the thyratron
turns off after its been triggered on: since AC voltage
periodically passes through a condition of 0 volts between
half-cycles, the current through an AC-powered load must
also periodically halt. This brief pause of current between
half-cycles gives the tube's gas time to cool, letting it return
to its normal "off" state. Conduction may resume only if
enough voltage is applied by the AC power source (some
other time in the wave's cycle) and if the DC control voltage
allows it.
An oscilloscope display of load voltage in such a circuit
would look something like Figure below.
Threshold voltage
Load voltage
AC supply voltage
Thyratron waveforms
As the AC supply voltage climbs from zero volts to its first
peak, the load voltage remains at zero (no load current) until
the threshold voltage is reached. At that point, the tube
switches "on" and begins to conduct, the load voltage now
following the AC voltage through the rest of the half cycle.
Load voltage exists (and thus load current) even when the
AC voltage waveform has dropped below the threshold value
of the tube. This is hysteresis at work: the tube stays in its
conductive mode past the point where it first turned on,
continuing to conduct until there the supply voltage drops
off to almost zero volts. Because thyratron tubes are one-
way (diode) devices, no voltage develops across the load
through the negative half-cycle of AC. In practical thyratron
circuits, multiple tubes arranged in some form of full-wave
rectifier circuit to facilitate full-wave DC power to the load.
The thyratron tube has been applied to a relaxation
oscillator circuit. [VTS] The frequency is controlled by a
small DC voltage between grid and cathode. (See Figure
below) This voltage-controlled oscillator is known as a VCO.
Relaxation oscillators produce a very non-sinusoidal output,
and they exist mostly as demonstration circuits (as is the
case here) or in applications where the harmonic rich
waveform is desirable. [MET]
Controlling
voltage
Voltage controlled thyratron relaxation oscillator
| speak of thyratron tubes in the past tense for good reason:
modern semiconductor components have obsoleted
thyratron tube technology for all but a few very special
applications. It is no coincidence that the word thyristor
bears so much similarity to the word thyratron, for this class
of semiconductor components does much the same thing:
use hAysteretically switch current on and off. It is these
modern devices that we now turn our attention to.
¢ REVIEW:
Electrical hysteresis, the tendency for a component to
remain "on" (conducting) after it begins to conduct and
to remain "off" (nonconducting) after it ceases to
conduct, helps to explain why lightning bolts exist as
momentary surges of current rather than continuous
discharges through the air.
e Simple gas-discharge tubes such as neon lamps exhibit
electrical hysteresis.
e More advanced gas-discharge tubes have been made
with control elements so that their "turn-on" voltage
could be adjusted by an external signal. The most
common of these tubes was called the thyratron.
e Simple oscillator circuits called relaxation oscillators
may be created with nothing more than a resistor-
Capacitor charging network and a hysteretic device
connected across the capacitor.
The Shockley Diode
Our exploration of thyristors begins with a device called the
four-layer diode, also Known as a PNPN diode, or a Shockley
diode after its inventor, William Shockley. This is not to be
confused with a Schottky diode, that two-layer metal-
semiconductor device known for its high switching speed. A
crude illustration of the Shockley diode, often seen in
textbooks, is a four-layer sandwich of P-N-P-N semiconductor
material, Figure below.
_ Anode
Cathode
aa
Shockley or 4-layer diode
Unfortunately, this simple illustration does nothing to
enlighten the viewer on how it works or why. Consider an
alternative rendering of the device's construction in Figure
below.
se
Anode
Cathode
—
Transistor equivalent of Shockley diode
Shown like this, it appears to be a set of interconnected
bipolar transistors, one PNP and the other NPN. Drawn using
standard schematic symbols, and respecting the layer
doping concentrations not shown in the last image, the
Shockley diode looks like this (Figure below)
Anode a
Cathode Cathode
Physical diagram Equivalent schematic Schematic symbol
Shockley diode: physical diagram, equivalent schematic
diagram, and schematic symbol.
Let's connect one of these devices to a source of variable
voltage and see what happens: (Figure below)
Powered Shockley diode equivalent circuit.
With no voltage applied, of course there will be no current.
As voltage is initially increased, there will still be no current
because neither transistor is able to turn on: both will be in
cutoff mode. To understand why this is, consider what it
takes to turn a bipolar junction transistor on: current
through the base-emitter junction. As you can see in the
diagram, base current through the lower transistor is
controlled by the upper transistor, and the base current
through the upper transistor is controlled by the lower
transistor. In other words, neither transistor can turn on until
the othertransistor turns on. What we have here, in
vernacular terms, is Known as a Catch-22.
So how can a Shockley diode ever conduct current, if its
constituent transistors stubbornly maintain themselves in a
state of cutoff? The answer lies in the behavior of rea/
transistors as opposed to /dea/ transistors. An ideal bipolar
transistor will never conduct collector current if no base
current flows, no matter how much or little voltage we apply
between collector and emitter. Real transistors, on the other
hand, have definite limits to how much collector-emitter
voltage each can withstand before one breaks down and
conduct. If two real transistors are connected in this fashion
to form a Shockley diode, each one will conduct if sufficient
voltage is applied by the battery between anode and
cathode to cause one of them to break down. Once one
transistor breaks down and begins to conduct, it will allow
base current through the other transistor, causing it to turn
on in a normal fashion, which then allows base current
through the first transistor. The end result is that both
transistors will be saturated, now keeping each other turned
on instead of off.
So, we can force a Shockley diode to turn on by applying
sufficient voltage between anode and cathode. As we have
seen, this will inevitably cause one of the transistors to turn
on, which then turns the other transistor on, ultimately
"latching" both transistors on where each will tend to
remain. But how do we now get the two transistors to turn
off again? Even if the applied voltage is reduced to a point
well below what it took to get the Shockley diode
conducting, it will remain conducting because both
transistors now have base current to maintain regular,
controlled conduction. The answer to this is to reduce the
applied voltage to a much lower point where too little
current flows to maintain transistor bias, at which point one
of the transistors will cutoff, which then halts base current
through the other transistor, sealing both transistors in the
"off" state as each one was before any voltage was applied
at all.
If we graph this sequence of events and plot the results on
an I/V graph, the hysteresis is evident. First, we will observe
the circuit as the DC voltage source (battery) is set to zero
voltage: (Figure below)
| Circuit
current
Applied voltage
Zero applied voltage; zero current
Next, we will steadily increase the DC voltage. Current
through the circuit is at or nearly at zero, as the breakdown
limit has not been reached for either transistor: (Figure
below)
| Circuit
current
Applied voltage
Some applied voltage; still no current
When the voltage breakdown limit of one transistor is
reached, it will begin to conduct collector current even
though no base current has gone through it yet. Normally,
this sort of treatment would destroy a bipolar junction
transistor, but the PNP junctions comprising a Shockley
diode are engineered to take this kind of abuse, similar to
the way a Zener diode is built to handle reverse breakdown
without sustaining damage. For the sake of illustration I'll
assume the lower transistor breaks down first, sending
current through the base of the upper transistor: (Figure
below)
Circuit
i; current
Applied voltage
More voltage applied; lower transistor breaks down
As the upper transistor receives base current, it turns on as
expected. This action allows the lower transistor to conduct
normally, the two transistors "sealing" themselves in the
"on" state. Full current is quickly seen in the circuit: (Figure
below)
Circuit
current
Applied voltage
Transistors are now fully conducting.
The positive feedback mentioned earlier in this chapter is
clearly evident here. When one transistor breaks down, it
allows current through the device structure. This current
may be viewed as the "output" signal of the device. Once an
output current is established, it works to hold both
transistors in saturation, thus ensuring the continuation of a
substantial output current. In other words, an output current
"feeds back" positively to the input (transistor base current)
to keep both transistors in the "on" state, thus reinforcing (or
regenerating) itself.
With both transistors maintained in a state of saturation with
the presence of ample base current, each will continue to
conduct even if the applied voltage is greatly reduced from
the breakdown level. The effect of positive feedback is to
keep both transistors in a state of saturation despite the loss
of input stimulus (the original, high voltage needed to break
down one transistor and cause a base current through the
other transistor): (Figure below)
Circuit
current
Applied voltage
Current maintained even when voltage is reduced
If the DC voltage source is turned down too far, though, the
circuit will eventually reach a point where there isn't enough
Current to sustain both transistors in saturation. As one
transistor passes less and less collector current, it reduces
the base current for the other transistor, thus reducing base
current for the first transistor. The vicious cycle continues
rapidly until both transistors fall into cutoff: (Figure below)
Circuit
current
Applied voltage
If voltage drops too low, both transistors shut off.
Here, positive feedback is again at work: the fact that the
cause/effect cycle between both transistors is "vicious" (a
decrease in current through one works to decrease current
through the other, further decreasing current through the
first transistor) indicates a positive relationship between
output (controlled current) and input (controlling current
through the transistors' bases).
The resulting curve on the graph is classically hysteretic: as
the input signal (voltage) is increased and decreased, the
output (current) does not follow the same path going down
as it did going up: (Figure below)
Circuit
current
Applied voltage
Hysteretic curve
Put in simple terms, the Shockley diode tends to stay on
once its turned on, and stay off once its turned off. No "in-
between" or "active" mode in its operation: it is a purely on
or off device, as are all thyristors.
A few special terms apply to Shockley diodes and all other
thyristor devices built upon the Shockley diode foundation.
First is the term used to describe its "on" state: /atched. The
word "latch" is reminiscent of a door lock mechanism, which
tends to keep the door closed once it has been pushed shut.
The term firing refers to the initiation of a latched state. To
get a Shockley diode to latch, the applied voltage must be
increased until breakover is attained. Though this action is
best described as transistor breakdown, the term breakover
is used instead because the result is a pair of transistors in
mutual saturation rather than destruction of the transistor. A
latched Shockley diode is re-set back into its nonconducting
state by reducing current through it until /ow-current
dropout occurs.
Note that Shockley diodes may be fired in a way other than
breakover: excessive vo/tage rise, or dv/dt. If the applied
voltage across the diode increases at a high rate of change,
it may trigger. This is able to cause latching (turning on) of
the diode due to inherent junction capacitances within the
transistors. Capacitors, as you may recall, oppose changes in
voltage by drawing or supplying current. If the applied
voltage across a Shockley diode rises at too fast a rate, those
tiny capacitances will draw enough current during that time
to activate the transistor pair, turning them both on. Usually,
this form of latching is undesirable, and can be minimized
by filtering high-frequency (fast voltage rises) from the
diode with series inductors and parallel resistor-capacitor
networks called snubbers: (Figure below)
Series inductor
Shockley RC "snubber"
diode
Both the series inductor and parallel resistor-capacitor
“snubber” circuit help minimize the Shockley diode's
exposure to excessively rising voltage.
The voltage rise limit of a Shockley diode is referred to as
the critical rate of voltage rise. Manufacturers usually
provide this specification for the devices they sell.
¢ REVIEW:
Shockley diodes are four-layer PNPN semiconductor
devices. These behave as a pair of interconnected PNP
and NPN transistors.
Like all thyristors, Shockley diodes tend to stay on once
turned on (/atchea), and stay off once turned off.
e To latch a Shockley diode exceed the anode-to-cathode
breakover voltage, or exceed the anode-to-cathode
critical rate of voltage rise.
e To cause a Shockley diode to stop conducting, reduce
the current going through it to a level below its /ow-
current dropout threshold.
The DIAC
Like all diodes, Shockley diodes are unidirectional devices;
that is, these only conduct current in one direction. If
bidirectional (AC) operation is desired, two Shockley diodes
may be joined in parallel facing different directions to form a
new kind of thyristor, the D/AC: (Figure below)
%
DIAC equivalent circuit DIAC schematic symbol
The DIAC
A DIAC operated with a DC voltage across it behaves exactly
the same as a Shockley diode. With AC, however, the
behavior is different from what one might expect. Because
alternating current repeatedly reverses direction, DIACs will
not stay latched longer than one-half cycle. If a DIAC
becomes latched, it will continue to conduct current only as
long as voltage is available to push enough current in that
direction. When the AC polarity reverses, as it must twice
per cycle, the DIAC will drop out due to insufficient current,
necessitating another breakover before it conducts again.
The result is the current waveform in Figure below.
Breakover voltage
DIAC current
AC supply voltage Breakover voltage
DIAC waveforms
DIACs are almost never used alone, but in conjunction with
other thyristor devices.
The Silicon-Controlled Rectifier (SCR)
Shockley diodes are curious devices, but rather limited in
application. Their usefulness may be expanded, however, by
equipping them with another means of latching. In doing so,
each becomes true amplifying devices (if only in an on/off
mode), and we refer to these as si/icon-controlled rectifiers,
or SCRs.
The progression from Shockley diode to SCR is achieved with
one small addition, actually nothing more than a third wire
connection to the existing PNPN structure: (Figure below)
Anode ie
Anode
Gate Gate x
Cathode
Cathode” Cathode
Physical diagram Equivalent schematic Schematic symbol
The Silicon-Controlled Rectifier (SCR)
If an SCR's gate is left floating (disconnected), it behaves
exactly as a Shockley diode. It may be latched by breakover
voltage or by exceeding the critical rate of voltage rise
between anode and cathode, just as with the Shockley
diode. Dropout is accomplished by reducing current until
one or both internal transistors fall into cutoff mode, also
like the Shockley diode. However, because the gate terminal
connects directly to the base of the lower transistor, it may
be used as an alternative means to latch the SCR. By
applying a small voltage between gate and cathode, the
lower transistor will be forced on by the resulting base
current, which will cause the upper transistor to conduct,
which then supplies the lower transistor's base with current
so that it no longer needs to be activated by a gate voltage.
The necessary gate current to initiate latch-up, of course,
will be much lower than the current through the SCR from
cathode to anode, so the SCR does achieve a measure of
amplification.
This method of securing SCR conduction is called triggering,
and it is by far the most common way that SCRs are latched
in actual practice. In fact, SCRs are usually chosen so that
their breakover voltage is far beyond the greatest voltage
expected to be experienced from the power source, so that it
can be turned on only by an intentional voltage pulse
applied to the gate.
It should be mentioned that SCRs may sometimes be turned
off by directly shorting their gate and cathode terminals
together, or by "reverse-triggering" the gate with a negative
voltage (in reference to the cathode), so that the lower
transistor is forced into cutoff. | say this is "sometimes"
possible because it involves shunting all of the upper
transistor's collector current past the lower transistor's base.
This current may be substantial, making triggered shut-off of
an SCR difficult at best. A variation of the SCR, called a
Gate-Turn-Off thyristor, or GTO, makes this task easier. But
even with a GTO, the gate current required to turn it off may
be as much as 20% of the anode (load) current! The
schematic symbol for a GTO is shown in the following
illustration: (Figure below)
Anode
Gate Xx
Cathode
The Gate Turn-Off thyristor (GTO)
SCRs and GTOs share the same equivalent schematics (two
transistors connected in a positive-feedback fashion), the
only differences being details of construction designed to
grant the NPN transistor a greater B than the PNP. This
allows a smaller gate current (forward or reverse) to exert a
greater degree of control over conduction from cathode to
anode, with the PNP transistor's latched state being more
dependent upon the NPN's than vice versa. The Gate-Turn-
Off thyristor is also Known by the name of Gate-Controlled
Switch, or GCS.
A rudimentary test of SCR function, or at least terminal
identification, may be performed with an ohmmeter.
Because the internal connection between gate and cathode
iS a Single PN junction, a meter should indicate continuity
between these terminals with the red test lead on the gate
and the black test lead on the cathode like this: (Figure
below)
gate x
cathode
[+ | [cue
Rudimentary test of SCR
All other continuity measurements performed on an SCR will
show "open" ("OL" on some digital multimeter displays). It
must be understood that this test is very crude and does not
constitute a comprehensive assessment of the SCR. It is
possible for an SCR to give good ohmmeter indications and
still be defective. Ultimately, the only way to test an SCR is
to subject it to a load current.
If you are using a multimeter with a "diode check" function,
the gate-to-cathode junction voltage indication you get may
or may not correspond to what's expected of a silicon PN
junction (approximately 0.7 volts). In some cases, you will
read a much lower junction voltage: mere hundredths of a
volt. This is due to an internal resistor connected between
the gate and cathode incorporated within some SCRs. This
resistor is added to make the SCR less susceptible to false
triggering by spurious voltage spikes, from circuit "noise" or
from static electric discharge. In other words, having a
resistor connected across the gate-cathode junction requires
that a strong triggering signal (substantial current) be
applied to latch the SCR. This feature is often found in larger
SCRs, not on small SCRs. Bear in mind that an SCR with an
internal resistor connected between gate and cathode will
indicate continuity in both directions between those two
terminals: (Figure below)
Anode
Gate
Gate-to-Cathode
resistor Cathode
Larger SCRs have gate to cathode resistor.
"Normal" SCRs, lacking this internal resistor, are sometimes
referred to as sensitive gate SCRs due to their ability to be
triggered by the slightest positive gate signal.
The test circuit for an SCR is both practical as a diagnostic
tool for checking suspected SCRs and also an excellent aid
to understanding basic SCR operation. A DC voltage source
is used for powering the circuit, and two pushbutton
switches are used to latch and unlatch the SCR, respectively:
(Figure below)
off
a SCR und
—_ unaer
= test
SCR testing circuit
Actuating the normally-open "on" pushbutton switch
connects the gate to the anode, allowing current from the
negative terminal of the battery, through the cathode-gate
PN junction, through the switch, through the load resistor,
and back to the battery. This gate current should force the
SCR to latch on, allowing current to go directly from cathode
to anode without further triggering through the gate. When
the "on" pushbutton is released, the load should remain
energized.
Pushing the normally-closed "off" pushbutton switch breaks
the circuit, forcing current through the SCR to halt, thus
forcing it to turn off (low-current dropout).
If the SCR fails to latch, the problem may be with the load
and not the SCR. A certain minimum amount of load current
is required to hold the SCR latched in the "on" state. This
minimum current level is called the holding current. A load
with too great a resistance value may not draw enough
current to keep an SCR latched when gate current ceases,
thus giving the false impression of a bad (unlatchable) SCR
in the test circuit. Holding current values for different SCRs
should be available from the manufacturers. Typical holding
current values range from 1 milliamp to 50 milliamps or
more for larger units.
For the test to be fully comprehensive, more than the
triggering action needs to be tested. The forward breakover
voltage limit of the SCR could be tested by increasing the
DC voltage supply (with no pushbuttons actuated) until the
SCR latches all on its own. Beware that a breakover test may
require very high voltage: many power SCRs have breakover
voltage ratings of 600 volts or more! Also, if a pulse voltage
generator is available, the critical rate of voltage rise for the
SCR could be tested in the same way: subject it to pulsing
supply voltages of different V/time rates with no pushbutton
switches actuated and see when it latches.
In this simple form, the SCR test circuit could suffice as a
start/stop control circuit for a DC motor, lamp, or other
practical load: (Figure below)
Motor off
SCR under
test
DC motor start/stop control circuit
Another practical use for the SCR in a DC circuit is asa
crowbar device for overvoltage protection. A "crowbar"
circuit consists of an SCR placed in parallel with the output
of a DC power supply, for placing a direct short-circuit on the
output of that supply to prevent excessive voltage from
reaching the load. Damage to the SCR and power supply is
prevented by the judicious placement of a fuse or
substantial series resistance ahead of the SCR to limit short-
circuit current: (Figure below)
Transformer
Ff pa
source
Fuse Load
_ Crowbar .
(triggering circuit
omitted for simplicity)
Crowbar circuit used in DC power supply
Some device or circuit sensing the output voltage will be
connected to the gate of the SCR, so that when an
overvoltage condition occurs, voltage will be applied
between the gate and cathode, triggering the SCR and
forcing the fuse to blow. The effect will be approximately the
Same as dropping a solid steel crowbar directly across the
output terminals of the power supply, hence the name of the
Circuit.
Most applications of the SCR are for AC power control,
despite the fact that SCRs are inherently DC (unidirectional)
devices. If bidirectional circuit current is required, multiple
SCRs may be used, with one or more facing each direction to
handle current through both half-cycles of the AC wave. The
primary reason SCRs are used at all for AC power control
applications is the unique response of a thyristor to an
alternating current. As we saw, the thyratron tube (the
electron tube version of the SCR) and the DIAC, a hysteretic
device triggered on during a portion of an AC half-cycle will
latch and remain on throughout the remainder of the half-
cycle until the AC current decreases to zero, as it must to
begin the next half-cycle. Just prior to the zero-crossover
point of the current waveform, the thyristor will turn off due
to insufficient current (this behavior is also Known as natural
commutation) and must be fired again during the next
cycle. The result is a circuit current equivalent to a "chopped
up" sine wave. For review, here is the graph of a DIAC's
response to an AC voltage whose peak exceeds the
breakover voltage of the DIAC: (Figure below)
Breakover voltage
DIAC current
AC supply voltage Breakover voltage
DIAC bidirectional response
With the DIAC, that breakover voltage limit was a fixed
quantity. With the SCR, we have control over exactly when
the device becomes latched by triggering the gate at any
point in time along the waveform. By connecting a suitable
control circuit to the gate of an SCR, we can "chop" the sine
wave at any point to allow for time-proportioned power
control to a load.
Take the circuit in Figure below as an example. Here, an SCR
is positioned in a circuit to control power to a load from an
AC source.
Load
AC
source SCR
SCR control of AC power
Being a unidirectional (one-way) device, at most we can only
deliver half-wave power to the load, in the half-cycle of AC
where the supply voltage polarity is positive on the top and
negative on the bottom. However, for demonstrating the
basic concept of time-proportional control, this simple circuit
is better than one controlling full-wave power (which would
require two SCRs).
With no triggering to the gate, and the AC source voltage
well below the SCR's breakover voltage rating, the SCR will
never turn on. Connecting the SCR gate to the anode
through a standard rectifying diode (to prevent reverse
current through the gate in the event of the SCR containing
a built-in gate-cathode resistor), will allow the SCR to be
triggered almost immediately at the beginning of every
positive half-cycle: (Figure below)
Load
AC
source
— Load current —
Gate connected directly to anode through a diode; nearly
complete half-wave current through load.
We can delay the triggering of the SCR, however, by
inserting some resistance into the gate circuit, thus
increasing the amount of voltage drop required before
enough gate current triggers the SCR. In other words, if we
make it harder for electrons to flow through the gate by
adding a resistance, the AC voltage will have to reach a
higher point in its cycle before there will be enough gate
current to turn the SCR on. The result is in Figure below.
Load
AC
source
Load current
AC source voltage
Resistance inserted in gate circuit; less than half-wave
current through load.
With the half-sine wave chopped up to a greater degree by
delayed triggering of the SCR, the load receives less average
power (power is delivered for less time throughout a cycle).
By making the series gate resistor variable, we can make
adjustments to the time-proportioned power: (Figure below)
Load
AC
source
trigger
threshold
Increasing the resistance raises the threshold level, causing
less power to be delivered to the load. Decreasing the
resistance lowers the threshold level, causing more power to
be delivered to the load.
Unfortunately, this control scheme has a significant
limitation. In using the AC source waveform for our SCR
triggering signal, we limit control to the first half of the
waveform's half-cycle. In other words, it is not possible for us
to wait until afterthe wave's peak to trigger the SCR. This
means we can turn down the power only to the point where
the SCR turns on at the very peak of the wave: (Figure
below)
Load
AC
source
trigger
threshold
Circuit at minimum power setting
Raising the trigger threshold any more will cause the circuit
to not trigger at all, since not even the peak of the AC power
voltage will be enough to trigger the SCR. The result will be
no power to the load.
An ingenious solution to this control dilemma is found in the
addition of a phase-shifting capacitor to the circuit: (Figure
below)
Load
AC
source
COOL
Capacitor voltage
Addition of a phase-shifting capacitor to the circuit
The smaller waveform shown on the graph is voltage across
the capacitor. For the sake of illustrating the phase shift, I'm
assuming a condition of maximum control resistance where
the SCR is not triggering at all with no load current, save for
what little current goes through the control resistor and
capacitor. This capacitor voltage will be phase-shifted
anywhere from 0° to 902 lagging behind the power source
AC waveform. When this phase-shifted voltage reaches a
high enough level, the SCR will trigger.
With enough voltage across the capacitor to periodically
trigger the SCR, the resulting load current waveform will
look something like Figure below)
Load
AC
source
trigger
oad thrag old
Capacitor voltage
Phase-shifted signal triggers SCR into conduction.
Because the capacitor waveform is still rising after the main
AC power waveform has reached its peak, it becomes
possible to trigger the SCR at a threshold level beyond that
peak, thus chopping the load current wave further than it
was possible with the simpler circuit. In reality, the capacitor
voltage waveform is a bit more complex that what is shown
here, its sinusoidal shape distorted every time the SCR
latches on. However, what I'm trying to illustrate here is the
delayed triggering action gained with the phase-shifting RC
network; thus, a simplified, undistorted waveform serves the
purpose well.
SCRs may also be triggered, or "fired," by more complex
circuits. While the circuit previously shown is sufficient for a
simple application like a lamp control, large industrial motor
controls often rely on more sophisticated triggering
methods. Sometimes, pulse transformers are used to couple
a triggering circuit to the gate and cathode of an SCR to
provide electrical isolation between the triggering and
power circuits: (Figure below)
pulse SCR
fanamanei
to triggering
circuit
Transformer coupling of trigger signal provides isolation.
to power
circuit
When multiple SCRs are used to control power, their
cathodes are often not electrically common, making it
difficult to connect a single triggering circuit to all SCRs
equally. An example of this is the controlled bridge rectifier
shown in Figure below.
SCR,
Load
Controlled bridge rectifier
In any bridge rectifier circuit, the rectifying diodes (in this
example, the rectifying SCRs) must conduct in opposite
pairs. SCR; and SCR3 must be fired simultaneously, and
SCR, and SCR, must be fired together as a pair. As you will
notice, though, these pairs of SCRs do not share the same
cathode connections, meaning that it would not work to
simply parallel their respective gate connections and
connect a single voltage source to trigger both: (Figure
below)
triggering
SCR, voltage
(pulse voltage
> source)
Load
This strategy will not work for triggering SCR> and SCR, as a
pair.
Although the triggering voltage source shown will trigger
SCRy, it will not trigger SCR» properly because the two
thyristors do not share a common cathode connection to
reference that triggering voltage. Pulse transformers
connecting the two thyristor gates to a common triggering
voltage source wi// work, however: (Figure below)
pulse
voltage
source
Transformer coupling of the gates allows triggering of SCR>
and SCR, .
Bear in mind that this circuit only shows the gate
connections for two out of the four SCRs. Pulse transformers
and triggering sources for SCR, and SCR3, as well as the
details of the pulse sources themselves, have been omitted
for the sake of simplicity.
Controlled bridge rectifiers are not limited to single-phase
designs. In most industrial control systems, AC power is
available in three-phase form for maximum efficiency, and
solid-state control circuits are built to take advantage of
that. A three-phase controlled rectifier circuit built with
SCRs, without pulse transformers or triggering circuitry
shown, would look like Figure below.
3-phase source
Controlled
rectifier
4
Load
Three-phase bridge SCR control of load
e REVIEW:
e A Silicon-Controlled Rectifier, or SCR, is essentially a
Shockley diode with an extra terminal added. This extra
terminal is called the gate, and it is used to trigger the
device into conduction (latch it) by the application of a
small voltage.
To trigger, or fire, an SCR, voltage must be applied
between the gate and cathode, positive to the gate and
negative to the cathode. When testing an SCR, a
momentary connection between the gate and anode is
sufficient in polarity, intensity, and duration to trigger it.
SCRs may be fired by intentional triggering of the gate
terminal, excessive voltage (breakdown) between anode
and cathode, or excessive rate of voltage rise between
anode and cathode. SCRs may be turned off by anode
current falling below the holding current value (low-
Current dropout), or by "reverse-firing" the gate
(applying a negative voltage to the gate). Reverse-firing
is only sometimes effective, and always involves high
gate current.
A variant of the SCR, called a Gate-Turn-Off thyristor
(GTO), is specifically designed to be turned off by means
of reverse triggering. Even then, reverse triggering
requires fairly high current: typically 20% of the anode
current.
SCR terminals may be identified by a continuity meter:
the only two terminals showing any continuity between
them at all should be the gate and cathode. Gate and
cathode terminals connect to a PN junction inside the
SCR, so a continuity meter should obtain a diode-like
reading between these two terminals with the red (+)
lead on the gate and the black (-) lead on the cathode.
Beware, though, that some large SCRs have an internal
resistor connected between gate and cathode, which will
affect any continuity readings taken by a meter.
SCRs are true rectifiers: they only allow current through
them in one direction. This means they cannot be used
alone for full-wave AC power control.
If the diodes in a rectifier circuit are replaced by SCRs,
you have the makings of a contro//ed rectifier circuit,
whereby DC power to a load may be time-proportioned
by triggering the SCRs at different points along the AC
power waveform.
The TRIAC
SCRs are unidirectional (one-way) current devices, making
them useful for controlling DC only. If two SCRs are joined in
back-to-back parallel fashion just like two Shockley diodes
were joined together to form a DIAC, we have a new device
known as the TRIAC: (Figure below)
Main Terminal 2
(MT,)
Main Terminal 2
(MT)
Gate Gate
Main Terminal 1
(MT,)
Main Terminal 1
(MT,)
TRIAC equivalent circuit TRIAC schematic symbol
The TRIAC SCR equivalent and, TRIAC schematic symbol
Because individual SCRs are more flexible to use in
advanced control systems, these are more commonly seen in
circuits like motor drives; TRIACs are usually seen in simple,
low-power applications like household dimmer switches. A
simple lamp dimmer circuit is shown in Figure below,
complete with the phase-shifting resistor-capacitor network
necessary for after-peak firing.
Lamp
AC
source
TRIAC phase-control of power
TRIACs are notorious for not firing symmetrically. This means
these usually won't trigger at the exact same gate voltage
level for one polarity as for the other. Generally speaking,
this is undesirable, because unsymmetrical firing results in a
current waveform with a greater variety of harmonic
frequencies. Waveforms that are symmetrical above and
below their average centerlines are comprised of only odd-
numbered harmonics. Unsymmetrical waveforms, on the
other hand, contain even-numbered harmonics (which may
or may not be accompanied by odd-numbered harmonics as
well).
In the interest of reducing total harmonic content in power
systems, the fewer and less diverse the harmonics, the
better -- one more reason individual SCRs are favored over
TRIACs for complex, high-power control circuits. One way to
make the TRIAC's current waveform more symmetrical is to
use a device external to the TRIAC to time the triggering
pulse. A DIAC placed in series with the gate does a fair job of
this: (Figure below)
Lamp
AC
source
DIAC improves symmetry of control
DIAC breakover voltages tend to be much more symmetrical
(the same in one polarity as the other) than TRIAC triggering
voltage thresholds. Since the DIAC prevents any gate
current until the triggering voltage has reached a certain,
repeatable level in either direction, the firing point of the
TRIAC from one half-cycle to the next tends to be more
consistent, and the waveform more symmetrical above and
below its centerline.
Practically all the characteristics and ratings of SCRs apply
equally to TRIACs, except that TRIACs of course are
bidirectional (can handle current in both directions). Not
much more needs to be said about this device except for an
important caveat concerning its terminal designations.
From the equivalent circuit diagram shown earlier, one
might think that main terminals 1 and 2 were
interchangeable. These are not! Although it is helpful to
imagine the TRIAC as being composed of two SCRs joined
together, it in fact is constructed from a single piece of
semiconducting material, appropriately doped and layered.
The actual operating characteristics may differ slightly from
that of the equivalent model.
This is made most evident by contrasting two simple circuit
designs, one that works and one that doesn't. The following
two circuits are a variation of the lamp dimmer circuit shown
earlier, the phase-shifting capacitor and DIAC removed for
simplicity's sake. Although the resulting circuit lacks the fine
control ability of the more complex version (with capacitor
and DIAC), it does function: (Figure below)
Lamp
AC
source
This circuit with the gate to MT> does function.
Suppose we were to swap the two main terminals of the
TRIAC around. According to the equivalent circuit diagram
shown earlier in this section, the swap should make no
difference. The circuit ought to work: (Figure below)
Lamp
AC
source
With the gate swapped to MT,, this circuit does not function.
However, if this circuit is built, it will be found that it does
not work! The load will receive no power, the TRIAC refusing
to fire at all, no matter how low or high a resistance value
the control resistor is set to. The key to successfully
triggering a TRIAC is to make sure the gate receives its
triggering current from the main terminal 2 side of the
circuit (the main terminal on the opposite side of the TRIAC
symbol from the gate terminal). Identification of the MT, and
MT, terminals must be done via the TRIAC's part number
with reference to a data sheet or book.
¢ REVIEW:
e A TRIAC acts much like two SCRs connected back-to-
back for bidirectional (AC) operation.
e TRIAC controls are more often seen in simple, low-power
circuits than complex, high-power circuits. In large
power control circuits, multiple SCRs tend to be favored.
e When used to control AC power to a load, TRIACs are
often accompanied by DIACs connected in series with
their gate terminals. The DIAC helps the TRIAC fire more
symmetrically (more consistently from one polarity to
another).
e Main terminals 1 and 2 on a TRIAC are not
interchangeable.
e To successfully trigger a TRIAC, gate current must come
from the main terminal 2 (MT>) side of the circuit!
Optothyristors
Like bipolar transistors, SCRs and TRIACs are also
manufactured as light-sensitive devices, the action of
impinging light replacing the function of triggering voltage.
Optically-controlled SCRs are often known by the acronym
LASCR, or Light Activated SCR. Its symbol, not surprisingly,
looks like Figure below.
Light Activated SCR
yy
LASCR
Light activated SCR
Optically-controlled TRIACs don't receive the honor of
having their own acronym, but instead are humbly known as
opto-TRIACs. Their schematic symbol is shown in Figure
below.
Opto-TRIAC
a
Opto-TRIAC
Optothyristors (a general term for either the LASCR or the
opto-TRIAC) are commonly found inside sealed
“optoisolator" modules.
The Unijunction Transistor (UJT)
Unijunction transistor: Although a unijunction transistor
is not a thyristor, this device can trigger larger thyristors
with a pulse at base B1. A unijunction transistor is composed
of a bar of N-type silicon having a P-type connection in the
middle. See Figure below(a). The connections at the ends of
the bar are known as bases B1 and B2; the P-type mid-point
is the emitter. With the emitter disconnected, the total
resistance Rego, a datasheet item, is the sum of Rp; and Rp>
as shown in Figure below(b). Rego ranges from 4-12kQ for
different device types. The intrinsic standoff ratio n is the
ratio of Rg; to Rego. It varies from 0.4 to 0.8 for different
devices. The schematic symbol is Figure below(c)
Repo = Rgi + Rp2
Ry>
: R
° N= R =
Ra BI B2 7
Bl
oe Rg
Bl ie Rgpo
(a) (b)
(c)
Unijunction transistor: (a) Construction, (b) Model, (c)
Symbol
The Unijunction emitter current vs voltage characteristic
curve (Figure below(a) ) shows that as V- increases, current
I- increases up Ip at the peak point. Beyond the peak point,
Current increases as voltage decreases in the negative
resistance region. The voltage reaches a minimum at the
valley point. The resistance of Rg), the saturation resistance
is lowest at the valley point.
lp and ly, are datasheet parameters; For a 2n2647, Ip and ly
are 2UA and 4mA, respectively. [AMS] Vp is the voltage drop
across Rg, plus a 0.7V diode drop; see Figure below(b). Vy is
estimated to be approximately 10% of Vpp.
Unijunction transistor: (a) emitter characteristic curve, (b)
model for Vp.
The relaxation oscillator in Figure below is an application of
the unijunction oscillator. Re charges C- until the peak point.
The unijunction emitter terminal has no effect on the
Capacitor until this point is reached. Once the capacitor
voltage, Ve, reaches the peak voltage point Vp, the lowered
emitter-basel E-B1 resistance quickly discharges the
capacitor. Once the capacitor discharges below the valley
point Vy, the E-RB1 resistance reverts back to high
resistance, and the capacitor is free to charge again.
2n2647 Rypo =4.7—9.1k 1 =0.68—0.82 Iy=8mA I,=20A
f= RCIMUAI-Hy) ~ (MO0kylOnF)In(IAl-0.75))
= 1.39kHz
Unijunction transistor relaxation oscillator and waveforms.
Oscillator drives SCR.
During capacitor discharge through the E-B1 saturation
resistance, a pulse may be seen on the external B1 and B2
load resistors, Figure above. The load resistor at B1 needs to
be low to not affect the discharge time. The external resistor
at B2 is optional. It may be replaced by a short circuit. The
approximate frequency is given by 1/f = T = RC. A more
accurate expression for frequency is given in Figure above.
The charging resistor Re must fall within certain limits. It
must be small enough to allow Ip to flow based on the Vpp
supply less Vp. It must be large enough to supply ly based on
the Vee supply less Vy. [MHW] The equations and an
example for a 2n2647:
202647 Rago =4.7—9.1k 1 =0.68—0.82 Iy=8mA Ip=2A
Vp=0.7+7Vap V, = 0.7 + 0.75(10) = 8.2V
Vy =0.10(V,,) V, =0.10(10) = 1V
Vap - Vy <R:< Van ~ Vp 10-1 <R:< 10 - 8.2
1, I, &mA 2uA
1.125k <R;< 900k
Programmable Unijunction Transistor (PUT): Although
the unijunction transistor is listed as obsolete (read
expensive if obtainable), the programmable unijunction
transistor is alive and well. It is inexpensive and in
production. Though it serves a function similar to the
unijunction transistor, the PUT is a three terminal thyristor.
The PUT shares the four-layer structure typical of thyristors
shown in Figure below. Note that the gate, an N-type layer
near the anode, is Known as an “anode gate”. Moreover, the
gate lead on the schematic symbol is attached to the anode
end of the symbol.
K
Programmable unijunction transistor: Characteristic curve,
internal construction, schematic symbol.
The characteristic curve for the programmable unijunction
transistor in Figure above is similar to that of the unijunction
transistor. This is a plot of anode current I, versus anode
voltage Vy. The gate lead voltage sets, programs, the peak
anode voltage Vp. As anode current inceases, voltage
increases up to the peak point. Thereafter, increasing
current results in decreasing voltage, down to the valley
point.
The PUT equivalent of the unijunction transistor is shown in
Figure below. External PUT resistors Rl and R2 replace
unijunction transistor internal resistors Rg; and Rp>,
respectively. These resistors allow the calculation of the
intrinsic standoff ratio n.
B2 Repo = Rl + R2
eee ee eee ee ——
Re VN= ia
B2 ! R1+R2
a: | Bl ae Vs= 1V pp
RI! Vp=Vr+Vs
ee Bl RI1-R2
Unijunction PUT equivalent ©~ RI+R2
PUT equivalent of unijunction transistor
Figure below shows the PUT version of the unijunction
relaxation oscillator Figure previous. Resistor R charges the
capacitor until the peak point, Figure previous, then heavy
conduction moves the operating point down the negative
resistance slope to the valley point. A current spike flows
through the cathode during capacitor discharge, developing
a voltage spike across the cathode resistors. After capacitor
discharge, the operating point resets back to the slope up to
the peak point.
PUT relaxation oscillator
Problem: What is the range of suitable values for R in
Figure above, a relaxation oscillator? The charging resistor
must be small enough to supply enough current to raise the
anode to Vp the peak point (Figure previous) while charging
the capacitor. Once V>p is reached, anode voltage decreases
as Current increases (negative resistance), which moves the
operating point to the valley. It is the job of the capacitor to
supply the valley current ly. Once it is discharged, the
operating point resets back to the upward slope to the peak
point. The resistor must be large enough so that it will never
supply the high valley current Ip. If the charging resistor ever
could supply that much current, the resistor would supply
the valley current after the capacitor was discharged and
the operating point would never reset back to the high
resistance condition to the left of the peak point.
We select the same Vpg=10V used for the unijunction
transistor example. We select values of Rl and R2 so that n
Is about 2/3. We calculate n and Vz. The parallel equivalent
of R1, R2 is Rg, which is only used to make selections from
Table below. Along with V,=10, the closest value to our 6.3,
we find V+=0.6V, in Table below and calculate Vp.
R1=27k R2=16k Vyy=10V
= = 0.6279
1 = RT+R2 = wig
Vo= 1Vap Vg = 0.6279(10) = 6.279V
.RI 7k. .
Ro= Ro= tO = 10K
R1 +R2 27k + 16k
Vp = V; + Vs
For R,=10k and V,=10V, V;-=0.6V
Vp=0.6+ 6.3 =6.9V
We also find Ip and ly, the peak and valley currents,
respectively in Table below. We still need Vy, the valley
voltage. We used 10% of Vep= 1V, in the previous
unijunction example. Consulting the datasheet, we find the
forward voltage V-=0.8V at IE=50mA. The valley current
ly=7 OWA is much less than Il-=50mA. Therefore, V\, must be
less than V-=0.8V. How much less? To be safe we set V\=OV.
This will raise the lower limit on the resistor range a little.
For Rg=10k and V,=10V, [p= 4.0nA
For Rg=10k and V,=10V, ly = 7OHA
V, =0.10(V,,) not used Vy =OV
Ver - Vv - - - 6.5
BB V <Ri< Vip Vp 10 0 <R:< 10 6.9
Ly Ip 7OWA 4uA
143k <Rp-< 755k
Choosing R > 143k guarantees that the operating point can
reset from the valley point after capacitor discharge. R <
755k allows charging up to Vp at the peak point.
Selected 2n6027 PUT parameters, adapted from 2n6027
datasheet. [ON1]
Conditions _| min |typical|max|units
a
Se Rese ia Se Ge 7
-—r
ee es
| Ms=20V,Re=iMegl fas 50 |_|
| Ms=20V,Re=10k fro [iso [|_|
ae er Re=2000/1500- fF |
Me ie=Soma fos VI
Figure below show the PUT relaxation oscillator with the final
resistor values. A practical application of a PUT triggering an
SCR is alSo shown. This circuit needs a Vpp unfiltered supply
(not shown) divided down from the bridge rectifier to reset
the relaxation oscillator after each power zero crossing. The
variable resistor should have a minimum resistor in series
with it to prevent a low pot setting from hanging at the
valley point.
R2
16k
R1
27k
PUT relaxation oscillator with component values. PUT drives
SCR lamp dimmer.
PUT timing circuits are said to be usable to 10KHZz. If a linear
ramp is required instead of an exponential ramp, replace the
charging resistor with a constant current source such as a
FET based constant current diode. A substitute PUT may be
built from a PNP and NPN silicon transistor as shown for the
SCS equivalent circuit in Figure below by omitting the
cathode gate and using the anode gate.
e REVIEW:
e A unijunction transistor consists of two bases (B1, B2)
attached to a resistive bar of silicon, and an emitter in
the center. The E-B1 junction has negative resistance
properties; it can switch between high and low
resistance.
e A PUT (programmable unijunction transistor) is a 3-
terminal 4-layer thyristor acting like a unijunction
transistor. An external resistor network “programs” n.
e The intrinsic standoff ratio is n=R1/(R1+R2) for a PUT;
substitute Rg, and Rg>, respectively, for a unijunction
transistor. The trigger voltage is determined by n.
e Unijunction transistors and programmable unijunction
transistors are applied to oscillators, timing circuits, and
thyristor triggering.
The Silicon-Controlled Switch (SCS)
If we take the equivalent circuit for an SCR and add another
external terminal, connected to the base of the top
transistor and the collector of the bottom transistor, we have
a device known as a silicon-controlled-switch, or SCS: (Figure
below)
Anode _ Anode _
—— ——
Anode Anode
Gate Anode
cathote Y Gate
Gate
Cathode
Anode
Gate
Cathode
Gate Cathode
Gate
Cathode” Cathode”
Physical diagram Equivalent schematic Schematic symbol
The Silicon-Controlled Switch(SCSs)
This extra terminal allows more control to be exerted over
the device, particularly in the mode of forced commutation,
where an external signal forces it to turn off while the main
current through the device has not yet fallen below the
holding current value. Note that the motor is in the anode
gate circuit in Figure below. This is correct, although it
doesn't look right. The anode lead is required to switch the
SCS off. Therefore the motor cannot be in series with the
anode.
SCS: Motor start/stop circuit, equivalent circuit with two
transistors.
When the "on" pushbutton switch is actuated, the voltage
applied between the cathode gate and the cathode, forward-
biases the lower transistor's base-emitter junction, and
turning it on. The top transistor of the SCS is ready to
conduct, having been supplied with a current path from its
emitter terminal (the SCS's anode terminal) through resistor
R> to the positive side of the power supply. As in the case of
the SCR, both transistors turn on and maintain each other in
the "on" mode. When the lower transistor turns on, it
conducts the motor's load current, and the motor starts and
runs.
The motor may be stopped by interrupting the power
supply, as with an SCR, and this is called natural
commutation. However, the SCS provides us with another
means of turning off: forced commutation by shorting the
anode terminal to the cathode. [GE1] If this is done (by
actuating the "off" pushbutton switch), the upper transistor
within the SCS will lose its emitter current, thus halting
current through the base of the lower transistor. When the
lower transistor turns off, it breaks the circuit for base
current through the top transistor (Securing its "off" state),
and the motor (making it stop). The SCS will remain in the
off condition until such time that the "on" pushbutton switch
is re-actuated.
e REVIEW:
e A silicon-controlled switch, or SCS, is essentially an SCR
with an extra gate terminal.
e Typically, the load current through an SCS is carried by
the anode gate and cathode terminals, with the cathode
gate and anode terminals sufficing as control leads.
e An SCS is turned on by applying a positive voltage
between the cathode gate and cathode terminals. It may
be turned off (forced commutation) by applying a
negative voltage between the anode and cathode
terminals, or simply by shorting those two terminals
together. The anode terminal must be kept positive with
respect to the cathode in order for the SCS to latch.
Field-effect-controlled thyristors
Two relatively recent technologies designed to reduce the
"driving" (gate trigger current) requirements of classic
thyristor devices are the MOS-gated thyristor and the MOS
Controlled Thyristor, or MCT.
The MOS-gated thyristor uses a MOSFET to initiate
conduction through the upper (PNP) transistor of a standard
thyristor structure, thus triggering the device. Since a
MOSFET requires negligible current to "drive" (cause it to
saturate), this makes the thyristor as a whole very easy to
trigger: (Figure below)
MOS-gated thyristor Anode
equivalent circuit
Gate _|
Cathode
MOS-gated thyristor equivalent circuit
Given the fact that ordinary SCRs are quite easy to "drive"
as it is, the practical advantage of using an even more
sensitive device (a MOSFET) to initiate triggering is
debatable. Also, placing a MOSFET at the gate input of the
thyristor now makes it /mpossib/e to turn it off by a reverse-
triggering signal. Only low-current dropout can make this
device stop conducting after it has been latched.
A device of arguably greater value would be a fully-
controllable thyristor, whereby a small gate signal could
both trigger the thyristor and force it to turn off. Such a
device does exist, and it is called the MOS Controlled
Thyristor, or MCT. |It uses a pair of MOSFETs connected to a
common gate terminal, one to trigger the thyristor and the
other to "untrigger" it: (Figure below)
MOS Controlled Thyristor Anode
(MCT) equivalent circuit
Gate
Cathode
MOS-controlled thyristor (MCT) equivalent circuit
A positive gate voltage (with respect to the cathode) turns
on the upper (N-channel) MOSFET, allowing base current
through the upper (PNP) transistor, which latches the
transistor pair in an "on" state. Once both transistors are
fully latched, there will be little voltage dropped between
anode and cathode, and the thyristor will remain latched as
long as the controlled current exceeds the minimum
(holding) current value. However, if a negative gate voltage
iS applied (with respect to the anode, which is at nearly the
same voltage as the cathode in the latched state), the lower
MOSFET will turn on and "short" between the lower (NPN)
transistor's base and emitter terminals, thus forcing it into
cutoff. Once the NPN transistor cuts off, the PNP transistor
will drop out of conduction, and the whole thyristor turns off.
Gate voltage has full control over conduction through the
MCT: to turn it on and to turn it off.
This device is still a thyristor, though. If zero voltage is
applied between gate and cathode, neither MOSFET will turn
on. Consequently, the bipolar transistor pair will remain in
whatever state it was last in (hysteresis). So, a brief positive
pulse to the gate turns the MCT on, a brief negative pulse
forces it off, and no applied gate voltage lets it remain in
whatever state it is already in. In essence, the MCT isa
latching version of the IGBT (Insulated Gate Bipolar
Transistor).
e REVIEW:
e A MOS-gated thyristor uses an N-channel MOSFET to
trigger a thyristor, resulting in an extremely low gate
Current requirement.
e A MOS Controlled Thyristor, or MCT, uses two MOSFETS
to exert full control over the thyristor. A positive gate
voltage triggers the device; a negative gate voltage
forces it to turn off. Zero gate voltage allows the
thyristor to remain in whatever state it was previously in
(off, or latched on).
Bibliography
1. [VTS]“Phattytron PT-1 Vacuum Tube Synthesizer”, The
Audio Playground Synthesizer Museum at
http://www.keyboardmuseum.com/ar/m/meta/ptl. htm!
2. [MET]“At last, a pitch source with tube power”,
METASONIX, PMB 109, 881 11th Street, Lakeport CA
95453 USA at http://www.metasonix.com/index.php?
option=com_content&task=view&id=14<emid=31
3. [GE1]“Silicon Contolled Switches”, GE Transistor Manual,
The General Electric Company, 1964, Figure 16.19(M).
4.[ON1] “2N6027, 2N6028 Programmable Unijunction
Transistor ”, datasheet at
http://www.onsemi.com/pub_link/Collateral/2N6027-
D.PDF
5. [AMS] “Unijunction Transistor ”, American
Microsemiconductor, at
http://www.americanmicrosemi.com/tutorials/unijunction
£htm
6. [MHW]Matthew H. Williams, “Unijunction Transistor ”, at
http://baec.tripod.com/DEC90/uni_tran.htm
Unijunction Transistor by
http://baec.tripod.com/DEC90/uni_tran.htm
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—||4/]l_—
—|}|+4]|—
Lessons In Electric Circuits -
- Volume Ill
Chapter 8
OPERATIONAL AMPLIFIERS
Introduction
Single-ended and differential amplifiers
The "operational" amplifier
Negative feedback
Divided feedback
An analogy for divided feedback
Voltage-to-current signal conversion
Averager and summer circuits
Building a differential amplifier
The instrumentation amplifier
Differentiator and integrator circuits
Positive feedback
Practical considerations
° Common-mode gain
o Offset voltage
o Bias current
o Drift
o Frequency response
o Input to output phase shift
perational amplifier models
ata
Contributors
O
D
Introduction
The operational amplifier is arguably the most useful single
device in analog electronic circuitry. With only a handful of
external components, it can be made to perform a wide variety
of analog signal processing tasks. It is also quite affordable,
most general-purpose amplifiers selling for under a dollar
apiece. Modern designs have been engineered with durability
in mind as well: several "op-amps" are manufactured that can
sustain direct short-circuits on their outputs without damage.
One key to the usefulness of these little circuits is in the
engineering principle of feedback, particularly negative
feedback, which constitutes the foundation of almost all
automatic control processes. The principles presented here in
operational amplifier circuits, therefore, extend well beyond
the immediate scope of electronics. It is well worth the
electronics student's time to learn these principles and learn
them well.
Single-ended and differential amplifiers
For ease of drawing complex circuit diagrams, electronic
amplifiers are often symbolized by a simple triangle shape,
where the internal components are not individually
represented. This symbology is very handy for cases where an
amplifier's construction is irrelevant to the greater function of
the overall circuit, and it is worthy of familiarization:
General amplifier circuit symbol
+V
supply
Input Output
~V supply
The +V and -V connections denote the positive and negative
sides of the DC power supply, respectively. The input and
output voltage connections are shown as single conductors,
because it is assumed that all signal voltages are referenced to
a common connection in the circuit called ground. Often (but
not always!), one pole of the DC power supply, either positive
or negative, is that ground reference point. A practical
amplifier circuit (showing the input voltage source, load
resistance, and power supply) might look like this:
Output 30V —
Rioad Je
Vv
input
‘IPs
Without having to analyze the actual transistor design of the
amplifier, you can readily discern the whole circuit's function:
to take an input signal (V;,), amplify it, and drive a load
resistance (Rigaqg). To complete the above schematic, it would
be good to specify the gains of that amplifier (Ay, Aj, Ap) and
the Q (bias) point for any needed mathematical analysis.
If it is necessary for an amplifier to be able to output true AC
voltage (reversing polarity) to the load, a sp/it DC power supply
may be used, whereby the ground point is electrically
"centered" between +V and -V. Sometimes the split power
supply configuration is referred to as a dua/ power supply.
V
input
The amplifier is still being supplied with 30 volts overall, but
with the split voltage DC power supply, the output voltage
across the load resistor can now swing from a theoretical
maximum of +15 volts to -15 volts, instead of +30 volts to 0
volts. This is an easy way to get true alternating current (AC)
output from an amplifier without resorting to capacitive or
inductive (transformer) coupling on the output. The peak-to-
peak amplitude of this amplifier's output between cutoff and
saturation remains unchanged.
By signifying a transistor amplifier within a larger circuit with a
triangle symbol, we ease the task of studying and analyzing
more complex amplifiers and circuits. One of these more
complex amplifier types that we'll be studying is called the
differential amplifier. Unlike normal amplifiers, which amplify a
single input signal (often called single-ended amplifiers),
differential amplifiers amplify the voltage difference between
two input signals. Using the simplified triangle amplifier
symbol, a differential amplifier looks like this:
Differential amplifier
OY cide
Input.
Output
Input,
-V
supply
The two input leads can be seen on the left-hand side of the
triangular amplifier symbol, the output lead on the right-hand
side, and the +V and -V power supply leads on top and bottom.
As with the other example, all voltages are referenced to the
circuit's ground point. Notice that one input lead is marked
with a (-) and the other is marked with a (+). Because a
differential amplifier amplifies the difference in voltage
between the two inputs, each input influences the output
voltage in opposite ways. Consider the following table of
input/output voltages for a differential amplifier with a voltage
gain of 4:
Voltage output equation: V.,,, = A,(Input, - Input,)
or
Vou = Ay(Input,,) - Input, »)
An increasingly positive voltage on the (+) input tends to drive
the output voltage more positive, and an increasingly positive
voltage on the (-) input tends to drive the output voltage more
negative. Likewise, an increasingly negative voltage on the (+)
input tends to drive the output negative as well, and an
increasingly negative voltage on the (-) input does just the
opposite. Because of this relationship between inputs and
polarities, the (-) input is commonly referred to as the /nverting
input and the (+) as the noninverting input.
It may be helpful to think of a differential amplifier as a
variable voltage source controlled by a sensitive voltmeter, as
such:
Bear in mind that the above illustration is only a mode! to aid
in understanding the behavior of a differential amplifier. It is
not a realistic schematic of its actual design. The "G" symbol
represents a galvanometer, a sensitive voltmeter movement.
The potentiometer connected between +V and -V provides a
variable voltage at the output pin (with reference to one side of
the DC power supply), that variable voltage set by the reading
of the galvanometer. It must be understood that any load
powered by the output of a differential amplifier gets its
current from the DC power source (battery), not the input
signal. The input signal (to the galvanometer) merely controls
the output.
This concept may at first be confusing to students new to
amplifiers. With all these polarities and polarity markings (-
and +) around, its easy to get confused and not know what the
output of a differential amplifier will be. To address this
potential confusion, here's a simple rule to remember:
ae
_ Differential +
input voltage = Output
voltage
ala
_—
_ Differential .
input voltage Output
—_ “t+ voltage
When the polarity of the differentia! voltage matches the
markings for inverting and noninverting inputs, the output will
be positive. When the polarity of the differential voltage
clashes with the input markings, the output will be negative.
This bears some similarity to the mathematical sign displayed
by digital voltmeters based on input voltage polarity. The red
test lead of the voltmeter (often called the "positive" lead
because of the color red's popular association with the positive
side of a power supply in electronic wiring) is more positive
than the black, the meter will display a positive voltage figure,
and vice versa:
blk —
a
_ Differential —evy + 6.00 V
input voltage — 6 Digital Voltmeter
—
+
dl 7
_ Differential — 6V - 6.00 V
Input voltage ae Digital Voltmeter
+
Just as a voltmeter will only display the voltage between its two
test leads, an ideal differential amplifier only amplifies the
potential difference between its two input connections, not the
voltage between any one of those connections and ground. The
output polarity of a differential amplifier, just like the signed
indication of a digital voltmeter, depends on the relative
polarities of the differential voltage between the two input
connections.
If the input voltages to this amplifier represented mathematical
quantities (as is the case within analog computer circuitry), or
physical process measurements (as is the case within analog
electronic instrumentation circuitry), you can see how a device
such as a differential amplifier could be very useful. We could
use it to compare two quantities to see which is greater (by the
polarity of the output voltage), or perhaps we could compare
the difference between two quantities (such as the level of
liquid in two tanks) and flag an alarm (based on the absolute
value of the amplifier output) if the difference became too
great. In basic automatic control circuitry, the quantity being
controlled (called the process variable) is compared with a
target value (called the setpoint), and decisions are made as to
how to act based on the discrepancy between these two
values. The first step in electronically controlling such a
scheme is to amplify the difference between the process
variable and the setpoint with a differential amplifier. In simple
controller designs, the output of this differential amplifier can
be directly utilized to drive the final control element (Such as a
valve) and keep the process reasonably close to setpoint.
e REVIEW:
e A "shorthand" symbol for an electronic amplifier is a
triangle, the wide end signifying the input side and the
narrow end signifying the output. Power supply lines are
often omitted in the drawing for simplicity.
e To facilitate true AC output from an amplifier, we can use
what is called a sp/it or dua/ power supply, with two DC
voltage sources connected in series with the middle point
grounded, giving a positive voltage to ground (+V) anda
negative voltage to ground (-V). Split power supplies like
this are frequently used in differential amplifier circuits.
e Most amplifiers have one input and one output. Differential
amplifiers have two inputs and one output, the output
signal being proportional to the difference in signals
between the two inputs.
e The voltage output of a differential amplifier is determined
by the following equation: Vout = Ay(Vnoninv - Vinv)
The "operational" amplifier
Long before the advent of digital electronic technology,
computers were built to electronically perform calculations by
employing voltages and currents to represent numerical
quantities. This was especially useful for the simulation of
physical processes. A variable voltage, for instance, might
represent velocity or force in a physical system. Through the
use of resistive voltage dividers and voltage amplifiers, the
mathematical operations of division and multiplication could
be easily performed on these signals.
The reactive properties of capacitors and inductors lend
themselves well to the simulation of variables related by
calculus functions. Remember how the current through a
capacitor was a function of the voltage's rate of change, and
how that rate of change was designated in calculus as the
derivative? Well, if voltage across a capacitor were made to
represent the velocity of an object, the current through the
capacitor would represent the force required to accelerate or
decelerate that object, the capacitor's capacitance
representing the object's mass:
ic=C $Y F=m $*
Where, Where,
i. = Instantaneous current F = Force applied to object
through capacitor
C = Capacitance in farads m = Mass of object
dv _ Rate of change of dv — Rate of change of
dt —_ voltage over time dt —_ velocity over time
This analog electronic computation of the calculus derivative
function is technically known as differentiation, and itis a
natural function of a capacitor's current in relation to the
voltage applied across it. Note that this circuit requires no
"programming" to perform this relatively advanced
mathematical function as a digital computer would.
Electronic circuits are very easy and inexpensive to create
compared to complex physical systems, so this kind of analog
electronic simulation was widely used in the research and
development of mechanical systems. For realistic simulation,
though, amplifier circuits of high accuracy and easy
configurability were needed in these early computers.
It was found in the course of analog computer design that
differential amplifiers with extremely high voltage gains met
these requirements of accuracy and configurability better than
single-ended amplifiers with custom-designed gains. Using
simple components connected to the inputs and output of the
high-gain differential amplifier, virtually any gain and any
function could be obtained from the circuit, overall, without
adjusting or modifying the internal circuitry of the amplifier
itself. These high-gain differential amplifiers came to be known
as operational amplifiers, or op-amps, because of their
application in analog computers’ mathematical operations.
Modern op-amps, like the popular model 741, are high-
performance, inexpensive integrated circuits. Their input
impedances are quite high, the inputs drawing currents in the
range of half a microamp (maximum) for the 741, and far less
for op-amps utilizing field-effect input transistors. Output
impedance is typically quite low, about 75 QO for the model 741,
and many models have built-in output short circuit protection,
meaning that their outputs can be directly shorted to ground
without causing harm to the internal circuitry. With direct
coupling between op-amps' internal transistor stages, they can
amplify DC signals just as well as AC (up to certain maximum
voltage-risetime limits). It would cost far more in money and
time to design a comparable discrete-transistor amplifier circuit
to match that kind of performance, unless high power
Capability was required. For these reasons, op-amps have all
but obsoleted discrete-transistor signal amplifiers in many
applications.
The following diagram shows the pin connections for single op-
amps (741 included) when housed in an 8-pin DIP (Dual Inline
Package) integrated circuit:
Typical 8-pin "DIP" op-amp
integrated circuit
No ay Offset
null
connection Output
Offset -V
null
Some models of op-amp come two to a package, including the
popular models TLO82 and 1458. These are called "dual" units,
and are typically housed in an 8-pin DIP package as well, with
the following pin connections:
Dual op-amp in 8-pin DIP
Operational amplifiers are also available four to a package,
usually in 14-pin DIP arrangements. Unfortunately, pin
assignments aren't as standard for these "quad" op-amps as
they are for the "dual" or single units. Consult the
manufacturer datasheet(s) for details.
Practical operational amplifier voltage gains are in the range of
200,000 or more, which makes them almost useless as an
analog differential amplifier by themselves. For an op-amp with
a voltage gain (Ay) of 200,000 and a maximum output voltage
swing of +15V/-15V, all it would take is a differential input
voltage of 75 uV (microvolts) to drive it to saturation or cutoff!
Before we take a look at how external components are used to
bring the gain down to a reasonable level, let's investigate
applications for the "bare" op-amp by itself.
One application is called the comparator. For all practical
purposes, we can Say that the output of an op-amp will be
saturated fully positive if the (+) input is more positive than
the (-) input, and saturated fully negative if the (+) input is
less positive than the (-) input. In other words, an op-amp's
extremely high voltage gain makes it useful as a device to
compare two voltages and change output voltage states when
one input exceeds the other in magnitude.
+V
LED
-V
In the above circuit, we have an op-amp connected as a
comparator, comparing the input voltage with a reference
voltage set by the potentiometer (Rj). If V;, drops below the
voltage set by R,, the op-amp's output will saturate to +V,
thereby lighting up the LED. Otherwise, if V;, is above the
reference voltage, the LED will remain off. If V;, is a voltage
signal produced by a measuring instrument, this comparator
circuit could function as a "low" alarm, with the trip-point set
by R,. Instead of an LED, the op-amp output could drive a
relay, a transistor, an SCR, or any other device capable of
switching power to a load such as a solenoid valve, to take
action in the event of a low alarm.
Another application for the comparator circuit shown is a
square-wave converter. Suppose that the input voltage applied
to the inverting (-) input was an AC sine wave rather than a
stable DC voltage. In that case, the output voltage would
transition between opposing states of saturation whenever the
input voltage was equal to the reference voltage produced by
the potentiometer. The result would be a square wave:
+V
Adjustments to the potentiometer setting would change the
reference voltage applied to the noninverting (+) input, which
would change the points at which the sine wave would cross,
changing the on/off times, or duty cycle of the square wave:
+V
It should be evident that the AC input voltage would not have
to be a sine wave in particular for this circuit to perform the
same function. The input voltage could be a triangle wave,
sawtooth wave, or any other sort of wave that ramped
smoothly from positive to negative to positive again. This sort
of comparator circuit is very useful for creating square waves of
varying duty cycle. This technique is sometimes referred to as
pulse-width modulation, or PWM (varying, or modulating a
waveform according to a controlling signal, in this case the
signal produced by the potentiometer).
Another comparator application is that of the bargraph driver.
If we had several op-amps connected as comparators, each
with its own reference voltage connected to the inverting
input, but each one monitoring the same voltage signal on
their noninverting inputs, we could build a bargraph-style
meter such as what is commonly seen on the face of stereo
tuners and graphic equalizers. As the signal voltage
(representing radio signal strength or audio sound level)
increased, each comparator would "turn on" in sequence and
send power to its respective LED. With each comparator
switching "on" at a different level of audio sound, the number
of LED's illuminated would indicate how strong the signal was.
+V
Simple bargraph driver circuit
In the circuit shown above, LED, would be the first to light up
as the input voltage increased in a positive direction. As the
input voltage continued to increase, the other LED's would
illuminate in succession, until all were lit.
This very same technology is used in some analog-to-digital
signal converters, namely the flash converter, to translate an
analog signal quantity into a series of on/off voltages
representing a digital number.
e REVIEW:
e A triangle shape is the generic symbol for an amplifier
circuit, the wide end signifying the input and the narrow
end signifying the output.
e Unless otherwise specified, a// voltages in amplifier circuits
are referenced to a common ground point, usually
connected to one terminal of the power supply. This way,
we can speak of a certain amount of voltage being "on" a
single wire, while realizing that voltage is a/ways measured
between two points.
e A differential amplifier is one amplifying the voltage
difference between two signal inputs. In such a circuit, one
input tends to drive the output voltage to the same
polarity of the input signal, while the other input does just
the opposite. Consequently, the first input is called the
noninverting (+) input and the second is called the
inverting (-) input.
e An operational amplifier (or op-amp for short) is a
differential amplifier with an extremely high voltage gain
(Ay = 200,000 or more). Its name hails from its original use
in analog computer circuitry (performing mathematical
operations).
e Op-amps typically have very high input impedances and
fairly low output impedances.
e Sometimes op-amps are used as signal comparators,
operating in full cutoff or saturation mode depending on
which input (inverting or noninverting) has the greatest
voltage. Comparators are useful in detecting "greater-than"
signal conditions (comparing one to the other).
e One comparator application is called the pulse-width
modulator, and is made by comparing a sine-wave AC
signal against a DC reference voltage. As the DC reference
voltage is adjusted, the square-wave output of the
comparator changes its duty cycle (positive versus
negative times). Thus, the DC reference voltage controls, or
modulates the pulse width of the output voltage.
Negative feedback
If we connect the output of an op-amp to its inverting input
and apply a voltage signal to the noninverting input, we find
that the output voltage of the op-amp closely follows that input
voltage (I've neglected to draw in the power supply, +V/-V
wires, and ground symbol for simplicity):
V Vout
in
As Vin increases, Voy will increase in accordance with the
differential gain. However, as Vo, increases, that output
voltage is fed back to the inverting input, thereby acting to
decrease the voltage differential between inputs, which acts to
bring the output down. What will happen for any given voltage
input is that the op-amp will output a voltage very nearly equal
to V,,, but just low enough so that there's enough voltage
difference left between V,, and the (-) input to be amplified to
generate the output voltage.
The circuit will quickly reach a point of stability (known as
equilibrium in physics), where the output voltage is just the
right amount to maintain the right amount of differential,
which in turn produces the right amount of output voltage.
Taking the op-amp's output voltage and coupling it to the
inverting input is a technique known as negative feedback,
and it is the key to having a self-stabilizing system (this is true
not only of op-amps, but of any dynamic system in general).
This stability gives the op-amp the capacity to work in its linear
(active) mode, as opposed to merely being saturated fully "on"
or "off" as it was when used as a comparator, with no feedback
at all.
Because the op-amp's gain is so high, the voltage on the
inverting input can be maintained almost equal to V;,,,. Let's say
that our op-amp has a differential voltage gain of 200,000. If
Vi, equals 6 volts, the output voltage will be
5.99997 0000149999 volts. This creates just enough
differential voltage (6 volts - 5.999970000149999 volts =
29.99985 uV) to cause 5.99997 0000149999 volts to be
manifested at the output terminal, and the system holds there
in balance. As you can see, 29.99985 UV is not a lot of
differential, so for practical calculations, we can assume that
the differential voltage between the two input wires is held by
negative feedback exactly at 0 volts.
The effects of negative feedback
29.99985 29.99985 LV
aT Vv
4 !
The effects of negative feedback
(rounded figures)
aa
pas
=
poets
One great advantage to using an op-amp with negative
feedback is that the actual voltage gain of the op-amp doesn't
matter, so long as its very large. If the op-amp's differential
gain were 250,000 instead of 200,000, all it would mean is that
the output voltage would hold just a little closer to V;,, (less
differential voltage needed between inputs to generate the
required output). In the circuit just illustrated, the output
voltage would still be (for all practical purposes) equal to the
non-inverting input voltage. Op-amp gains, therefore, do not
have to be precisely set by the factory in order for the circuit
designer to build an amplifier circuit with precise gain.
Negative feedback makes the system self-correcting. The
above circuit as a whole will simply follow the input voltage
with a stable gain of 1.
Going back to our differential amplifier model, we can think of
the operational amplifier as being a variable voltage source
controlled by an extremely sensitive nu// detector, the kind of
meter movement or other sensitive measurement device used
in bridge circuits to detect a condition of balance (zero volts).
The "potentiometer" inside the op-amp creating the variable
voltage will move to whatever position it must to "balance" the
inverting and noninverting input voltages so that the "null
detector" has zero voltage across it:
As the "potentiometer" will move to provide an output voltage
necessary to satisfy the "null detector" at an "indication" of
zero volts, the output voltage becomes equal to the input
voltage: in this case, 6 volts. If the input voltage changes at all,
the "potentiometer" inside the op-amp will change position to
hold the "null detector" in balance (indicating zero volts),
resulting in an output voltage approximately equal to the input
voltage at all times.
This will hold true within the range of voltages that the op-amp
can output. With a power supply of +15V/-1L5V, and an ideal
amplifier that can swing its output voltage just as far, it will
faithfully "follow" the input voltage between the limits of +15
volts and -15 volts. For this reason, the above circuit is known
as a voltage follower. Like its one-transistor counterpart, the
common-collector ("emitter-follower") amplifier, it has a
voltage gain of 1, a high input impedance, a low output
impedance, and a high current gain. Voltage followers are also
known as voltage buffers, and are used to boost the current-
sourcing ability of voltage signals too weak (too high of source
impedance) to directly drive a load. The op-amp model shown
in the last illustration depicts how the output voltage is
essentially isolated from the input voltage, so that current on
the output pin is not supplied by the input voltage source at
all, but rather from the power supply powering the op-amp.
It should be mentioned that many op-amps cannot swing their
output voltages exactly to +V/-V power supply rail voltages.
The model 741 is one of those that cannot: when saturated, its
output voltage peaks within about one volt of the +V power
supply voltage and within about 2 volts of the -V power supply
voltage. Therefore, with a split power supply of +15/-15 volts, a
741 op-amp's output may go as high as +14 volts or as low as
-13 volts (approximately), but no further. This is due to its
bipolar transistor design. These two voltage limits are known as
the positive saturation voltage and negative saturation
voltage, respectively. Other op-amps, such as the model 3130
with field-effect transistors in the final output stage, have the
ability to swing their output voltages within millivolts of either
power supply ra// voltage. Consequently, their positive and
negative saturation voltages are practically equal to the supply
voltages.
REVIEW:
Connecting the output of an op-amp to its inverting (-)
input is called negative feedback. This term can be broadly
applied to any dynamic system where the output signal is
"fed back" to the input somehow so as to reach a point of
equilibrium (balance).
When the output of an op-amp is direct/y connected to its
inverting (-) input, a vo/tage follower will be created.
Whatever signal voltage is impressed upon the
noninverting (+) input will be seen on the output.
An op-amp with negative feedback will try to drive its
output voltage to whatever level necessary so that the
differential voltage between the two inputs is practically
zero. The higher the op-amp differential gain, the closer
that differential voltage will be to zero.
Some op-amps cannot produce an output voltage equal to
their supply voltage when saturated. The model 741 is one
of these. The upper and lower limits of an op-amp's output
voltage swing are known as positive saturation voltage and
negative saturation voltage, respectively.
Divided feedback
If we add a voltage divider to the negative feedback wiring so
that only a fraction of the output voltage is fed back to the
inverting input instead of the full amount, the output voltage
will be a multiple of the input voltage (please bear in mind that
the power supply connections to the op-amp have been
omitted once again for simplicity's sake):
The effects of divided negative feedback
All voltage figures shown in
reference to groun
6v¥ —
If Ry and R> are both equal and V,, is 6 volts, the op-amp will
output whatever voltage is needed to drop 6 volts across R, (to
make the inverting input voltage equal to 6 volts, as well,
keeping the voltage difference between the two inputs equal to
zero). With the 2:1 voltage divider of R; and Rj, this will take
12 volts at the output of the op-amp to accomplish.
Another way of analyzing this circuit is to start by calculating
the magnitude and direction of current through Rj, Knowing
the voltage on either side (and therefore, by subtraction, the
voltage across Rj), and R's resistance. Since the left-hand side
of Rj is connected to ground (0 volts) and the right-hand side
is at a potential of 6 volts (due to the negative feedback
holding that point equal to V;,), we can see that we have 6
volts across Rj. This gives us 6 mA of current through R, from
left to right. Because we know that both inputs of the op-amp
have extremely high impedance, we can safely assume they
won't add or subtract any current through the divider. In other
words, we can treat R; and R> as being in series with each
other: all of the electrons flowing through R, must flow through
R>. Knowing the current through R> and the resistance of R3,
we can calculate the voltage across R> (6 volts), and its
polarity. Counting up voltages from ground (0 volts) to the
right-hand side of R>, we arrive at 12 volts on the output.
Upon examining the last illustration, one might wonder, "where
does that 6 mA of current go?" The last illustration doesn't
show the entire current path, but in reality it comes from the
negative side of the DC power supply, through ground, through
R,, through R>, through the output pin of the op-amp, and then
back to the positive side of the DC power supply through the
output transistor(s) of the op-amp. Using the null
detector/potentiometer model of the op-amp, the current path
looks like this:
The 6 volt signal source does not have to supply any current
for the circuit: it merely commands the op-amp to balance
voltage between the inverting (-) and noninverting (+) input
pins, and in so doing produce an output voltage that is twice
the input due to the dividing effect of the two 1 kOQ resistors.
We can change the voltage gain of this circuit, overall, just by
adjusting the values of R; and R> (changing the ratio of output
voltage that is fed back to the inverting input). Gain can be
calculated by the following formula:
R,
l
Note that the voltage gain for this design of amplifier circuit
can never be less than 1. If we were to lower R> to a value of
zero ohms, our circuit would be essentially identical to the
voltage follower, with the output directly connected to the
inverting input. Since the voltage follower has a gain of 1, this
sets the lower gain limit of the noninverting amplifier. However,
the gain can be increased far beyond 1, by increasing R> in
proportion to Rj.
Also note that the polarity of the output matches that of the
input, just as with a voltage follower. A positive input voltage
results in a positive output voltage, and vice versa (with
respect to ground). For this reason, this circuit is referred to as
a noninverting amplifier.
Just as with the voltage follower, we see that the differential
gain of the op-amp is irrelevant, so long as its very high. The
voltages and currents in this circuit would hardly change at all
if the op-amp's voltage gain were 250,000 instead of 200,000.
This stands as a stark contrast to single-transistor amplifier
circuit designs, where the Beta of the individual transistor
greatly influenced the overall gains of the amplifier. With
negative feedback, we have a self-correcting system that
amplifies voltage according to the ratios set by the feedback
resistors, not the gains internal to the op-amp.
Let's see what happens if we retain negative feedback through
a voltage divider, but apply the input voltage at a different
location:
All voltage figures shown in
= reference to ground
By grounding the noninverting input, the negative feedback
from the output seeks to hold the inverting input's voltage at 0
volts, as well. For this reason, the inverting input is referred to
in this circuit as a virtual ground, being held at ground
potential (0 volts) by the feedback, yet not directly connected
to (electrically common with) ground. The input voltage this
time is applied to the left-hand end of the voltage divider (R, =
R> = 1 kO again), so the output voltage must swing to -6 volts
in order to balance the middle at ground potential (0 volts).
Using the same techniques as with the noninverting amplifier,
we can analyze this circuit's operation by determining current
magnitudes and directions, starting with R,, and continuing on
to determining the output voltage.
We can change the overall voltage gain of this circuit, overall,
just by adjusting the values of R, and R> (changing the ratio of
output voltage that is fed back to the inverting input). Gain
can be calculated by the following formula:
Note that this circuit's voltage gain can be less than 1,
depending solely on the ratio of R> to Rj. Also note that the
output voltage is always the opposite polarity of the input
voltage. A positive input voltage results in a negative output
voltage, and vice versa (with respect to ground). For this
reason, this circuit is referred to as an inverting amplifier.
Sometimes, the gain formula contains a negative sign (before
the R>/R, fraction) to reflect this reversal of polarities.
These two amplifier circuits we've just investigated serve the
purpose of multiplying or dividing the magnitude of the input
voltage signal. This is exactly how the mathematical operations
of multiplication and division are typically handled in analog
computer circuitry.
REVIEW:
By connecting the inverting (-) input of an op-amp directly
to the output, we get negative feedback, which gives us a
voltage follower circuit. By connecting that negative
feedback through a resistive voltage divider (feeding back
a fraction of the output voltage to the inverting input), the
output voltage becomes a multiple of the input voltage.
A negative-feedback op-amp circuit with the input signal
going to the noninverting (+) input is called a noninverting
amplifier. The output voltage will be the same polarity as
the input. Voltage gain is given by the following equation:
Ay = (R>/R,) + 1
A negative-feedback op-amp circuit with the input signal
going to the "bottom" of the resistive voltage divider, with
the noninverting (+) input grounded, is called an inverting
amplifier. |ts output voltage will be the opposite polarity of
the input. Voltage gain is given by the following equation:
Ay = -R>/Ry
An analogy for divided feedback
A helpful analogy for understanding divided feedback amplifier
circuits is that of a mechanical lever, with relative motion of
the lever's ends representing change in input and output
voltages, and the fulcrum (pivot point) representing the
location of the ground point, real or virtual.
Take for example the following noninverting op-amp circuit. We
know from the prior section that the voltage gain of a
noninverting amplifier configuration can never be less than
unity (1). If we draw a lever diagram next to the amplifier
schematic, with the distance between fulcrum and lever ends
representative of resistor values, the motion of the lever will
signify changes in voltage at the input and output terminals of
the amplifier:
Physicists call this type of lever, with the input force (effort)
applied between the fulcrum and output (load), a third-class
lever. It is characterized by an output displacement (motion) at
least as large than the input displacement -- a "gain" of at least
1 -- and in the same direction. Applying a positive input
voltage to this op-amp circuit is analogous to displacing the
"input" point on the lever upward:
<
out
vies = (Vin)
Due to the displacement-amplifying characteristics of the lever,
the "output" point will move twice as far as the "input" point,
and in the same direction. In the electronic circuit, the output
voltage will equal twice the input, with the same polarity.
Applying a negative input voltage is analogous to moving the
lever downward from its level "Zero" position, resulting in an
amplified output displacement that is also negative:
Vin
aa
If we alter the resistor ratio R5/R;, we change the gain of the
Op-amp circuit. In lever terms, this means moving the input
point in relation to the fulcrum and lever end, which similarly
changes the displacement "gain" of the machine:
Vv
out
|x R,>}«—— R, _l
jal a a
{}
Vv V ist = AV.)
in
Now, any input signal will become amplified by a factor of four
instead of by a factor of two:
Vout = AV in)
Inverting op-amp circuits may be modeled using the lever
analogy as well. With the inverting configuration, the ground
point of the feedback voltage divider is the op-amp's inverting
input with the input to the left and the output to the right. This
is mechanically equivalent to a first-class lever, where the
input force (effort) is on the opposite side of the fulcrum from
the output (load):
With equal-value resistors (equal-lengths of lever on each side
of the fulcrum), the output voltage (displacement) will be
equal in magnitude to the input voltage (displacement), but of
the opposite polarity (direction). A positive input results in a
negative output:
Changing the resistor ratio R>/R,; changes the gain of the
amplifier circuit, just as changing the fulcrum position on the
lever changes its mechanical displacement "gain." Consider
the following example, where R> is made twice as large as R}:
With the inverting amplifier configuration, though, gains of less
than 1 are possible, just as with first-class levers. Reversing R>
and R, values is analogous to moving the fulcrum to its
complementary position on the lever: one-third of the way from
the output end. There, the output displacement will be one-half
the input displacement:
Voltage-to-current signal conversion
In instrumentation circuitry, DC signals are often used as
analog representations of physical measurements such as
temperature, pressure, flow, weight, and motion. Most
commonly, DC current signals are used in preference to DC
voltage signals, because current signals are exactly equal in
magnitude throughout the series circuit loop carrying current
from the source (measuring device) to the load (indicator,
recorder, or controller), whereas voltage signals in a parallel
circuit may vary from one end to the other due to resistive wire
losses. Furthermore, current-sensing instruments typically have
low impedances (while voltage-sensing instruments have high
impedances), which gives current-sensing instruments greater
electrical noise immunity.
In order to use current as an analog representation of a
physical quantity, we have to have some way of generating a
precise amount of current within the signal circuit. But how do
we generate a precise current signal when we might not know
the resistance of the loop? The answer is to use an amplifier
designed to hold current to a prescribed value, applying as
much or as little voltage as necessary to the load circuit to
maintain that value. Such an amplifier performs the function of
a current source. An op-amp with negative feedback is a
perfect candidate for such a task:
4 to 20 mA
250.2 sghented =
load
+H
Vin 1 to 5 volt signal range
The input voltage to this circuit is assumed to be coming from
some type of physical transducer/amplifier arrangement,
calibrated to produce 1 volt at 0 percent of physical
measurement, and 5 volts at 100 percent of physical
measurement. The standard analog current signal range is 4
mA to 20 mA, signifying 0% to 100% of measurement range,
respectively. At 5 volts input, the 250 O (precision) resistor will
have 5 volts applied across it, resulting in 20 mA of current in
the large loop circuit (with Rjgaq). It does not matter what
resistance value Rjgag is, or how much wire resistance is present
in that large loop, so long as the op-amp has a high enough
power supply voltage to output the voltage necessary to get 20
mA flowing through Rjgag. The 250 Q resistor establishes the
relationship between input voltage and output current, in this
case creating the equivalence of 1-5 V in / 4-20 mA out. If we
were converting the 1-5 volt input signal to a 10-50 mA output
signal (an older, obsolete instrumentation standard for
industry), we'd use a 100 O precision resistor instead.
Another name for this circuit is transconductance amplifier. |n
electronics, transconductance is the mathematical ratio of
current change divided by voltage change (Al / A V), and it is
measured in the unit of Siemens, the same unit used to express
conductance (the mathematical reciprocal of resistance:
current/voltage). In this circuit, the transconductance ratio is
fixed by the value of the 250 O resistor, giving a linear current-
out/voltage-in relationship.
e REVIEW:
e In industry, DC current signals are often used in preference
to DC voltage signals as analog representations of physical
quantities. Current in a series circuit is absolutely equal at
all points in that circuit regardless of wiring resistance,
whereas voltage in a parallel-connected circuit may vary
from end to end because of wire resistance, making
current-signaling more accurate from the "transmitting" to
the "receiving" instrument.
e Voltage signals are relatively easy to produce directly from
transducer devices, whereas accurate current signals are
not. Op-amps can be used to "convert" a voltage signal
into a current signal quite easily. In this mode, the op-amp
will output whatever voltage is necessary to maintain
current through the signaling circuit at the proper value.
Averager and summer circuits
If we take three equal resistors and connect one end of each to
a common point, then apply three input voltages (one to each
of the resistors' free ends), the voltage seen at the common
point will be the mathematical average of the three.
"Passive averager" circuit
With equal value resistors:
3
This circuit is really nothing more than a practical application
of Millman's Theorem:
<
<
|.x
—_—
—
|
This circuit is Commonly known as a passive averager, because
it generates an average voltage with non-amplifying
components. Passive simply means that it is an unamplified
circuit. The large equation to the right of the averager circuit
comes from Millman's Theorem, which describes the voltage
produced by multiple voltage sources connected together
through individual resistances. Since the three resistors in the
averager circuit are equal to each other, we can simplify
Millman's formula by writing Rj, Rz, and R3 simply as R (one,
equal resistance instead of three individual resistances):
Vout =
+ +
Vv; +V,+V
R
Vout = ny
aan
R
. Vi + V2. +V
If we take a passive averager and use it to connect three input
voltages into an op-amp amplifier circuit with a gain of 3, we
can turn this averaging function into an addition function. The
result is called a noninverting summer circuit:
1kQ 2 kQ
With a voltage divider composed of a 2 KQ/ 1 KO combination,
the noninverting amplifier circuit will have a voltage gain of 3.
By taking the voltage from the passive averager, which is the
sum of Vj, V>, and V3 divided by 3, and multiplying that
average by 3, we arrive at an output voltage equal to the sum
of Vi, V>, and V3:
V,+V,+V;
Vour = 3 a ae
Vou = Vi + V.+ V3
Much the same can be done with an inverting op-amp
amplifier, using a passive averager as part of the voltage
divider feedback circuit. The result is called an inverting
summer circuit:
R <— |,
Now, with the right-hand sides of the three averaging resistors
connected to the virtual ground point of the op-amp's inverting
input, Millman's Theorem no longer directly applies as it did
before. The voltage at the virtual ground is now held at 0 volts
by the op-amp's negative feedback, whereas before it was free
to float to the average value of Vj, V2, and V3. However, with
all resistor values equal to each other, the currents through
each of the three resistors will be proportional to their
respective input voltages. Since those three currents will add
at the virtual ground node, the algebraic sum of those currents
through the feedback resistor will produce a voltage at Vout
equal to V; + V> + V3, except with reversed polarity. The
reversal in polarity is what makes this circuit an inverting
summer:
Vou = -(V, + V3 + V3)
Summer (adder) circuits are quite useful in analog computer
design, just as multiplier and divider circuits would be. Again,
it is the extremely high differential gain of the op-amp which
allows us to build these useful circuits with a bare minimum of
components.
e REVIEW:
e A summer circuit is one that sums, or adds, multiple analog
voltage signals together. There are two basic varieties of
Op-amp summer circuits: noninverting and inverting.
Building a differential amplifier
An op-amp with no feedback is already a differential amplifier,
amplifying the voltage difference between the two inputs.
However, its gain cannot be controlled, and it is generally too
high to be of any practical use. So far, our application of
negative feedback to op-amps has resulting in the practical
loss of one of the inputs, the resulting amplifier only good for
amplifying a single voltage signal input. With a little ingenuity,
however, we can construct an op-amp circuit maintaining both
voltage inputs, yet with a controlled gain set by external
resistors.
vi
out
If all the resistor values are equal, this amplifier will have a
differential voltage gain of 1. The analysis of this circuit is
essentially the same as that of an inverting amplifier, except
that the noninverting input (+) of the op-amp is at a voltage
equal to a fraction of V3, rather than being connected directly
to ground. As would stand to reason, V> functions as the
noninverting input and V, functions as the inverting input of
the final amplifier circuit. Therefore:
Von = V>- Vi
If we wanted to provide a differential gain of anything other
than 1, we would have to adjust the resistances in both upper
and lower voltage dividers, necessitating multiple resistor
changes and balancing between the two dividers for
symmetrical operation. This is not always practical, for obvious
reasons.
Another limitation of this amplifier design is the fact that its
input impedances are rather low compared to that of some
other op-amp configurations, most notably the noninverting
(single-ended input) amplifier. Each input voltage source has
to drive current through a resistance, which constitutes far less
impedance than the bare input of an op-amp alone. The
solution to this problem, fortunately, is quite simple. All we
need to do is "buffer" each input voltage signal through a
voltage follower like this:
Now the V, and V> input lines are connected straight to the
inputs of two voltage-follower op-amps, giving very high
impedance. The two op-amps on the left now handle the
driving of current through the resistors instead of letting the
input voltage sources (whatever they may be) do it. The
increased complexity to our circuit is minimal for a substantial
benefit.
The instrumentation amplifier
As suggested before, it is beneficial to be able to adjust the
gain of the amplifier circuit without having to change more
than one resistor value, as is necessary with the previous
design of differential amplifier. The so-called instrumentation
builds on the last version of differential amplifier to give us
that capability:
out
This intimidating circuit is constructed from a buffered
differential amplifier stage with three new resistors linking the
two buffer circuits together. Consider all resistors to be of equal
value except for Rgain. The negative feedback of the upper-left
op-amp causes the voltage at point 1 (top of Rgain) to be equal
to V;. Likewise, the voltage at point 2 (bottom of Rgain) is held
to a value equal to V>. This establishes a voltage drop across
Rgain equal to the voltage difference between V, and Vp. That
voltage drop causes a current through Rgajn, and since the
feedback loops of the two input op-amps draw no current, that
same amount of current through Rgain Must be going through
the two "R" resistors above and below it. This produces a
voltage drop between points 3 and 4 equal to:
V34=(V>-V d+ RR)
gain
The regular differential amplifier on the right-hand side of the
circuit then takes this voltage drop between points 3 and 4,
and amplifies it by a gain of 1 (assuming again that all "R"
resistors are of equal value). Though this looks like a
cumbersome way to build a differential amplifier, it has the
distinct advantages of possessing extremely high input
impedances on the V, and V> inputs (because they connect
straight into the noninverting inputs of their respective op-
amps), and adjustable gain that can be set by a single resistor.
Manipulating the above formula a bit, we have a general
expression for overall voltage gain in the instrumentation
amplifier:
2R
R
eain
Ay =(1 + )
Though it may not be obvious by looking at the schematic, we
can change the differential gain of the instrumentation
amplifier simply by changing the value of one resistor: Rgain.
Yes, we could still change the overall gain by changing the
values of some of the other resistors, but this would necessitate
balanced resistor value changes for the circuit to remain
symmetrical. Please note that the lowest gain possible with the
above circuit is obtained with Rgai, completely open (infinite
resistance), and that gain value is 1.
e REVIEW:
e An instrumentation amplifier is a differential op-amp circuit
providing high input impedances with ease of gain
adjustment through the variation of a single resistor.
Differentiator and integrator circuits
By introducing electrical reactance into the feedback loops of
Op-amp amplifier circuits, we can cause the output to respond
to changes in the input voltage over time. Drawing their names
from their respective calculus functions, the integrator
produces a voltage output proportional to the product
(multiplication) of the input voltage and time; and the
differentiator (not to be confused with differentia/) produces a
voltage output proportional to the input voltage's rate of
change.
Capacitance can be defined as the measure of a capacitor's
opposition to changes in voltage. The greater the capacitance,
the more the opposition. Capacitors oppose voltage change by
creating current in the circuit: that is, they either charge or
discharge in response to a change in applied voltage. So, the
more capacitance a capacitor has, the greater its charge or
discharge current will be for any given rate of voltage change
across it. The equation for this is quite simple:
Changing
DC Z :
voltage
_qo dv
i=C ae
The dv/at fraction is a calculus expression representing the rate
of voltage change over time. If the DC supply in the above
circuit were steadily increased from a voltage of 15 volts toa
voltage of 16 volts over a time span of 1 hour, the current
through the capacitor would most likely be very small, because
of the very low rate of voltage change (dv/dt = 1 volt / 3600
seconds). However, if we steadily increased the DC supply from
15 volts to 16 volts over a shorter time span of 1 second, the
rate of voltage change would be much higher, and thus the
charging current would be much higher (3600 times higher, to
be exact). Same amount of change in voltage, but vastly
different rates of change, resulting in vastly different amounts
of current in the circuit.
To put some definite numbers to this formula, if the voltage
across a 47 UF capacitor was changing at a linear rate of 3 volts
per second, the current "through" the capacitor would be (47
UF)(3 V/s) = 141 UA.
We can build an op-amp circuit which measures change in
voltage by measuring current through a capacitor, and outputs
a voltage proportional to that current:
Differentiator
C
OV R
Vin —
OV
Nat
OV
The right-hand side of the capacitor is held to a voltage of 0
volts, due to the "virtual ground" effect. Therefore, current
"through" the capacitor is solely due to change in the input
voltage. A steady input voltage won't cause a current through
C, but a changing input voltage will.
Capacitor current moves through the feedback resistor,
producing a drop across it, which is the same as the output
voltage. A linear, positive rate of input voltage change will
result in a steady negative voltage at the output of the op-amp.
Conversely, a linear, negative rate of input voltage change will
result in a steady positive voltage at the output of the op-amp.
This polarity inversion from input to output is due to the fact
that the input signal is being sent (essentially) to the inverting
input of the op-amp, so it acts like the inverting amplifier
mentioned previously. The faster the rate of voltage change at
the input (either positive or negative), the greater the voltage
at the output.
The formula for determining voltage output for the
differentiator is as follows:
_ -RC dvi,
i dt
Applications for this, besides representing the derivative
calculus function inside of an analog computer, include rate-of-
change indicators for process instrumentation. One such rate-
of-change signal application might be for monitoring (or
controlling) the rate of temperature change in a furnace, where
too high or too low of a temperature rise rate could be
detrimental. The DC voltage produced by the differentiator
circuit could be used to drive a comparator, which would signal
an alarm or activate a control if the rate of change exceeded a
pre-set level.
In process control, the derivative function is used to make
control decisions for maintaining a process at setpoint, by
monitoring the rate of process change over time and taking
action to prevent excessive rates of change, which can lead to
an unstable condition. Analog electronic controllers use
variations of this circuitry to perform the derivative function.
On the other hand, there are applications where we need
precisely the opposite function, called integration in calculus.
Here, the op-amp circuit would generate an output voltage
proportional to the magnitude and duration that an input
voltage signal has deviated from 0 volts. Stated differently, a
constant input signal would generate a certain rate of change
in the output voltage: differentiation in reverse. To do this, all
we have to do is swap the capacitor and resistor in the previous
circuit:
Integrator
R OV =
OV
out
OV
As before, the negative feedback of the op-amp ensures that
the inverting input will be held at 0 volts (the virtual ground). If
the input voltage is exactly 0 volts, there will be no current
through the resistor, therefore no charging of the capacitor,
and therefore the output voltage will not change. We cannot
guarantee what voltage will be at the output with respect to
ground in this condition, but we can say that the output
voltage will be constant.
However, if we apply a constant, positive voltage to the input,
the op-amp output will fall negative at a linear rate, in an
attempt to produce the changing voltage across the capacitor
necessary to maintain the current established by the voltage
difference across the resistor. Conversely, a constant, negative
voltage at the input results in a linear, rising (positive) voltage
at the output. The output voltage rate-of-change will be
proportional to the value of the input voltage.
The formula for determining voltage output for the integrator is
as follows:
dV ou —o Vin
dt RC
or
t
V.
Vor=l- =i dt+ce
out 0 RC c
Where,
c = Output voltage at start time (t=0)
One application for this device would be to keep a "running
total" of radiation exposure, or dosage, if the input voltage was
a proportional signal supplied by an electronic radiation
detector. Nuclear radiation can be just as damaging at low
intensities for long periods of time as it is at high intensities for
short periods of time. An integrator circuit would take both the
intensity (input voltage magnitude) and time into account,
generating an output voltage representing total radiation
dosage.
Another application would be to integrate a signal representing
water flow, producing a signal representing total quantity of
water that has passed by the flowmeter. This application of an
integrator is sometimes called a tota/izer in the industrial
instrumentation trade.
e REVIEW:
e A differentiator circuit produces a constant output voltage
for a steadily changing input voltage.
e An integrator circuit produces a steadily changing output
voltage for a constant input voltage.
¢ Both types of devices are easily constructed, using reactive
components (usually capacitors rather than inductors) in
the feedback part of the circuit.
Positive feedback
As we've seen, negative feedback is an incredibly useful
principle when applied to operational amplifiers. It is what
allows us to create all these practical circuits, being able to
precisely set gains, rates, and other significant parameters with
just a few changes of resistor values. Negative feedback makes
all these circuits stable and self-correcting.
The basic principle of negative feedback is that the output
tends to drive in a direction that creates a condition of
equilibrium (balance). In an op-amp circuit with no feedback,
there is no corrective mechanism, and the output voltage will
saturate with the tiniest amount of differential voltage applied
between the inputs. The result is a comparator:
With negative feedback (the output voltage "fed back"
somehow to the inverting input), the circuit tends to prevent
itself from driving the output to full saturation. Rather, the
output voltage drives only as high or as low as needed to
balance the two inputs’ voltages:
Negative feedback
out
Whether the output is directly fed back to the inverting (-)
input or coupled through a set of components, the effect is the
same: the extremely high differential voltage gain of the op-
amp will be "tamed" and the circuit will respond according to
the dictates of the feedback "loop" connecting output to
inverting input.
Another type of feedback, namely positive feedback, also finds
application in op-amp circuits. Unlike negative feedback, where
the output voltage is "fed back" to the inverting (-) input, with
positive feedback the output voltage is somehow routed back
to the noninverting (+) input. In its simplest form, we could
connect a straight piece of wire from output to noninverting
input and see what happens:
Positive feedback
out
The inverting input remains disconnected from the feedback
loop, and is free to receive an external voltage. Let's see what
happens if we ground the inverting input:
out
OV
With the inverting input grounded (maintained at zero volts),
the output voltage will be dictated by the magnitude and
polarity of the voltage at the noninverting input. If that voltage
happens to be positive, the op-amp will drive its output
positive as well, feeding that positive voltage back to the
noninverting input, which will result in full positive output
saturation. On the other hand, if the voltage on the
noninverting input happens to start out negative, the op-amp's
output will drive in the negative direction, feeding back to the
noninverting input and resulting in full negative saturation.
What we have here is a circuit whose output is bistable: stable
in one of two states (saturated positive or saturated negative).
Once it has reached one of those saturated states, it will tend
to remain in that state, unchanging. What is necessary to get it
to switch states is a voltage placed upon the inverting (-) input
of the same polarity, but of a slightly greater magnitude. For
example, if our circuit is saturated at an output voltage of +12
volts, it will take an input voltage at the inverting input of at
least +12 volts to get the output to change. When it changes,
it will saturate fully negative.
So, an op-amp with positive feedback tends to stay in whatever
output state its already in. It "latches" between one of two
states, saturated positive or saturated negative. Technically,
this is known as hysteresis.
Hysteresis can be a useful property for a comparator circuit to
have. As we've seen before, comparators can be used to
produce a square wave from any sort of ramping waveform
(sine wave, triangle wave, sawtooth wave, etc.) input. If the
incoming AC waveform is noise-free (that is, a "pure"
waveform), a simple comparator will work just fine.
+V
out
-V
Square wave
output voltage
voltage
AC input
voltage
A "clean" AC input waveform produces predictable
transition points on the output voltage square wave
However, if there exist any anomalies in the waveform such as
harmonics or "spikes" which cause the voltage to rise and fall
significantly within the timespan of a single cycle, a
comparator's output might switch states unexpectedly:
+V
-V
Square wave
output voltage
AC input
voltage
Any time there is a transition through the reference voltage
level, no matter how tiny that transition may be, the output of
the comparator will switch states, producing a square wave
with "glitches."
If we add a little positive feedback to the comparator circuit,
we will introduce hysteresis into the output. This hysteresis will
cause the output to remain in its current state unless the AC
input voltage undergoes a major change in magnitude.
+V
out
Positive feedback
resistor
What this feedback resistor creates is a dual-reference for the
comparator circuit. The voltage applied to the noninverting (+)
input as a reference which to compare with the incoming AC
voltage changes depending on the value of the op-amp's
output voltage. When the op-amp output is saturated positive,
the reference voltage at the noninverting input will be more
positive than before. Conversely, when the op-amp output is
saturated negative, the reference voltage at the noninverting
input will be more negative than before. The result is easier to
understand on a graph:
DC reference voltages
as el center
square wave
tput voltage
AC input
voltage
When the op-amp output is saturated positive, the upper
reference voltage is in effect, and the output won't drop to a
negative saturation level unless the AC input rises above that
upper reference level. Conversely, when the op-amp output is
saturated negative, the lower reference voltage is in effect, and
the output won't rise to a positive saturation level unless the
AC input drops be/ow that lower reference level. The result is a
clean square-wave output again, despite significant amounts of
distortion in the AC input signal. In order for a "glitch" to cause
the comparator to switch from one state to another, it would
have to be at least as big (tall) as the difference between the
upper and lower reference voltage levels, and at the right point
in time to cross both those levels.
Another application of positive feedback in op-amp circuits is
in the construction of oscillator circuits. An oscillator is a device
that produces an alternating (AC), or at least pulsing, output
voltage. Technically, it is known as an astable device: having
no stable output state (no equilibrium whatsoever). Oscillators
are very useful devices, and they are easily made with just an
Op-amp and a few external components.
Oscillator circuit using positive feedback
V wn IS a Square wave just like V,.;, only taller
refs
When the output is saturated positive, the V,a¢ will be positive,
and the capacitor will charge up in a positive direction. When
Vramp exceeds Vyer by the tiniest margin, the output will
saturate negative, and the capacitor will charge in the opposite
direction (polarity). Oscillation occurs because the positive
feedback is instantaneous and the negative feedback is
delayed (by means of an RC time constant). The frequency of
this oscillator may be adjusted by varying the size of any
component.
e REVIEW:
e Negative feedback creates a condition of equilibrium
(balance). Positive feedback creates a condition of
hysteresis (the tendency to "latch" in one of two extreme
states).
e An oscillator is a device producing an alternating or pulsing
output voltage.
Practical considerations
Real operational have some imperfections compared to an
“ideal” model. A real device deviates from a perfect difference
amplifier. One minus one may not be Zero. It may have have an
offset like an analog meter which is not zeroed. The inputs may
draw current. The characteristics may drift with age and
temperature. Gain may be reduced at high frequencies, and
phase may shift from input to output. These imperfection may
cause no noticable errors in some applications, unacceptable
errors in others. In some cases these errors may be
compensated for. Sometimes a higher quality, higher cost
device is required.
Common-mode gain
As stated before, an ideal differential amplifier only amplifies
the voltage difference between its two inputs. If the two inputs
of a differential amplifier were to be shorted together (thus
ensuring zero potential difference between them), there should
be no change in output voltage for any amount of voltage
applied between those two shorted inputs and ground:
out
V.. should remain the same
V regardless of V.
common-mode__ ommon-mode
dl
Voltage that is common between either of the inputs and
ground, aS "Viommon-mode_ !S in this case, is called common-
mode voltage. As we vary this common voltage, the perfect
differential amplifier's output voltage should hold absolutely
steady (no change in output for any arbitrary change in
common-mode input). This translates to a common-mode
voltage gain of zero.
Change in V
out
~ Change in V,,
... lfchange inV,,,=0...
ee
Change in V,,
Ay =0
The operational amplifier, being a differential amplifier with
high differential gain, would ideally have zero common-mode
gain as well. In real life, however, this is not easily attained.
Thus, common-mode voltages will invariably have some effect
on the op-amp's output voltage.
The performance of a real op-amp in this regard is most
commonly measured in terms of its differential voltage gain
(how much it amplifies the difference between two input
voltages) versus its common-mode voltage gain (how much it
amplifies a common-mode voltage). The ratio of the former to
the latter is called the common-mode rejection ratio,
abbreviated as CMRR:
Differential A,,
CMRR = —____-
Common-mode A,,
An ideal op-amp, with zero common-mode gain would have an
infinite CMRR. Real op-amps have high CMRRs, the ubiquitous
741 having something around 70 dB, which works out to a little
over 3,000 in terms of a ratio.
Because the common mode rejection ratio in a typical op-amp
is so high, common-mode gain is usually not a great concern in
circuits where the op-amp is being used with negative
feedback. If the common-mode input voltage of an amplifier
circuit were to suddenly change, thus producing a
corresponding change in the output due to common-mode
gain, that change in output would be quickly corrected as
negative feedback and differential gain (being much greater
than common-mode gain) worked to bring the system back to
equilibrium. Sure enough, a change might be seen at the
output, but it would be a lot smaller than what you might
expect.
A consideration to keep in mind, though, is common-mode gain
in differential op-amp circuits such as instrumentation
amplifiers. Outside of the op-amp's sealed package and
extremely high differential gain, we may find common-mode
gain introduced by an imbalance of resistor values. To
demonstrate this, we'll run a SPICE analysis on an
instrumentation amplifier with inputs shorted together (no
differential voltage), imposing a common-mode voltage to see
what happens. First, we'll run the analysis showing the output
voltage of a perfectly balanced circuit. We should expect to see
no change in output voltage as the common-mode voltage
changes:
; Prrcuaes
(jumper
wire)
instrumentation amplifier
vl 10
rinl 1 0 9el12
rjump 1 4 le-12
rin2 4 0 9el12
el 3 0 1 2 999k
e2 6 0 4 5 999k
e3 9 0 8 7 999k
rload 9 0 10k
rl 2 3 10k
rgain 2 5 10k
r2 5 6 10k
r3 3 7 10k
r4 7 9 10k
r5 6 8 10k
r6 8 0 10k
.dc vl 0 10 1
.print dc v(9)
.end
vl v(9)
0.000E+00 0.000E+00
1.000E+00 1.355E-16
2.000E+00 2.710E-16
3.000E+00 0.000E+00
v(9)
4.000E+00 5.421E-16
mode
5.000E+00 0.000E+00
6.000E+00 0.000E+00
7 .Q000E+00 0.000E+00
8.000E+00 1.084E-15
9.000E+00 1.084E-15
1.000E+01 0.000E+00
As you can see, the output voltage
hardly changes at all for a common-
input voltage (vl) that sweeps from 0
to 10 volts.
Aside from very small deviations (actually due to quirks of
SPICE rather than real behavior of the circuit), the output
remains stable where it should be: at 0 volts, with zero input
voltage differential. However, let's introduce a resistor
imbalance in the circuit, increasing the value of Rs from 10,000
Q to 10,500 Q, and see what happens (the netlist has been
omitted for brevity -- the only thing altered is the value of Rs):
vl v(9)
0.000E+00 0.000E+00
1.000E+00 -2.439E-02
2.Q000E+00 -4.878E-02
3.000E+00 -7.317E-02 This time we see a significant
variation
4.000E+00 -9.756E-02 (from 0 to 0.2439 volts) in output
voltage
5 .000E+00 -1.220E-01 as the common-mode input voltage
Sweeps
6.000E+00 -1.463E-01 from 0 to 10 volts as it did before.
7 .Q000E+00 -1.707E-01
8.000E+00 -1.951E-01
9.000E+00 -2.195E-01
1.000E+01 -2.439E-01
Our input voltage differential is still zero volts, yet the output
voltage changes significantly as the common-mode voltage is
changed. This is indicative of a common-mode gain, something
we're trying to avoid. More than that, its a common-mode gain
of our own making, having nothing to do with imperfections in
the op-amps themselves. With a much-tempered differential
gain (actually equal to 3 in this particular circuit) and no
negative feedback outside the circuit, this common-mode gain
will go unchecked in an instrument signal application.
There is only one way to correct this common-mode gain, and
that is to balance all the resistor values. When designing an
instrumentation amplifier from discrete components (rather
than purchasing one in an integrated package), it is wise to
provide some means of making fine adjustments to at least one
of the four resistors connected to the final op-amp to be able to
"trim away" any such common-mode gain. Providing the means
to "trim" the resistor network has additional benefits as well.
Suppose that all resistor values are exactly as they should be,
but a common-mode gain exists due to an imperfection in one
of the op-amps. With the adjustment provision, the resistance
could be trimmed to compensate for this unwanted gain.
One quirk of some op-amp models is that of output /atch-up,
usually caused by the common-mode input voltage exceeding
allowable limits. If the common-mode voltage falls outside of
the manufacturer's specified limits, the output may suddenly
"latch" in the high mode (saturate at full output voltage). In
JFET-input operational amplifiers, latch-up may occur if the
common-mode input voltage approaches too closely to the
negative power supply rail voltage. On the TLO82 op-amp, for
example, this occurs when the common-mode input voltage
comes within about 0.7 volts of the negative power supply rail
voltage. Such a situation may easily occur in a single-supply
circuit, where the negative power supply rail is ground (0
volts), and the input signal is free to swing to O volts.
Latch-up may also be triggered by the common-mode input
voltage exceeding power supply rail voltages, negative or
positive. As a rule, you should never allow either input voltage
to rise above the positive power supply rail voltage, or sink
below the negative power supply rail voltage, even if the op-
amp in question is protected against latch-up (as are the 741
and 1458 op-amp models). At the very least, the op-amp's
behavior may become unpredictable. At worst, the kind of
latch-up triggered by input voltages exceeding power supply
voltages may be destructive to the op-amp.
While this problem may seem easy to avoid, its possibility is
more likely than you might think. Consider the case of an
operational amplifier circuit during power-up. If the circuit
receives full input signal voltage before its own power supply
has had time enough to charge the filter capacitors, the
common-mode input voltage may easily exceed the power
supply rail voltages for a short time. If the op-amp receives
signal voltage from a circuit supplied by a different power
source, and its own power source fails, the signal voltage(s)
may exceed the power supply rail voltages for an indefinite
amount of time!
Offset voltage
Another practical concern for op-amp performance is vo/tage
offset. That is, effect of having the output voltage something
other than zero volts when the two input terminals are shorted
together. Remember that operational amplifiers are differential
amplifiers above all: they're supposed to amplify the difference
in voltage between the two input connections and nothing
more. When that input voltage difference is exactly zero volts,
we would (ideally) expect to have exactly zero volts present on
the output. However, in the real world this rarely happens.
Even if the op-amp in question has zero common-mode gain
(infinite CMRR), the output voltage may not be at zero when
both inputs are shorted together. This deviation from zero is
called offset.
+15 V
Vout = +14.7 V (saturated +)
A perfect op-amp would output exactly zero volts with both its
inputs shorted together and grounded. However, most op-amps
off the shelf will drive their outputs to a saturated level, either
negative or positive. In the example shown above, the output
voltage is saturated at a value of positive 14.7 volts, just a bit
less than +V (+15 volts) due to the positive saturation limit of
this particular op-amp. Because the offset in this op-amp is
driving the output to a completely saturated point, there's no
way of telling how much voltage offset is present at the output.
If the +V/-V split power supply was of a high enough voltage,
who knows, maybe the output would be several hundred volts
one way or the other due to the effects of offset!
For this reason, offset voltage is usually expressed in terms of
the equivalent amount of /nout voltage differential producing
this effect. In other words, we imagine that the op-amp is
perfect (no offset whatsoever), and a small voltage is being
applied in series with one of the inputs to force the output
voltage one way or the other away from zero. Being that op-
amp differential gains are so high, the figure for "input offset
voltage" doesn't have to be much to account for what we see
with shorted inputs:
+15 V
V un = +14.7 V (saturated +)
Input offset voltage
(internal to the real op-amp,
external to this ideal op-amp)
Offset voltage will tend to introduce slight errors in any op-amp
circuit. So how do we compensate for it? Unlike common-mode
gain, there are usually provisions made by the manufacturer to
trim the offset of a packaged op-amp. Usually, two extra
terminals on the op-amp package are reserved for connecting
an external "trim" potentiometer. These connection points are
labeled offset nu// and are used in this general way:
+15 V
out
-15V
Potentiometer adjusted so that
V4. = 0 volts with inputs shorted together
On single op-amps such as the 741 and 3130, the offset null
connection points are pins 1 and 5 on the 8-pin DIP package.
Other models of op-amp may have the offset null connections
located on different pins, and/or require a slightly difference
configuration of trim potentiometer connection. Some op-amps
don't provide offset null pins at all! Consult the manufacturer's
specifications for details.
Bias current
Inputs on an op-amp have extremely high input impedances.
That is, the input currents entering or exiting an op-amp's two
input signal connections are extremely small. For most
purposes of op-amp circuit analysis, we treat them as though
they don't exist at all. We analyze the circuit as though there
was absolutely zero current entering or exiting the input
connections.
This idyllic picture, however, is not entirely true. Op-amps,
especially those op-amps with bipolar transistor inputs, have to
have some amount of current through their input connections
in order for their internal circuits to be properly biased. These
currents, logically, are called bias currents. Under certain
conditions, op-amp bias currents may be problematic. The
following circuit illustrates one of those problem conditions:
+V
Thermocouple V
out
-V
At first glance, we see no apparent problems with this circuit. A
thermocouple, generating a small voltage proportional to
temperature (actually, a voltage proportional to the difference
in temperature between the measurement junction and the
"reference" junction formed when the alloy thermocouple wires
connect with the copper wires leading to the op-amp) drives
the op-amp either positive or negative. In other words, this is a
kind of comparator circuit, comparing the temperature
between the end thermocouple junction and the reference
junction (near the op-amp). The problem is this: the wire loop
formed by the thermocouple does not provide a path for both
input bias currents, because both bias currents are trying to go
the same way (either into the op-amp or out of it).
Thermocouple
This comparator circuit won’t work
In order for this circuit to work properly, we must ground one of
the input wires, thus providing a path to (or from) ground for
both currents:
+V
Thermocouple Vv
out
This comparator circuit will work
Not necessarily an obvious problem, but a very real one!
Another way input bias currents may cause trouble is by
dropping unwanted voltages across circuit resistances. Take
this circuit for example:
Voltage drop due
to bias current:
out
V Ibias BF
af Voltage at (+) op-amp input
= will not be exactly equal to V,,
We expect a voltage follower circuit such as the one above to
reproduce the input voltage precisely at the output. But what
about the resistance in series with the input voltage source? If
there is any bias current through the noninverting (+) input at
all, it will drop some voltage across R;,, thus making the
voltage at the noninverting input unequal to the actual V,,
value. Bias currents are usually in the microamp range, so the
voltage drop across R,, won't be very much, unless R,,, is very
large. One example of an application where the input
resistance (R,,) would be very large is that of pH probe
electrodes, where one electrode contains an ion-permeable
glass barrier (a very poor conductor, with millions of Q of
resistance).
If we were actually building an op-amp circuit for pH electrode
voltage measurement, we'd probably want to use a FET or
MOSFET (IGFET) input op-amp instead of one built with bipolar
transistors (for less input bias current). But even then, what
Slight bias currents may remain can cause measurement errors
to occur, so we have to find some way to mitigate them
through good design.
One way to do so is based on the assumption that the two
input bias currents will be the same. In reality, they are often
close to being the same, the difference between them referred
to as the /nput offset current. |f they are the same, then we
should be able to cancel out the effects of input resistance
voltage drop by inserting an equal amount of resistance in
series with the other input, like this:
out
With the additional resistance added to the circuit, the output
voltage will be closer to V;, than before, even if there is some
offset between the two input currents.
For both inverting and noninverting amplifier circuits, the bias
current compensating resistor is placed in series with the
noninverting (+) input to compensate for bias current voltage
drops in the divider network:
Noninverting amplifier with
compensating resistor
R
comp
in —
| Reomp = R, Mf R,
Inverting amplifier with
compensating resistor
R
comp
Ream = R, // R,
comp
In either case, the compensating resistor value is determined
by calculating the parallel resistance value of R; and R>5. Why
is the value equal to the paral/le/ equivalent of R; and R>?
When using the Superposition Theorem to figure how much
voltage drop will be produced by the inverting (-) input's bias
current, we treat the bias current as though it were coming
from a current source inside the op-amp and short-circuit all
voltage sources (V;, and Voy). This gives two parallel paths for
bias current (through R, and through R3>, both to ground). We
want to duplicate the bias current's effect on the noninverting
(+) input, so the resistor value we choose to insert in series
with that input needs to be equal to Rj in parallel with R>.
A related problem, occasionally experienced by students just
learning to build operational amplifier circuits, is caused by a
lack of a common ground connection to the power supply. It is
imperative to proper op-amp function that some terminal of the
DC power supply be common to the "ground" connection of the
input signal(s). This provides a complete path for the bias
currents, feedback current(s), and for the load (output) current.
Take this circuit illustration, for instance, showing a properly
grounded power supply:
Here, arrows denote the path of electron flow through the
power supply batteries, both for powering the op-amp's
internal circuitry (the "potentiometer" inside of it that controls
output voltage), and for powering the feedback loop of
resistors R; and R>. Suppose, however, that the ground
connection for this "split" DC power supply were to be
removed. The effect of doing this is profound:
A power supply ground is essential to circuit operation!
broken
connection
=
|
No electrons may flow in or out of the op-amp's output
terminal, because the pathway to the power supply is a "dead
end." Thus, no electrons flow through the ground connection to
the left of R;, neither through the feedback loop. This
effectively renders the op-amp useless: it can neither sustain
current through the feedback loop, nor through a grounded
load, since there is no connection from any point of the power
supply to ground.
The bias currents are also stopped, because they rely on a path
to the power supply and back to the input source through
ground. The following diagram shows the bias currents (only),
as they go through the input terminals of the op-amp, through
the base terminals of the input transistors, and eventually
through the power supply terminal(s) and back to ground.
Bias current paths shown, through power supply
lias
lias
<———
GY aoa | |
Without a ground reference on the power supply, the bias
currents will have no complete path for a circuit, and they will
halt. Since bipolar junction transistors are current-controlled
devices, this renders the input stage of the op-amp useless as
well, as both input transistors will be forced into cutoff by the
complete lack of base current.
REVIEW:
Op-amp inputs usually conduct very small currents, called
bias currents, needed to properly bias the first transistor
amplifier stage internal to the op-amps' circuitry. Bias
currents are small (in the microamp range), but large
enough to cause problems in some applications.
Bias currents in both inputs must have paths to flow to
either one of the power supply "rails" or to ground. It is not
enough to just have a conductive path from one input to
the other.
To cancel any offset voltages caused by bias current
flowing through resistances, just add an equivalent
resistance in series with the other op-amp input (called a
compensating resistor). This corrective measure is based
on the assumption that the two input bias currents will be
equal.
e Any inequality between bias currents in an op-amp
constitutes what is called an input offset current.
e It is essential for proper op-amp operation that there bea
ground reference on some terminal of the power supply, to
form complete paths for bias currents, feedback current(s),
and load current.
Drift
Being semiconductor devices, op-amps are subject to slight
changes in behavior with changes in operating temperature.
Any changes in op-amp performance with temperature fall
under the category of op-amp adrift. Drift parameters can be
specified for bias currents, offset voltage, and the like. Consult
the manufacturer's data sheet for specifics on any particular
Op-amp.
To minimize op-amp drift, we can select an op-amp made to
have minimum drift, and/or we can do our best to keep the
operating temperature as stable as possible. The latter action
may involve providing some form of temperature control for the
inside of the equipment housing the op-amp(s). This is not as
strange as it may first seem. Laboratory-standard precision
voltage reference generators, for example, are sometimes
known to employ "ovens" for keeping their sensitive
components (such as zener diodes) at constant temperatures.
If extremely high accuracy is desired over the usual factors of
cost and flexibility, this may be an option worth looking at.
e REVIEW:
e Op-amps, being semiconductor devices, are susceptible to
variations in temperature. Any variations in amplifier
performance resulting from changes in temperature is
known as adrift. Drift is best minimized with environmental
temperature control.
Frequency response
With their incredibly high differential voltage gains, op-amps
are prime candidates for a phenomenon known as feedback
oscillation. You've probably heard the equivalent audio effect
when the volume (gain) on a public-address or other
microphone amplifier system is turned too high: that high
pitched squeal resulting from the sound waveform "feeding
back" through the microphone to be amplified again. An op-
amp circuit can manifest this same effect, with the feedback
happening electrically rather than audibly.
A case example of this is seen in the 3130 op-amp, if it is
connected as a voltage follower with the bare minimum of
wiring connections (the two inputs, output, and the power
supply connections). The output of this op-amp will self-
oscillate due to its high gain, no matter what the input voltage.
To combat this, a small compensation capacitor must be
connected to two specially-provided terminals on the op-amp.
The capacitor provides a high-impedance path for negative
feedback to occur within the op-amp's circuitry, thus
decreasing the AC gain and inhibiting unwanted oscillations. If
the op-amp is being used to amplify high-frequency signals,
this compensation capacitor may not be needed, but it is
absolutely essential for DC or low-frequency AC signal
operation.
Some op-amps, such as the model 741, have a compensation
capacitor built in to minimize the need for external
components. This improved simplicity is not without a cost:
due to that capacitor's presence inside the op-amp, the
negative feedback tends to get stronger as the operating
frequency increases (that capacitor's reactance decreases with
higher frequencies). As a result, the op-amp's differential
voltage gain decreases as frequency goes up: it becomes a less
effective amplifier at higher frequencies.
Op-amp manufacturers will publish the frequency response
curves for their products. Since a sufficiently high differential
gain is absolutely essential to good feedback operation in op-
amp circuits, the gain/frequency response of an op-amp
effectively limits its "bandwidth" of operation. The circuit
designer must take this into account if good performance is to
be maintained over the required range of signal frequencies.
e REVIEW:
e Due to capacitances within op-amps, their differential
voltage gain tends to decrease as the input frequency
increases. Frequency response curves for op-amps are
available from the manufacturer.
Input to output phase shift
In order to illustrate the phase shift from input to output of an
operational amplifier (op-amp), the OPA227 was tested in our
lab. The OPA227 was constructed in a typical non-inverting
configuration (Figure below).
RAE iM
OPA227 Non-inverting stage
The circuit configuration calls for a signal gain of =34 V/V or
=50 dB. The input excitation at Vsrc was set to 10 mVp, and
three frequencies of interest: 2.2 kHz, 22 kHz, and 220 MHz.
The OPA227's open loop gain and phase curve vs. frequency is
shown in Figure below.
OPEN-LOOP GAIN/PHASE vs FREQUENCY
Ag, (dB)
Phase (°)
10 1 10 100 tk 10K 100k 1M 10M 100M
Frequency (Hz)
I
I
I
I
I
I
ll
C
Ay and ® vs. Frequency plot
To help predict the closed loop phase shift from input to output,
we can use the open loop gain and phase curve. Since the
circuit configuration calls for a closed loop gain, or 1/8, of =50
dB, the closed loop gain curve intersects the open loop gain
curve at approximately 22 kHz. After this intersection, the
closed loop gain curve rolls off at the typical 20 dB/decade for
voltage feedback amplifiers, and follows the open loop gain
curve.
What is actually at work here is the negative feedback from the
closed loop modifies the open loop response. Closing the loop
with negative feedback establishes a closed loop pole at 22
kHz. Much like the dominant pole in the open loop phase
curve, we will expect phase shift in the closed loop response.
How much phase shift will we see?
Since the new pole is now at 22 KHz, this is also the -3 dB point
as the pole starts to roll off the closed loop again at 20 dB per
decade as stated earlier. As with any pole in basic control
theory, phase shift starts to occur one decade in frequency
before the pole, and ends at 90° of phase shift one decade in
frequency after the pole. So what does this predict for the
closed loop response in our circuit?
This will predict phase shift starting at 2.2 kHz, with 45° of
phase shift at the -3 dB point of 22 kHz, and finally ending with
90° of phase shift at 220 kHz. The three Figures shown below
are oscilloscope captures at the frequencies of interest for our
OPA227 circuit. Figure below is set for 2.2 kHz, and no
noticeable phase shift is present. Figure below is set for 220
kHz, and =45° of phase shift is recorded. Finally, Figure below
is set for 220 MHz, and the expected =90° of phase shift is
recorded. The scope plots were captured using a LeCroy 44x
Wavesurfer. The final scope plot used a x1 probe with the
trigger set to HF reject.
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OPA227 Av=500B @ 2.2 kHz
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OPA227 Av=500B @ 22 kHz
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OPA227 Av=500B @ 220 kHz
Operational amplifier models
While mention of operational amplifiers typically provokes
visions of semiconductor devices built as integrated circuits on
a miniature silicon chip, the first op-amps were actually
vacuum tube circuits. The first commercial, general purpose
operational amplifier was manufactured by the George A.
Philbrick Researches, Incorporated, in 1952. Designated the
K2-W, it was built around two twin-triode tubes mounted in an
assembly with an octal (8-pin) socket for easy installation and
servicing in electronic equipment chassis of that era. The
assembly looked something like this:
The Philbrick Researches
op-amp, model K2-W
approx.
4 inches
The schematic diagram shows the two tubes, along with ten
resistors and two capacitors, a fairly simple circuit design even
by 1952 standards:
The Philbrick Researches op-amp, mode! K2-W
+300 V
. - +
S 220k0 Ske S 680 ko $
+ NE-68
+ A
12AX7 12AX7 75 pF
Inverting (-)
input
wu
Noninverting (+)
Output
Input
120 kQ a MQ
-300 V
In case you're unfamiliar with the operation of vacuum tubes,
they operate similarly to N-channel depletion-type IGFET
transistors: that is, they conduct more current when the control
grid (the dashed line) is made more positive with respect to the
cathode (the bent line near the bottom of the tube symbol),
and conduct less current when the control grid is made less
positive (or more negative) than the cathode. The twin triode
tube on the left functions as a differential pair, converting the
differential inputs (inverting and noninverting input voltage
signals) into a single, amplified voltage signal which is then fed
to the control grid of the left triode of the second triode pair
through a voltage divider (1 MQ -- 2.2 MQ). That triode
amplifies and inverts the output of the differential pair for a
larger voltage gain, then the amplified signal is coupled to the
second triode of the same dual-triode tube in a noninverting
amplifier configuration for a larger current gain. The two neon
"glow tubes" act as voltage regulators, similar to the behavior
of semiconductor zener diodes, to provide a bias voltage in the
coupling between the two single-ended amplifier triodes.
With a dual-supply voltage of +300/-300 volts, this op-amp
could only swing its output +/- 50 volts, which is very poor by
today's standards. It had an open-loop voltage gain of 15,000
to 20,000, a slew rate of +/- 12 volts/usecond, a maximum
output current of 1 mA, a quiescent power consumption of over
3 watts (not including power for the tubes’ filaments!), and
cost about $24 in 1952 dollars. Better performance could have
been attained using a more sophisticated circuit design, but
only at the expense of greater power consumption, greater
cost, and decreased reliability.
With the advent of solid-state transistors, op-amps with far less
quiescent power consumption and increased reliability became
feasible, but many of the other performance parameters
remained about the same. Take for instance Philbrick's model
P55A, a general-purpose solid-state op-amp circa 1966. The
P55A sported an open-loop gain of 40,000, a slew rate of 1.5
volt/usecond and an output swing of +/- 11 volts (at a power
supply voltage of +/- 15 volts), a maximum output current of
2.2 MA, and a cost of $49 (or about $21 for the "utility grade"
version). The P55A, as well as other op-amps in Philbrick's
lineup of the time, was of discrete-component construction, its
constituent transistors, resistors, and capacitors housed ina
solid "brick" resembling a large integrated circuit package.
It isn't very difficult to build a crude operational amplifier using
discrete components. A schematic of one such circuit is shown
in Figure below.
Output
input (+) (-) input
A simple operational
amplifier made from
discrete components
A simple operational amplifier made from discrete
components.
While its performance is rather dismal by modern standards, it
demonstrates that complexity is not necessary to create a
minimally functional op-amp. Transistors Q3 and Q, form the
heart of another differential pair circuit, the semiconductor
equivalent of the first triode tube in the K2-W schematic. As it
was in the vacuum tube circuit, the purpose of a differential
pair is to amplify and convert a differential voltage between
the two input terminals to a single-ended output voltage.
With the advent of integrated-circuit (IC) technology, op-amp
designs experienced a dramatic increase in performance,
reliability, density, and economy. Between the years of 1964
and 1968, the Fairchild corporation introduced three models of
IC op-amps: the 702, 709, and the still-popular 741. While the
741 is now considered outdated in terms of performance, it is
still a favorite among hobbyists for its simplicity and fault
tolerance (short-circuit protection on the output, for instance).
Personal experience abusing many 741 op-amps has led me to
the conclusion that it is a hard chip to kill...
The internal schematic diagram for a model 741 op-amp is
shown in Figure below.
+V Internal schematic of a model 741 operational amplifier
> > >
Qs P| Qo Qn» Qu i
(-) Input
(-) INP . Qs
(+) input Q, 3 ” i: Ry
Quy Output
C, = LN Qs,
. oa
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Q; Q Qh i Qu a On
offset null —+ A
offset null
Qy
Q
RS R, R, SR, ae SR, Q,,
Schematic diagram of a model 741 op-amp.
By integrated circuit standards, the 741 is a very simple
device: an example of small-scale integration, or SSI
technology. It would be no small matter to build this circuit
using discrete components, so you can see the advantages of
even the most primitive integrated circuit technology over
discrete components where high parts counts are involved.
For the hobbyist, student, or engineer desiring greater
performance, there are literally hundreds of op-amp models to
choose from. Many sell for less than a dollar apiece, even retail!
Special-purpose instrumentation and radio-frequency (RF) op-
amps may be quite a bit more expensive. In this section | will
showcase several popular and affordable op-amps, comparing
and contrasting their performance specifications. The
venerable 741 is included as a "benchmark" for comparison,
although it is, as | said before, considered an obsolete design.
Widely used operational amplifiers
i Power : Bias
current
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50
00
5
2
5
2
5
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2
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20,
00
050
800
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Listed in Table above are but a few of the low-cost operational
amplifier models widely available from electronics suppliers.
Most of them are available through retail supply stores such as
Radio Shack. All are under $1.00 cost direct from the
manufacturer (year 2001 prices). As you can see, there is
substantial variation in performance between some of these
units. Take for instance the parameter of input bias current: the
CA3130 wins the prize for lowest, at 0.05 nA (or 50 pA), and
the LM833 has the highest at slightly over 1 WA. The model
CA3130 achieves its incredibly low bias current through the
use of MOSFET transistors in its input stage. One manufacturer
advertises the 3130's input impedance as 1.5 tera-ohms, or 1.5
x 1014 O! Other op-amps shown here with low bias current
Pree
figures use JFET input transistors, while the high bias current
models use bipolar input transistors.
While the 741 is specified in many electronic project
schematics and showcased in many textbooks, its performance
has long been surpassed by other designs in every measure.
Even some designs originally based on the 741 have been
improved over the years to far surpass original design
specifications. One such example is the model 1458, two op-
amps in an 8-pin DIP package, which at one time had the exact
same performance specifications as the single 741. In its latest
incarnation it boasts a wider power supply voltage range, a
slew rate 50 times as great, and almost twice the output
current capability of a 741, while still retaining the output
short-circuit protection feature of the 741. Op-amps with JFET
and MOSFET input transistors far exceed the 741's
performance in terms of bias current, and generally manage to
beat the 741 in terms of bandwidth and slew rate as well.
My own personal recommendations for op-amps are as such:
when low bias current is a priority (Such as in low-speed
integrator circuits), choose the 3130. For general-purpose DC
amplifier work, the 1458 offers good performance (and you get
two op-amps in the space of one package). For an upgrade in
performance, choose the model 353, as it is a pin-compatible
replacement for the 1458. The 353 is designed with JFET input
circuitry for very low bias current, and has a bandwidth 4 times
are great as the 1458, although its output current limit is lower
(but still short-circuit protected). It may be more difficult to find
on the shelf of your local electronics supply house, but it is just
as reasonably priced as the 1458.
If low power supply voltage is a requirement, | recommend the
model 324, as it functions on as low as 3 volts DC. Its input bias
current requirements are also low, and it provides four op-amps
in a single 14-pin chip. Its major weakness is speed, limited to
1 MHz bandwidth and an output slew rate of only 0.25 volts per
us. For high-frequency AC amplifier circuits, the 318 is a very
good "general purpose" model.
Special-purpose op-amps are available for modest cost which
provide better performance specifications. Many of these are
tailored for a specific type of performance advantage, such as
maximum bandwidth or minimum bias current. Take for
instance the op-amps, both designed for high bandwidth in
Table below.
High bandwidth operational amplifiers
Devices/|/Power Bandwidth Bias
package|supply current
foun) 1 oe) 0)
1
1_t0/14232 44,000
1/5/14 1900 40,000
The CLC404 lists at $21.80 (almost as much as George
Philbrick's first commercial op-amp, albeit without correction
for inflation), while the CLC425 is quite a bit less expensive at
$3.23 per unit. In both cases high speed is achieved at the
expense of high bias currents and restrictive power supply
voltage ranges. Some op-amps, designed for high power output
are listed in Table below.
High current operational amplifiers
Devices/|Power Bandwidth Bias Output
package|supply current current
(count) | (V) (MHz) (nA) (mA)
1
5 / 80). 1000
Sey)
Yes, the LM12CL actually has an output current rating of 13
amps (13,000 milliamps)! It lists at $14.40, which is not a lot of
money, considering the raw power of the device. The LM7171,
on the other hand, trades high current output ability for fast
voltage output ability (a high slew rate). It lists at $1.19, about
as low as some "general purpose" op-amps.
Amplifier packages may also be purchased as complete
application circuits as opposed to bare operational amplifiers.
The Burr-Brown and Analog Devices corporations, for example,
both long known for their precision amplifier product lines,
offer instrumentation amplifiers in pre-designed packages as
well as other specialized amplifier devices. In designs where
high precision and repeatability after repair is important, it
might be advantageous for the circuit designer to choose such
a pre-engineered amplifier "block" rather than build the circuit
from individual op-amps. Of course, these units typically cost
quite a bit more than individual op-amps.
Data
Parametrical data for all semiconductor op-amp models except
the CA3130 comes from National Semiconductor's online
resources, available at this website: [*]. Data for the CA3130
comes from Harris Semiconductor's CA3130/CA3130A
datasheet (file number 817.4).
Contributors
Contributors to this chapter are listed in chronological order of
their contributions, from most recent to first. See Appendix 2
(Contributor List) for dates and contact information.
Wayne Little (June 2007): Author, “Input to output phase
shift” subsection, in “Practical considerations” section.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—|| +4]
—| | +4/l—
Lessons In Electric Circuits
-- Volume Ill
Chapter 9
PRACTICAL ANALOG
SEMICONDUCTOR CIRCUITS
ElectroStatic Discharge
o ESD Damage Prevention
o Storage and Transportation of ESD sensitive
component and boards
o Conclusion
Power Supply circuits
o Power Supply types
o Power Supply Introduction
o Linear power supplies
Amplifier circuits -- PENDING
Oscillator circuits -- INCOMPLETE
o Varactor multiplier
e Phase-locked loops -- PENDING
e Radio circuits -- INCOMPLETE
e Computational circuits
e Measurement circuits -- INCOMPLETE
e Control circuits -- PENDING
e Contributors
e Bibliography
*& INCOMPLETE ***
ElectroStatic Discharge
Volume | chapter 1.1 discusses static electricity, and how it is
created. This has a lot more significance than might be first
assumed, as control of static electricity plays a large part in
modern electronics and other professions. An ElectroStatic
Discharge event is when a static charge is bled off in an
uncontrolled fashion, and will be referred to as ESD hereafter.
ESD comes in many forms, it can be as small as 50 volts of
electricity being equalized up to tens of thousands of volts.
The actual power is extremely small, so small that no danger
is generally offered to someone who is in the discharge path
of ESD. It usually takes several thousand volts for a person to
even notice ESD in the form of a spark and the familiar zap
that accompanies it. The problem with ESD is even a small
discharge that can go completely unnoticed can ruin
semiconductors. A static charge of thousands of volts is
common, however the reason it is not a threat is there is no
current of any substantial duration behind it. These extreme
voltages do allow ionization of the air and allow other
materials to break down, which is the root of where the
damage comes from.
ESD is not a new problem. Black powder manufacturing and
other pyrotechnic industries have always been dangerous If
an ESD event occurs in the wrong circumstance. During the
era of tubes (AKA valves) ESD was a nonexistent issue for
electronics, but with the advent of semiconductors, and the
increase in miniaturization, it has become much more
serious.
Damage to components can, and usually do, occur when the
part is in the ESD path. Many parts, such as power diodes,
are very robust and can handle the discharge, but if a part
has a small or thin geometry as part of their physical
structure then the voltage can break down that part of the
semiconductor. Currents during these events become quite
high, but are in the nanosecond to microsecond time frame.
Part of the component is left permanently damaged by this,
which can cause two types of failure modes. Catastrophic is
the easy one, leaving the part completely nonfunctional. The
other can be much more serious. Latent damage may allow
the problem component to work for hours, days or even
months after the initial damage before catastrophic failure.
Many times these parts are referred to as "walking wounded",
since they are working but bad. Figure below is shown an
example of latent ("walking wounded") ESD damage. If these
components end up in a life support role, such as medical or
military use, then the consequences can be grim. For most
hobbyists it is an inconvenience, but it can be an expensive
one.
Even components that are considered fairly rugged can be
damaged by ESD. Bipolar transistors, the earliest of the solid
state amplifiers, are not immune, though less susceptible.
Some of the newer high speed components can be ruined
with as little as 3 volts. There are components that might not
be considered at risk, such as some specialized resistors and
Capacitors manufactured using MOS (Metal Oxide
Semiconductor) technology, that can be damaged via ESD.
Images Courtesy of Bunny Studios LCC
This is an example of latent ESD,
also known as“ Walking Wounded”,
This three terminal regulator IC
worked about an hour after the
initial ESD damage.
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ESD Damage Prevention
Before ESD can be prevented it is important to understand
what causes it. Generally materials around the workbench
can be broken up into 3 categories. These are ESD
Generative, ESD Neutral, and ESD Dissipative (or ESD
Conductive). ESD Generative materials are active static
generators, such as most plastics, cat hair, and polyester
clothing. ESD Neutral materials are generally insulative, but
don't tend to generate or hold static charges very well.
Examples of this include wood, paper, and cotton. This is not
to say they can not be static generators or an ESD hazard,
but the risk is somewhat minimized by other factors. Wood
and wood products, for example, tend to hold moisture,
which can make them slightly conductive. This is true of a lot
of organic materials. A highly polished table would not fall
under this category, because the gloss is usually plastic, or
varnish, which are highly efficient insulators. ESD Conductive
materials are pretty obvious, they are the metal tools laying
around. Plastic handles can be a problem, but the metal will
bleed a static charge away as fast as it is generated if it is on
a grounded surface. There are a lot of other materials, such
as some plastics, that are designed to be conductive. They
would fall under the heading of ESD Dissipative. Dirt and
concrete are also conductive, and fall under the ESD
Dissipative heading.
There are a lot of activities that generate static, which you
need to be aware of as part of an ESD control regimen. The
simple act of pulling tape off a dispenser can generate
extreme voltage. Rolling around in a chair is another static
generator, as is scratching. In fact, any activity that allows 2
or more surfaces to rub against each other is pretty certain to
generate some static charge. This was mentioned in the
beginning of this book, but real world examples can be
subtle. This is why a method for continuously bleeding off
this voltage is needed. Things that generate huge amounts
of static should be avoided while working on components.
Plastic is usually associated with the generation of static.
This has been gotten around in the form of conductive
plastics. The usual way to make conductive plastic is an
additive that changes the electrical characteristics of the
plastic from an insulator to a conductor, although it will likely
still have a resistance of millions of ohms per square inch.
Plastics have been developed that can be used as conductors
is in low weight applications, such as those in the airline
industries. These are specialist applications, and are not
generally associated with ESD control.
It is not all bad news for ESD protection. The human body isa
pretty decent conductor. High humidity in the air will also
allow a static charge to dissipate harmlessly away, as well as
making ESD Neutral materials more conductive. This is why
cold winter days, where the humidity inside a house can be
quite low, can increase the number of sparks on a doorknob.
Summer, or rainy days, you would have to work quite hard to
generate a substantial amount of static. Industry clean rooms
and factory floors go the effort to regulate both temperature
and humidity for this reason. Concrete floors are also
conductive, so there may be some existing components in
the home that can aid in setting up protections.
To establish ESD protection there has to be a standard
voltage level that everything is referenced to. Such a level
exists in the form of ground. There are very good safety
reasons that ground is used around the house in outlets. In
some ways this relates to static, but not directly. It does give
us a place to dump our excess electrons, or acquire some if
we are short, to neutralize any charges our bodies and tools
might acquire. If everything on a workbench is connected
directly or indirectly to ground via a conductor then static
will dissipate long before an ESD event has a chance to
occur.
A good grounding point can be made several different ways.
In houses with modern wiring that is up to code the ground
pin on the AC plug in can be used, or the screw that holds
the outlets cover plate on. This is because house wiring
actually has a wire or spike going into the earth somewhere
where the power is tapped from the main power lines. For
people whose house wiring isn't quite right a spike driven
into the earth at least 3 feet or a simple electrical connection
to metal plumbing (worst option) can be used. The main
thing is to establish an electrical path to the earth outside
the house.
Ten megohms is considered a conductor in the world of ESD
control. Static electricity is voltage with no real current, and
if a charge is bled off seconds after being generated it is
nullified. Generally a 1 to 10 megohm resistor is used to
connect any ESD protection for this reason. It has the benefit
of slowing the discharge rate during an ESD event, which
increases the likelihood of a component surviving
undamaged. The faster the discharge, the higher the current
spike going though the component. Another reason such a
resistance is considered desirable is if the user is accidentally
shorted to high voltage, such as household current, it won't
be the ESD protections that kill them.
A large industry has grown up around controlling ESD in the
electronics industry. The staple of any electronics
construction is the workbench with a static conductive or
dissipative surface. This surface can be bought commercially,
or home made in the form of a sheet of metal or foil. In the
case of a metal surface it might be a good idea to lay thin
paper on top, although it is not necessary if you are not
doing any powered tests on the surface. The commercial
version is usually some form of conductive plastic whose
resistance is high enough not to be a problem, which is a
better solution. If you are making your own surface for the
workbench be sure to add the 10 megohm resistor to ground,
otherwise you have no protection at all.
The other big item that needs ESD grounded is you. People
are walking static generators. Your body being conductive it
is relatively easy to ground it though, this is usually done
with a wrist strap. Commercial versions already have the
resistor built in, and have a wide strap to offer a good contact
surface with your skin. Disposable versions can be bought for
a few dollars. A metal watchband is also a good ESD
protection connection point. Just add a wire (with the
resistor) to your grounding point. Most industries take the
issue seriously enough to use real time monitors that will
sound an alarm if the operator is not properly grounded.
10Mo
10Ma 10Moa
10Ma
Correctly grounded station, the Incorrectly grounded station, the
table surface and wrist strap wrist strap is connected to the
each have their own path to table surface, in the event of
ground. ESD in the wrist strap it will
also raise the table's potential.
This is better than no protection.
Another way of grounding yourself is a heel strap. A
conductive plastic part is wrapped around the heel of your
shoe, with a conductive plastic strap going up and under
your sock for good contact with the skin. It only works on
floors with conductive wax or concrete. The method will keep
a person from generating large charges that can overwhelm
other ESD protections, and is not considered adequate in and
of itself. You can get the same effect by walking barefoot on a
concrete floor.
Yet another ESD protection is to wear ESD conductive
smocks. Like the heel strap, this is a secondary protection,
not meant to replace the wrist strap. They are meant to short
circuit any charges that your clothes may generate.
Moving air can also generate substantial static charges.
When you blow dust off your electronics their will be static
generated. An industrial solution to the problem to this issue
is two fold: Firstly, air guns have a small, well shielded
radioactive material implanted within the air gun to ionize
the air. lonized air is a conductor, and will bleed off static
charges quite well. Secondly, use high voltage electricity to
ionize the air coming out of a fan, which has the same effect
as the air gun. This will effectively help a workstation reduce
the potential for ESD generation by a large amount.
Another ESD protection is the simplest of all, distance. Many
industries have rules stating all Neutral and Generative
materials will be at least 12 inches or more from any work in
progress.
The user can also reduce the possibility of ESD damage by
simply not removing the part out of its protective packaging
until it is time to insert it into the circuit. This will reduce the
likelihood of ESD exposure, and while the circuit will still be
vulnerable, the component will have some minor protection
from the rest of the components, as the other components
will offer different discharge paths for ESD.
Storage and Transportation of ESD sensitive
component and boards
It does no good to follow ESD protections on the workbench if
the parts are being damaged while storing or carrying them.
The most common method is to use a variation of a Faraday
cage, an ESD bag. An ESD bag surrounds the component
with a conductive shield, and usually has a non static
generating insulative layer inside. In permanent Faraday
cages this shield is grounded, as in the case of RFI rooms, but
with portable containers this isn't practical. By putting a ESD
bag on a grounded surface the same thing is accomplished.
Faraday cages work by routing the electric charge around the
contents and grounding them immediately. A car struck by
lightning is an extreme example of a Faraday cage.
Static bags are by far the most common method of storing
components and boards. They are made using extremely thin
layers of metal, so thin as to be almost transparent. A bag
with a hole, even small ones, or one that is not folded on top
to seal the content from outside charges is ineffective.
Another method of protecting parts in storage is totes or
tubes. In these cases the parts are put into conductive boxes,
with a lid of the same material. This effectively forms a
Faraday cage. A tube is meant for ICs and other devices with
a lot of pins, and stores the parts in a molded conductive
plastic tube that keeps the parts safe both mechanically and
electrically.
bs DQ &
These are some of the more common
logos associated with anti-static labels.
They are used to inform the user that the
contents are static sensitive.
Conclusion
ESD can be a minor unfelt event measuring a few volts, ora
massive event presenting real dangers to operators. All ESD
protections can be overwhelmed by circumstance, but this
can be circumvented by awareness of what it is and how to
prevent it. Many projects have been built with no ESD
protections at all and worked well. Given that protecting
these projects is a minor inconvenience it is better to make
the effort.
Industry takes the problem very seriously, as both a potential
life threatening issue and a quality issue. Someone who buys
an expensive piece of electronics or high tech hardware is
not going to be happy if they have to return it in 6 months.
When a reputation is on the line it is easier to do the right
thing.
Power Supply circuits
There are three major kinds of power supplies: unregulated
(also called brute force), linear regulated, and switching. A
fourth type of power supply circuit called the ripple-
regulated, is a hybrid between the "brute force" and
"switching" designs, and merits a subsection to itself.
Unregulated
An unregulated power supply is the most rudimentary type,
consisting of a transformer, rectifier, and low-pass filter.
These power supplies typically exhibit a lot of ripple voltage
(i.e. rapidly-varying instability) and other AC "noise"
superimposed on the DC power. If the input voltage varies,
the output voltage will vary by a proportional amount. The
advantage of an unregulated supply is that its cheap, simple,
and efficient.
See Rectifier circuits in the Diodes chapter for the various
configurations of the rectifiers used in unregulated power
supplies. Note that those circuits are unfiltered, A low pass
filter is normally added to the output of the rectifier circuit to
remove some of the ripple.
A linear regulated supply is simply a "brute force"
(unregulated) power supply followed by a transistor circuit
operating in its "active," or "linear" mode, hence the name
linear regulator. (Obvious in retrospect, isn't it?) A typical
linear regulator is designed to output a fixed voltage fora
wide range of input voltages, and it simply drops any excess
input voltage to allow a maximum output voltage to the load.
This excess voltage drop results in significant power
dissipation in the form of heat. If the input voltage gets too
low, the transistor circuit will lose regulation, meaning that it
will fail to keep the voltage steady. It can only drop excess
voltage, not make up for a deficiency in voltage from the
brute force section of the circuit. Therefore, you have to keep
the input voltage at least 1 to 3 volts higher than the desired
output, depending on the regulator type. This means the
power equivalent of at /east 1 to 3 volts multiplied by the full
load current will be dissipated by the regulator circuit,
generating a lot of heat. This makes linear regulated power
supplies rather inefficient. Also, to get rid of all that heat
they have to use large heat sinks which makes them large,
heavy, and expensive.
Switching
A switching regulated power supply ("switcher") is an effort
to realize the advantages of both brute force and linear
regulated designs (small, efficient, and cheap, but also
"clean," stable output voltage). Switching power supplies
work on the principle of rectifying the incoming AC power
line voltage into DC, re-converting it into high-frequency
square-wave AC through transistors operated as on/off
switches, stepping that AC voltage up or down by using a
lightweight transformer, then rectifying the transformer's AC
output into DC and filtering for final output. Voltage
regulation is achieved by altering the "duty cycle" of the DC-
to-AC inversion on the transformer's primary side. In addition
to lighter weight because of a smaller transformer core,
switchers have another tremendous advantage over the prior
two designs: this type of power supply can be made so
totally independent of the input voltage that it can work on
any electric power system in the world; these are called
"universal" power supplies.
The downside of switchers is that they are more complex,
and due to their operation they tend to generate a lot of
high-frequency AC "noise" on the power line. Most switchers
also have significant ripple voltage on their outputs. With the
cheaper types, this noise and ripple can be as bad as for an
unregulated power supply; such low-end switchers aren't
worthless, because they still provide a stable average output
voltage, and there's the "universal" input capability.
Expensive switchers are ripple-free and have noise nearly as
low as for some a linear types; these switchers tend to be as
expensive as linear supplies. The reason to use an expensive
switcher instead of a good linear is if you need universal
power system compatibility or high efficiency. High
efficiency, light weight, and small size are the reasons
switching power supplies are almost universally used for
powering digital computer circuitry.
Ripple regulated
A ripple-regulated power supply is an alternative to the linear
regulated design scheme: a "brute force" power supply
(transformer, rectifier, filter) constitutes the "front end" of the
circuit, but a transistor operated strictly in its on/off
(saturation/cutoff) modes transfers DC power to a large
Capacitor as needed to maintain the output voltage between
a high and a low setpoint. As in switchers, the transistor in a
ripple regulator never passes current while in its "active," or
"linear," mode for any substantial length of time, meaning
that very little energy will be wasted in the form of heat.
However, the biggest drawback to this regulation scheme is
the necessary presence of some ripple voltage on the output,
as the DC voltage varies between the two voltage control
setpoints. Also, this ripple voltage varies in frequency
depending on load current, which makes final filtering of the
DC power more difficult.
Ripple regulator circuits tend to be quite a bit simpler than
switcher circuitry, and they need not handle the high power
line voltages that switcher transistors must handle, making
them safer to work on.
Power Supply Introduction
Power supply circuits are a class of circuits that are designed
to convert electrical energy for some load. Every power
supply consists of at least three parts:
e An input power source, which delivers power at some
voltage or range of voltages V1
e A load, which requires power delivered at some voltage
or range of voltages V2
e Conversion circuitry, which receives voltage V1 as an
input and generates voltage V2 as an output
Some devices are simple enough that they can operate
properly without any modifications to the voltage and
current provided by the input source. For example, the
lightbulb inside a low-cost flashlight is designed to emit light
when connected in series with a few batteries, meaning the
entire conversion circuit is just wires. In a similar way,
household incandescent lightbulbs are designed to operate
properly when connected to an AC source, operated at a well-
regulated voltage and line frequency. But for the majority of
electronic devices, it is impractical to operate an entire
circuit at voltages commonly available. Computers, cell
phones, car stereos, aircraft sensors, traffic lights, and
pacemakers all have elements which require drastically
different voltages than those delivered by any common
power source. Well-designed power supply circuits convert
almost all of the energy supplied by batteries, solar cells, AC
lines and other power sources to voltage levels suitable for
the operation of intricate electronic devices.
These are some of the typical considerations when designing
a power supply circuit:
Efficiency
Efficiency is defined as the output power divided by the total
input power. The maximum theoretical efficiency of a circuit
is 100%, and this makes sense: the only place output power
can come from in a power supply is the input power source.
Energy that is consumed in the conversion process, and is
not delivered as output power, is called power loss. All power
supply circuits have some losses, even if those losses are
very small. Maximizing efficiency and minimizing losses is of
key importance in power supply design. Highly efficient
devices can last longer on a single battery charge, cost less
money to operate from a utility AC line, and generate less
heat.
Heat
Power loss is dissipated away from a power supply circuit as
heat. Very small semiconductor components may only be
able to dissipate a few hundred milliwatts before they
become too hot and fail. On the other hand, very large power
supplies can convert multiple kilowatts of power, and
routinely see tens of watts dissipated across only a few
components. Further complicating issues, many power
supplies are designed to operate in hot or cold environments,
where temperatures can vary by over 100° C. At hotter
temperatures, devices must be thermally derated to avoid
overheating, which significantly reduces the maximum
output power available. At colder temperatures, considerable
deviations in component values can be expected, and rapid
changes in loading can lead to thermal shock effects, where
repeated heating and cooling stresses components to failure.
In most cases, cold temperature performance can be
guaranteed with proper component selection; removing
waste heat and preventing damage from overheating receive
much greater consideration.
In order to prevent component failures, high dissipation
components are usually connected to heat sinks. Sometimes
the only heat sink needed is a solid connection to a copper
plane in a printed circuit board. But for anything beyond a
few watts, components need to be connected to a separate,
thermally conductive metal block. By putting long metal fins
on these blocks, the surface area can be boosted to increase
convective heat transfer. A fan can also be used to increase
airflow. Some designs even use water or oil traveling through
the block to more effectively remove waste heat.
As a general rule, most semiconductors begin experiencing
damage when the circuit's internal temperature reaches
150°C, though some devices are designed to withstand even
higher temperatures. Other components such as inductors
and capacitors are available in a wide range of operating
temperatures and tolerances, with a premium charged for
more extreme temperatures and tighter tolerances.
Size
In some devices such as cell phones or smart watches, there
can be dozens or hundreds of components made to fit within
only a few square centimeters. Power supply circuits in these
types of devices must be small leave room for other, feature-
rich components. In other devices such as aircraft
electronics, the power requirements are large enough that
many components must be attached to a heat sink. This can
add significant weight to the overall design, which reduces
fuel economy of the aircraft. Size is directly related to the
amount of power being converted, and the efficiency of the
conversion. The more power being converted, the larger the
components must be to spread out self-heating and to
withstand the high voltages used for larger power
conversions. Improvements in efficiency can help to reduce
supply size, since less heat sinking is required.
Cost
Unsurprisingly, cost is a critical factor. Generally, as both
power and efficiency are increased, the cost of the power
supply increases as well. This cost increase comes from a
combination of expensive but well-optimized components,
increased complexity leading to longer design and test
cycles, and costs associated with regulatory compliance. As
In any engineering challenge, power supply design is a
tradeoff of acceptable performance and cost. Since all
electronic devices require one or more power supply circuits,
aggressive cost optimization is common. In high-volume
manufacturing, saving even a few cents per product can
reduce build costs by thousands of dollars.
Line Regulation
Line regulation is a measure of how well a power supply
circuit can respond to changes in input source voltage. Many
input power sources present a wide voltage range to a power
supply input: battery voltages can vary by 30% or more
across one charge cycle, solar cell voltages vary
proportionally to incident sunlight, and AC line voltages can
(on rare occasion) deviate by as much as 20% in either
direction. Line regulation is defined as the output voltage at
the maximum/minimum input voltage, minus the output
voltage at the nominal input voltage. It can also be given as
a percentage of the nominal output voltage value. An ideal
power supply has perfect line regulation, OV or +0%
change. It is not uncommon for modern power supplies to see
values < +5mV or < +0.1%.
Load Regulation
Load regulation is a measure of how well a power supply
circuit can respond to changes in output loading. As output
power increases, heating from power loss causes changes in
reference parameters used by the circuit to control the
output. Power supply designers use carefully designed
reference circuits to minimize the effects of temperature
variations, but observable effects still exist. Load regulation
is defined as the output voltage at full load, minus the output
voltage at no load. It can also be given as a percentage of the
nominal output voltage value. An ideal power supply has
perfect load regulation, OV or 0% change. Modern power
supply circuits can achieve values similar to line regulation.
Ripple Rejection
For many power supply circuits with an AC line as input, the
line frequency is coupled through the supply to the output.
Some power supply circuits specify a ripple rejection, usually
in dB, which is defined as the magnitude of a specific
frequency on the output (commonly 100HZz or 120Hz)
relative to the magnitude of that same specific frequency on
the input.
Quiescent Current
Even at no load, some power is required to keep a power
supply in regulation. The housekeeping current used to
power the control circuitry of the supply is called the
quiescent current. This value has a wide range, spanning
from hundreds of milliamps all the way down to hundreds of
nanoamps.
Output Impedance
An ideal voltage source has zero output impedance. Practical
converters see some small output impedance, which tends to
grow at higher frequencies. For a power supply to effectively
regulate against loads that change in milliseconds or less,
low output impedance is mandatory. Otherwise, sudden
changes in load current will produce severe changes in
output voltage. Nearly all converters can easily achieve
output impedances of less than an ohm; < 10mQ at DC is not
uncommon.
Output Voltage Noise
Electrons flowing in resistors and transistors are susceptible
to thermodynamic events, statistical fluctuations in current
density, and other complex particle-scale phenomena. These
tendencies manifest in all circuits, including power supply
circuits, as noise on the output voltage. Although the
average value of a power supply output is constant, noise
can cause the output to experience millivolt excursions on a
microsecond or submicrosecond scale. For lower power
analog circuits that depend on tightly regulated power
supply voltages such as high-resolution analog-to-digital
converters or high-frequency oscillators, power supply noise
can cripple performance. Because noise sources tend to bea
function of frequency, noise is commonly listed as a value
integrated over a frequency range (in RMS Volts), or is
specified as a plot of noise spectral density comparing noise
(in Volts/Hz) vs. frequency. Wideband (10Hz to 100kHz)
integrated noise can be controlled to < 10UVrms, and noise
at high frequencies can approach < 10nV/Hz.
Higher power designs tend to introduce noise on the order of
tens or hundreds of millivolts, concentrated at specific
frequencies, as a function of their construction. Though some
designs exist which can tightly control even high-power
supply noise, they are costly and are therefore reserved for
specialized test and measurement equipment. Virtually all
practical power supplies above a few watts will generate
millivolts of noise on the output, and for many types of load
this does not affect device performance in the slightest. It is
common to use an effective but noisy power supply for
insensitive loads, and as the input to a second, quieter power
supply.
Linear power supplies
There are two major types of power supplies whose output
behavior can be determined according to linear equations:
shunt regulators, and series regulators. Shunt regulators
(pictured in Figure below are so named because they shunt
away unnecessary load current to keep the output in
regulation. In a shunt regulator, high quiescent current is
necessary, since the shunt must be able to redirect the full
load current at no load conditions. This can lead to high
power dissipation, especially for appreciably large full load
currents. On the bright side, they are relatively simple, often
made of entirely passive elements, and can be reduced to
two-terminal devices.
Vout
Shunt Control
Shunt regulators
Vout = Vin Ii,R
ctrl = tin ~ tout
Series regulators, in contrast, regulate the input current with
a pass element to control the output current delivered to a
load pictured in Figure below. This can be utilized to reduce
quiescent current to almost nothing at light or no load,
though some current must always pass through the series
element to ensure proper output voltage regulation.
However, the series regulator must control both the voltage
drop across and the current through the series element to
regulate the output voltage, and no passive element can be
used to guarantee this behavior. Because of the need for
some form of output sensing circuitry, a three terminal
solution is almost always required.
Vout
Series
Control
Series regulators
Vout= (in * ToudR
ctrl = tin
For the sake of illustrating the common terms seen in power
supply design, consider the following specification: Suppose
there is a need to take a static 15V output from a converter,
and step it down to a 5V level. The input voltage may vary by
as much as +3V, and the output current must be 500mA
maximum. In the following examples, several basic
topologies will be explored, and the relative strengths and
weaknesses of the different approaches compared.
Example Supply: Resistor Divider
Ry
Vout
Resistor divider power supply
The humble resistor divider circuit of Figure above is perhaps
the simplest power supply circuit. While its behavior is
entirely linear, it is hard to say whether such a supply should
be considered a shunt or series regulator, since the output
voltage is a function of both the shunt and series elements.
For the purposes of this first example, the distinction is
unimportant. The nonidealities of this circuit, especially when
constrained by the specification above, make it useful to
conceptualize the common terms used in power supply
design.
The resistor values must be small to simultaneously allow
500mA through the pass element without causing too much
of a voltage drop, and maintain a nominal output voltage of
5V from a 15V supply. The values 15Q and 7.5Q are selected
for Rl and R2, respectively. The behavior of the circuit can be
described by a system of equations, first using ohm's law at
the output, second by using the standard divider equation
considering R2 and the load in parallel:
V
Foot!
V.,R.
in
Vout= (R, ole 3R,)
out
By selecting values for Vin and one other parameter, and
solving for the remaining unknowns, the performance of this
circuit may be interrogated. The key points are summarized
below.
Efficiency: To achieve the maximum output current of
500mA with nominal input conditions (Vin = 15V), solving
the system of equations gives:
Vout = 25V
L —
Load Power P; = 2.5V x 0.5A = 1.25W
Meanwhile, the input power is the input voltage multiplied
by the input current, and the input current is found as:
(1SV -2.5V)
150 = ().866A
Therefore,
Input Power P; = I5V x 0.866A = 12.5W
Our efficiency at full load is then:
P, 1.25W
= = 10%
PI 12.5W
It can also be shown using calculus that the maximum
efficiency is achieved with a load resistance of 5Q, yielding
only 10.1% efficiency. These values are unimpressive.
Interestingly, a quick calculation will reveal that this
maximum efficiency is the same, regardless of input voltage.
This makes sense, since output power is ratiometric with
input power for the same circuit.
Quiescent current: At no load, the circuit draws:
ISV
25Q, = 667mA
This is more than the maximum output by some margin, and
Is very wasteful compared to what might be achievable with
other topologies. Worse still, the total current only increases
with increasing load.
Heat: At no load, the circuit dissipates 1OW power, and at
full load this increases to 12.5W. Under short circuit
conditions, this increases to 15W, all dissipated in R1. Both
R1 and R2 would need to be large wirewound resistors, or
would require active cooling, for this supply to function at
ambient temperature. Performance above ambient
temperature is more difficult.
Load regulation: At full load, the output voltage drops from
5V to 2.5V. From the load regulation equation, we find that
this supply has the following load regulation:
(2.5V - 5V)
5V = -0.5V or -50%
This is atrocious. Any improvement in load regulation is also
practically infeasible; to make the parallel combination of R2
and the load negligibly different from the load, even at full
load, the value of Rl and R2 would need to be further
decreased by more than an order of magnitude, which would
necessarily increase the quiescent current and decrease the
efficiency by the same degree. It is unreasonable to require
over LOOW of power dissipation to maintain a reasonable
load regulation from a resistor divider. Ideally, it shouldn't
even take milliwatts.
Line regulation: At 18V with no load, the output voltage is:
7.SQ
I8V x 5550
=6V
Meanwhile, at 12V, the output voltage is:
7.5Q .
| A ee 750. =4V
This corresponds to a line regulation of +1V, or 20%. This is
quite terrible.
While on the subject of input voltage variations, consider
that the values of quiescent current and load regulation will
change for different input voltages. As the input voltage
increases, the quiescent current increases, and the heat
generation increases. The output will deliver 500mA to a 7Q
load at 3.5V. At <15V the output voltage decreases
substantially, delivering 500mA to a 3Q load at 1.5V. Since
the output voltage is directly proportional to the input
voltage, the output is dependent on a stable input voltage,
which is not always possible.
Output Impedance: By small signal analysis, the voltage
source at Vin is shorted, and the output impedance is plainly
the parallel combination of R1||R2, or 5Q. Since this output
impedance is static across all changes in input voltage and
output current, it is understandable why the output voltage
varies so much with every change in input and load
conditions.
Output Noise: Although this resistor will be affected by
thermal noise, standard 1/f noise, and excess noise due to
resistor construction, ultimately noise is unlikely to be the
biggest concern in this design, and true noise analysis will be
saved for more deserving circuits.
The performance of this circuit as a power supply is nothing
short of abysmal. In fairness to the resistor divider, the most
common use for such a circuit is voltage division into high-
impedance loads, such as amplifier input pins and transistor
gates. For these high-impedance load conditions, the divider
may be treated as very close to ideal, and as such it is not
often thought of as a power supply circuit. Nevertheless,
when operating conditions begin to change (such as with
supply voltage variations or even small load current
increases), high impedance amplifier inputs and transistors
can still be made to misbehave.
In this impractical example it should be clear that a resistor
divider is unsuited for any serious power delivery, with
completely unusable line and load regulation and horrible
overall efficiency. However, with only a minor modification,
this circuit can be augmented with vastly improved line and
load regulation. This is explored in the following example.
Example Supply: Zener Divider
Ry
Vout
Zener Divder Power Supply
The circuit of Figure above is a Zener divider (Zener diodes
are discussed in chapter 3). By substituting a reverse-biased
Zener diode in place of R2 in the previous circuit, the shifting
Zener impedance above a certain reverse current knee point
can be exploited to guarantee a stable output voltage over
different line and load conditions. Keeping in mind that
Zener diodes can only be constructed with certain reverse
voltages, the closest stable output to 5V is chosen, giving a
Zener voltage of 5.1V. At no load, all available current will be
passed through the Zener diode. By choosing this load
current to be slightly over 500mA at maximum load (say by
1OmA), regulation can be ensured even when the the full
load current is delivered to the load. R1 is selected for all
voltages within the tolerance of the input voltage range: the
worst case, at 12V, requires that:
(12V -5.1V)
R
I
R, = 13.5
510mA =
As long as the Zener diode has current through it, a load of
10.2Q can now be attached, with any input supply voltage in
the specified range, and 500mA will be delivered to it. To
prove this assertion, test the behavior at Vin = 12V and Vin
= 18V:
(12V -5.1V)
Lin 12V = 13.50 = 51 ImA; L = Tin 2V ” I, = ] ImA
linsv = a eaaae = 956mA; I, = Ij, oy - I, = 465mA
In theory, this design should therefore be capable of meeting
all the requirements. A closer examination of the affected
parameters offers some caveats.
Efficiency: The Zener regulator efficiency differs depending
on input voltage and output loading. The best case efficiency
for any input voltage is at full load, and the best case
efficiency for any load is at the minimum input voltage. In
this condition, we find that:
P, =5.1V x 500mA = 2.55W
P;= 12V x 511mA = 6.132W
255W
6.132w = 41.6%
This is better than the resistor divider, but not by much, and
only at one extreme corner of operation. At the other corner
the results are less impressive:
P,= 18V x 965mA = 17.2W
2.5) W
Quiescent current: At no load, the full operating current of
the Zener regulator must travel through the Zener diode.
Best case, this is always more than the maximum output
current; worst case, it can be much greater. At 17V, this
regulator consumes almost double the maximum output
current!
Heat: Since the Zener regulator quiescent current is always
greater than the maximum operating current, the worst case
power dissipation leads to a great deal of heat dissipated in
both R1 and in the diode. However, as the load current
increases, the Zener diode dissipates less and less power,
since the current and therefore the power must be diverted
from the diode to the output load. Meanwhile, Rl power
dissipation remains almost constant across loading, but
benefits from a lower input voltage. If for any reason the
output current exceeds the quiescent current (Such as during
a short circuit), the power dissipation in Rl increases above
the typical worst case operating point, requiring a larger
component or better cooling to endure this stress. Even
under normal operating conditions, R1 still dissipates enough
to require a large wirewound resistor and probably some form
of active cooling:
(18V-5.1V)" _,,
sa
It is worth noting, in passing, that at worst case the Zener
diode must dissipate close to 5W; while there exist Zener
diodes capable of this, 5W is an uncommonly large value for
a Zener diode. With smaller maximum load current
requirements, low power Zeners may be used at substantially
decreased costs.
Line and load regulation: From an ideal standpoint, the
Zener voltage is always 5.1V, across all line and load
conditions. In reality, however, the Zener diode has some
temperature related effects which cause the Zener voltage to
change. Worse still, the temperature effects do not all act in
the same direction. Low voltage Zener diodes behave
predominantly according to the Zener effect, an electron
tunneling process, which has a negative temperature
coefficient (Zener voltage decreases with increasing heat).
Higher voltage Zener diodes behave predominantly
according to the avalanche effect, a form of current
multiplication that has a positive temperature coefficient
(Zener voltage increases with increasing heat). At around 4V
to 6V, and dependent on the Zener current, the temperature
coefficients of these two mechanisms will combine and can
occasionally cancel out almost entirely. Unfortunately, there
is still some effect at 5.1V; A 5W rated 1N5338B, for
example, can see a difference of almost 0.4V across
temperature, typically increasing in voltage.
A basic approximation of this effect can explain the difficulty.
With a 15V supply voltage, at no load the Zener current and
power are found to be:
1 _ (ISV-5.1V)
2 13.52
P,=5.1V x 733mA = 3.74W
= 733mA
Assuming the change in Zener voltage is up to 0.4V at 5W
for the given Zener diode, and the change is both linear and
positive, the Zener voltage may increase in response to
increasing junction temperature by as much as:
3.74W
y St =f 9
0.4W x SW 0.3V
This changes the Zener current and power to:
1 _ (ISV-5.1V)
z~ “13.50
P,=5.1V x 733mA = 3.74W
= 733mA
Iteration shows this change in Zener voltage with
temperature eventually stabilizes; still, the output is far from
its ideal value. As load current increases, Zener current
decreases, returning the output voltage to a lower value. Line
and load regulation are difficult to estimate precisely, since
the exact location of the Zener knee, the effect of process
variation on the temperature coefficient, and the variation of
the temperature coefficient with Zener current cannot always
be predicted. Both are frequently verified experimentally or
with a spice simulation. Broadly speaking, with a maximum
specified regulation swing of about 0.4V, and assuming this
can be either positive or negative for a 5.1V Zener diode, the
combined line and load regulation can be stated as:
0.4V
: — +7 2G
spy = 17.8%
Output Impedance: For the small signal analysis, voltage
sources are shorted. The impedance looking into the output
is just R1||RZ. But RZ is a dynamic value, based on the Zener
voltage (Vout) and the Zener current (IZ).
lz = lin 7 Fe
= (Vi, 2 Vout)
in ~~ R,
= __oul
Ry
Noni _ N ois R, x Ry
Zout = R, | | (Viav V out) _ Vout (V., x Ry ~ Vout % R,)
R, Ry
In the limit as RL approaches infinity, Zout becomes Vout x
R1 / Vin, approximately 4.590 at L5V input. Interestingly,
from this equation we can discover that the output
impedance increases as a function of increasing load, to a
maximum of Rl at a dead short across the output. The output
is regulated because the output impedance continuously
changes to match the level of loading.
Output Noise: The topic of output noise for Zener diodes is
complicated. Due to the different mechanisms of Zener diode
behavior, there are different sources of noise for different
Zener voltages and currents. Some attempt to simplify these
topics will be made here.
Low voltage Zener diodes operate on the Zener effect, where
discrete electrons tunnel across a barrier. Since this is a
discrete, random process centered around a mean value, it
follows a Poisson distribution and generates corresponding
shot noise. The noise level is proportional to the square root
of the number of discrete events. Thus, as current increases,
shot noise increases as well. For a given Zener current lz and
Zener voltage Vz and recalling the electron elementary
charge gq = 1.6 x 10-19 coulombs, the shot noise is:
—<—<—s: V
E, = V2xqxIL x = 5.66x10'"x Tr [NV me/VHz]
Zz Z
At no load, this effect is almost negligible, since it is inversely
proportional to lz. But at full load, lz shrinks considerably. The
noise at full load for V,, = 12V is 7x worse than the noise at
no load. At 18V, since the difference in lz at no load and at
full load is smaller, the effect is much less pronounced.
High voltage Zener diodes operate on the avalanche effect,
where one carrier collides with many others and causes an
avalanche multiplication of carrier movements, resulting in
wide-bandwidth noise that can exceed simple shot noise by
orders of magnitude. In fact, the equation is almost identical,
but depends to some extent on the recombination lifetime of
each new electron in the avalanche. Without getting too
deeply into the physics, it is usually sufficient to introduce
some large multiplier to the original shot noise equation.
Whereas a low voltage Zener diode might measure its
wideband noise in the hundreds of nV, a high voltage Zener
diode might measure its wideband noise in hundreds of
uvolts or even low millivolts.
To keep a Zener diode at the lowest possible noise, there are
only two requirements: first, use a low voltage Zener diode,
to minimize avalanche noise; second, use a large Zener
current, even at full load. Though increasing the current
increases the power dissipation, potentially leading to
greater thermal noise, remember that thermal noise is
proportional to Zener impedance, and that Zener impedance
shrinks faster than absolute temperature grows. In power
supply design Zener noise is once again rarely an issue, since
other regulators can be created with less noise, more
efficiency, and better line and load regulation.
In general, shunt regulators are used in cases where the
power dissipation is negligible, and the load current is small
(tens of milliamps or fewer). More complex shunt regulators
can incorporate compensation schemes which minimize the
effects of line, load, and temperature variations. The exact
mechanisms of these compensation schemes are beyond the
scope of this discussion, but line and load regulation values
of <1% are achievable with shunt regulation schemes, over a
very wide range of temperatures and input voltages.
Amplifier circuits -- PENDING
Note, Q3 and Q, in Figure below are complementary, NPN and
PNP respectively. This circuit works well for moderate power
audio amplifiers. For an explanation of this circuit see “Direct
coupled complementary-pair,” Ch 4.
R,
39kQ
input
ne
220 nF
Cs
4000 uF
Direct coupled complementary symmetry 3 w audio
amplifier. After Mullard. [MUL]
Oscillator circuits -- INCOMPLETE
Phase shift oscillator. R,C 1, RoC>, and R3C3 each provide 60°
of phase shift.
The phase shift oscillator of Figure above produces a
sinewave output in the audio frequency range. Resistive
feedback from the collector would be negative feedback due
to 180° phasing (base to collector phase inversion). However,
the three 60° RC phase shifters ( R}C,, RoC>, and R3C3)
provide an additional 180° for a total of 360°. This in-phase
feedback constitutes positive feedback. Oscillations result if
transistor gain exceeds feedback network losses.
Varactor multiplier
A Varactor or variable capacitance diode with a nonlinear
Capacitance vs frequency characteristic distorts the applied
sinewave f1 in Figure below, generating harmonics, f3.
I Vite
RF blocking
hok
Resonant
inductor
varactor
diode DC blocking
| | capacitor
Varactor diode, having a nonlinear capacitance vs voltage
Characteristic, serves in frequency multiplier.
capacitance
voltage —_
The fundamental filter passes f1, blocking the harmonics
from returning to the generator. The choke passes DC, and
blocks radio frequencies (RF) from entering the V,j,, supply.
The harmonic filter passes the desired harmonic, say the 3rd,
to the output, f3. The capacitor at the bottom of the inductor
is a large value, low reactance, to block DC but ground the
inductor for RF. The varicap diode in parallel with the indctor
constitutes a parallel resonant network. It is tuned to the
desired harmonic. Note that the reverse bias, Vpia<, is fixed.
The varicap multiplier is primarily used to generate
microwave signals which cannot be directly produced by
oscillators. The lumped circuit representation in Figure above
is actually stripline or waveguide sections. Frequenies up to
hundreds of gHz may be produced by varactor multipliers.
Phase-locked loops -- PENDING
Radio circuits -- INCOMPLETE
(a) Crystal radio. (b) Modulated RF at antenna. (c) Rectified
RF at diode cathode, without C2 filter capacitor. (d)
Demodualted audio to headphones.
An antenna ground system, tank circuit, peak detector, and
headphones are the the main components of a crystal radio.
See Figure above (a). The antenna absorbs transimtted radio
signals (b) which flow to ground via the other components.
The combination of Cl and L1 comprise a resonant circuit,
refered to as a tank circuit. Its purpose is to select one out of
many available radios signals. The variable capacitor Cl
allows for tuning to the various signals. The diode passes the
positive half cycles of the RF, removing the negative half
cycles (c). C2 is sized to filter the radio frequencies from the
RF envelope (c), passing audio frequencies (d) to the
headset. Note that no power supply is required for a crystal
radio. A germanium diode, which has a lower forward voltage
drop provides greater sensitvity than a silicon diode.
While 20000 magnetic headphones are shown above, a
ceramic earphone, sometimes called a crystal earphone, is
more sensitive. The ceramic earphone is desirable for all but
the strongest radio signals
The circuit in Figure below produces a stronger output than
the crystal detector. Since the transistor is not biased in the
linear region (no base bias resistor), it only conducts for
positive half cycles of RF input, detecting the audio
modulation. An advantage of a transistor detector is
amplification in addition to detection. This more powerful
circuit can readily drive 2000Q magnetic headphones. Note
the transistor is a germanuim PNP device. This is probably
more sensitive, due to the lower 0.2V Vp-, compared with
silicon. However, a silicon device should still work. Reverse
battery polarity for NPN silicon devices.
20002 double headphones
Coil - #34 AWG magnet wire
close wound over | in. length on
| 1/4 in. dia. form. Tap 1/4 in.
from bottom.
TR One, one transistor radio. No-bias-resistor causes
operation as a detector. After Stoner, Figure 4.4A. [DLS]
The 2000Q headphones are no longer a widely available
item. However, the low impedance earbuds commonly used
with portable audio equipment may be substituted when
paired with a suitable audio transformer. See Volume 6
Experiments, AC Circuits, Sensitive audio detector for details.
The circuit in Figure below adds an audio amplifier to the
crystal detector for greater headphone volume. The original
circuit used a germanium diode and transistor. [DLS] A
schottky diode may be substituted for the germanium diode.
A silicon transistor may be used if the base-bias resistor is
changed according to the table.
20002 double
headphones
Resistor
1.5V
Ge 47k 220k
Si 120k 1Meg
Coil - #34 AWG magnet
500 + wire close wound over
~ 1 in. length on | 1/4 in.
dia. form. Tap 1/4 in.
pF L
Crystal radio with one transistor audio amplifer, base-bias.
After Stoner, Figure 4.3A. [DLS]
— from bottom.
For more crystal radio circuits, simple one-transistor radios,
and more advanced low transistor count radios, see Wenzel
[CW1]
Regency TR1: First mass produced transistor radio, 1954.
The circuit in Figure below is an integrated circuit AM radio
containing all the active radio frequency circuitry within a
single IC. All capacitors and inductors, along with a few
resistors, are external to the IC. The 370 Pf variable capacitor
tunes the desired RF signal. The 320 pF variable capacitor
tunes the local oscillator 455 KHz above the RF input signal.
The RF signal and local oscillator frequencies mix producing
the sun and difference of the two at pin 15. The external 455
KHz ceramic filter between pins 15 and 12, selects the 455
KHz difference frequency. Most of the amplification is in the
intermediate frequency (IF) amplifier between pins 12 and 7.
A diode at pin 7 recovers audio from the IF. Some automatic
gain control (AGC) is recovered and filtered to DC and fed
back into pin 9.
= Ceramic filter
IC radio, After Signetics [SIG]
Figure below shows conventional mecahnical tuning (a) of
the RF input tuner and the local oscillator with varactor diode
tuning (b). The meshed plates of a dual variable capacitor
make for a bulky component. It is ecconomic to replace it
with varicap tuning diodes. Increasing the reverse bias Viyne
decreases capacitance which increases frequency. Viune could
be produced by a potentiometer.
weer em ee em me em em em ew ee em eee
Jp - - Vec —
’ 320pF +Vtune 970K . | .
(b)
IC radio comparison of (a) mechanical tuning to (b) electronic
varicap diode tuning.[SIG]
Figure below shows an even lower parts count AM radio. Sony
engineers have included the intermediate frequency (IF)
bandpass filter within the 8-pin IC. This eliminates external IF
transformers and an IF ceramic filter. L-C tuning components
are still required for the radio frequency (RF) input and the
local oscillator. Though, the variable capacitors could be
replaced by varicap tuning diodes.
Compact IC radio eliminates external IF filters. After Sony
[SNE]
Figure below shows a low-parts-count FM radio based on a
TDA7 021T integrated circuit by NXP Wireless. The bulky
external IF filter transformers have been replaced by R-C
filters. The resistors are integrated, the capacitors external.
This circuit has been simplified from Figure 5 in the NXP
Datasheet. See Figure 5 or 8 of the datasheet for the omitted
signal strength circuit. The simple tuning circuit is from the
Figure 5 Test Circuit. Figure 8 has a more elaborate tuner.
Datasheet Figure 8 shows a stereo FM radio with an audio
amplifier for driving a speaker. [NXP]
220 ‘
100 «| 3.3 220 _| ‘pF Field :
ap ap pk strengt
16 15 14 13 12 1] 10 9
a
- = A -
a Ye a
2 3 + 5 6 7 )
| 2
= 100
si ~ = 10 365 nF =
3, ]10 |100 nF nH | 40 is
+ nF nF pF
IC FM radio, signal strength circuit not shown. After NXP
Wireless Figure 5. [NXP]
For a construction project, the simplified FM Radio in Figure
above is recommended. For the 56nH inductor, wind 8 turns
of #22 AWG bare wire or magnet wire on a 0.125 inch drill bit
or other mandrel. Remove the mandrel and strech to 0.6 inch
length. The tuning capacitor may be a miniature trimmer
Capacitor.
Figure below is an example of a common-base (CB) RF
amplifier. It is a good illustration because it looks like a CB for
lack of a bias network. Since there is no bias, this is a class C
amplifier. The transistor conducts for less than 180° of the
input signal because at least 0.7 V bias would be required for
180° class B. The common-base configuration has higher
power gain at high RF frequencies than common-emitter.
This is a power amplifier (3/4 W) as opposed to a small signal
amplifier. The input and output m-networks match the emitter
and collector to the 50 Q input and output coaxial
terminations, respectively. The output m-network also helps
filter harmonics generated by the class C amplifier. Though,
more sections would likely be required by modern radiated
emissions standards.
100pF gt oND863 25nH 100pF
Class C common-base 750 mW RF power amplifier. L1 = #10
Cu wire 1/2 turn, 5/8 in. ID by 3/4 in. high. L2 = #14 tinned
Cu wire 1 1/2 turns, 1/2 in. ID by 1/3 in. spacing. After Texas
Instruments [TX1]
An example of a high gain common-base RF amplifier is
shown in Figure below. The common-base circuit can be
pushed to a higher frequency than other configurations. This
IS a common base configuration because the transistor bases
are grounded for AC by 1000 pF capacitors. The capacitors
are necessary (unlike the class C, Figure previous) to allow
the 1KQ-4KQ voltage divider to bias the transistor base for
class A operation. The 500Q resistors are emitter bias
resistors. They stablize the collector current. The 8500
resistors are collector DC loads. The three stage amplifier
provides an overall gain of 38 dB at 100 MHz with a 9 MHz
bandwidth.
68 4-30 4-30 4-30
1OnH 80nH ~— 1000 80nH ~~ 1000 =
2N1141 pl 2NL 141 pl 2N1141 » * fe)
(*) 3090 004 )
100nH
500 500 }
: 1000
= - pr | =
= IK
4K
2 2
nl |
100uH REC 2
100unH REC 2
nl’ | nb |
Class A common-base small-signal high gain amplifier. After
Texas Instruments [TX2]
A cascode amplifier has a wide bandwath like a common-
base amplifier and a moderately high input impedance like a
common emitter arrangement. The biasing for this cascode
amplifier (Figure below) is worked out in an example problem
Ch 4.
Class A cascode small-signal high gain amplifier.
This circuit (Figure above) is simulated in the “Cascode”
section of the BJT chapter Ch 4 .. Use RF or microwave
transistors for best high frequency response.
PIN diode T/R switch disconnects receiver from antenna
during transmit.
left antenna right antenna
PIN diode attenuator: PIN diodes function as voltage variable
resistors. After Lin [LCC].
The PIN diodes are arranged in a m-attenuator network. The
anti-series diodes cancel some harmonic distortion compared
with a single series diode. The fixed 1.25 V supply forward
biases the parallel diodes, which not only conducting DC
current from ground via the resistors, but also, conduct RF to
ground through the diodes' capacitors. The control voltage
Veontroy INCreases current through the parallel diodes as it
increases. This decreases the resistance and attenuation,
passing more RF from input to output. Attenuation is about 3
dB at Voontrol= 5 V. Attenuation is 40 dB at Voontrgi= 1 V with
flat frequency response to 2 gHz. At Voontroi= 0-5 V,
attenuation is 80 dB at 10 MHz. However, the frequency
response varies too much to use. [LCC]
Computational circuits
When someone mentions the word "computer," a digital
device is what usually comes to mind. Digital circuits
represent numerical quantities in binary format: patterns of
L's and O's represented by a multitude of transistor circuits
operating in saturated or cutoff states. However, analog
circuitry may also be used to represent numerical quantities
and perform mathematical calculations, by using variable
voltage signals instead of discrete on/off states.
Here is a simple example of binary (digital) representation
versus analog representation of the number "twenty-five:"
A digital circuit representing the number 25:
|
16+8+1=25
An analog circuit representing the number 25:
Voltmeter
I
Digital circuits are very different from circuits built on analog
principles. Digital computational circuits can be incredibly
complex, and calculations must often be performed in
sequential "steps" to obtain a final answer, much as a human
being would perform arithmetical calculations in steps with
pencil and paper. Analog computational circuits, on the other
hand, are quite simple in comparison, and perform their
calculations in continuous, real-time fashion. There is a
disadvantage to using analog circuitry to represent numbers,
though: imprecision. The digital circuit shown above is
representing the number twenty-five, precisely. The analog
circuit shown above may or may not be exactly calibrated to
25.000 volts, but is subject to "drift" and error.
In applications where precision is not critical, analog
computational circuits are very practical and elegant. Shown
here are a few op-amp circuits for performing analog
computation:
Analog summer (adder) circuit
1 kQ 1 kQ
Output
Output = Input, + Input,
Analog subtractor circuit
R R
Input)
Output
R R
Input,,
Output = Input,,, - Input, )
Analog averager circuit
R -. *— Output
Input, > (Buffer optional)
Input,
Input, + Input,
Output = 7
Analog inverter (sign reverser) circuit
R R
Input
Output
Output = - Input
Analog "multiply-by-constant" circuit
K
Output
Input
Output = (K)(Input)
Analog "divide-by-constant” circuit
Input
*— Output
Input
Output =
‘ K
Analog inverting "multiply/divide-
by-constant" circuit
Input
Output
Output = - (K)(Input)
Each of these circuits may be used in modular fashion to
create a circuit capable of multiple calculations. For instance,
suppose that we needed to subtract a certain fraction of one
variable from another variable. By combining a divide-by-
constant circuit with a subtractor circuit, we could obtain the
required function:
K
Divide-by-constant
Subtractor
R
Output
Input,
Input, -
Output = Input, - x
Devices called analog computers used to be common in
universities and engineering shops, where dozens of op-amp
circuits could be "patched" together with removable jumper
wires to model mathematical statements, usually for the
purpose of simulating some physical process whose
underlying equations were known. Digital computers have
made analog computers all but obsolete, but analog
computational circuitry cannot be beaten by digital in terms
of sheer elegance and economy of necessary components.
Analog computational circuitry excels at performing the
calculus operations integration and differentiation with
respect to time, by using capacitors in an op-amp feedback
loop. To fully understand these circuits' operation and
applications, though, we must first grasp the meaning of
these fundamental calculus concepts. Fortunately, the
application of op-amp circuits to real-world problems
involving calculus serves as an excellent means to teach
basic calculus. In the words of John I. Smith, taken from his
outstanding textbook, Modern Operational Circuit Design:
"A note of encouragement is offered to certain readers:
integral calculus is one of the mathematical disciplines
that operational [amplifier] circuitry exploits and, in the
process, rather demolishes as a barrier to
understanding." (pg. 4)
Mr. Smith's sentiments on the pedagogical value of analog
circuitry as a learning tool for mathematics are not unique.
Consider the opinion of engineer George Fox Lang, in an
article he wrote for the August 2000 issue of the journal
Sound and Vibration, entitled, "Analog was not a Computer
Trademark!":
"Creating a real physical entity (a circuit) governed by a
particular set of equations and interacting with it
provides unique insight into those mathematical
statements. There is no better way to develop a "gut
feel" for the interplay between physics and mathematics
than to experience such an interaction. The analog
computer was a powerful interdisciplinary teaching tool;
its obsolescence is mourned by many educators in a
variety of fields." (pg. 23)
Differentiation is the first operation typically learned by
beginning calculus students. Simply put, differentiation is
determining the instantaneous rate-of-change of one variable
as it relates to another. In analog differentiator circuits, the
independent variable is time, and so the rates of change
we're dealing with are rates of change for an electronic signal
(voltage or current) with respect to time.
Suppose we were to measure the position of a car, traveling
in a direct path (no turns), from its starting point. Let us call
this measurement, x. If the car moves at a rate such that its
distance from "start" increases steadily over time, its position
will plot on a graph as a /inear function (straight line):
—
Position
Time —>~
If we were to calculate the derivative of the car's position
with respect to time (that is, determine the rate-of-change of
the car's position with respect to time), we would arrive ata
quantity representing the car's velocity. The differentiation
function is represented by the fractional notation d/d, so
when differentiating position (x) with respect to time (f), we
denote the result (the derivative) as dx/dt:
—
Position Velocity
dx
dt
Time —~ Time —>~
For a linear graph of x over time, the derivate of position
(dx/dt), otherwise and more commonly known as velocity, will
be a flat line, unchanging in value. The derivative of a
mathematical function may be graphically understood as its
slope when plotted on a graph, and here we can see that the
position (x) graph has a constant slope, which means that its
derivative (dx/dt) must be constant over time.
Now, suppose the distance traveled by the car increased
exponentially over time: that is, it began its travel in slow
movements, but covered more additional distance with each
passing period in time. We would then see that the derivative
of position (dx/dt), otherwise known as velocity (v), would not
be constant over time, but would increase:
@Baar==
—
Position Velocity
dx
dt
Time —> Time —>
The height of points on the velocity graph correspond to the
rates-of-change, or slope, of points at corresponding times on
the position graph:
Position Velocity
dx
dt
Tine —~ Tine —~
What does this have to do with analog electronic circuits?
Well, if we were to have an analog voltage signal represent
the car's position (think of a huge potentiometer whose wiper
was attached to the car, generating a voltage proportional to
the car's position), we could connect a differentiator circuit to
this signal and have the circuit continuously ca/culate the
car's velocity, displaying the result via a voltmeter connected
to the differentiator circuit's output:
Differentiator
, dx
x Velocity + dt
Position | _
> = —)
—
Recall from the last chapter that a differentiator circuit
outputs a voltage proportional to the input voltage's rate-of
change over time (d/dt). Thus, if the input voltage is
changing over time at a constant rate, the output voltage will
be at a constant value. If the car moves in such a way that its
elapsed distance over time builds up at a steady rate, then
that means the car is traveling at a constant velocity, and
the differentiator circuit will output a constant voltage
proportional to that velocity. If the car's elapsed distance
over time changes in a non-steady manner, the differentiator
circuit's output will likewise be non-steady, but always ata
level representative of the input's rate-of-change over time.
Note that the voltmeter registering velocity (at the output of
the differentiator circuit) is connected in "reverse" polarity to
the output of the op-amp. This is because the differentiator
circuit shown is /nverting: outputting a negative voltage for a
positive input voltage rate-of-change. If we wish to have the
voltmeter register a positive value for velocity, it will have to
be connected to the op-amp as shown. As impractical as it
may be to connect a giant potentiometer to a moving object
such as an automobile, the concept should be clear: by
electronically performing the calculus function of
differentiation on a signal representing position, we obtain a
signal representing velocity.
Beginning calculus students learn symbolic techniques for
differentiation. However, this requires that the equation
describing the original graph be known. For example,
calculus students learn how to take a function such as y = 3x
and find its derivative with respect to x (d/dx), 3, simply by
manipulating the equation. We may verify the accuracy of
this manipulation by comparing the graphs of the two
functions:
Ps aac slope = 3
i
Nonlinear functions such as y = 3x? may also be
differentiated by symbolic means. In this case, the derivative
of y = 3x¢ with respect to x is 6x:
In real life, though, we often cannot describe the behavior of
any physical event by a simple equation like y = 3x, and so
symbolic differentiation of the type learned by calculus
students may be impossible to apply to a physical
measurement. If someone wished to determine the derivative
of our hypothetical car's position (dx/dt = velocity) by
symbolic means, they would first have to obtain an equation
describing the car's position over time, based on position
measurements taken from a real experiment -- a nearly
impossible task unless the car is operated under carefully
controlled conditions leading to a very simple position graph.
However, an analog differentiator circuit, by exploiting the
behavior of a capacitor with respect to voltage, current, and
time / = C(dv/dt), naturally differentiates any real signal in
relation to time, and would be able to output a signal
corresponding to instantaneous velocity (dx/dt) at any
moment. By logging the car's position signal along with the
differentiator's output signal using a chart recorder or other
data acquisition device, both graphs would naturally present
themselves for inspection and analysis.
We may take the principle of differentiation one step further
by applying it to the velocity signal using another
differentiator circuit. In other words, use it to calculate the
rate-of-change of velocity, which we know is the rate-of-
change of position. What practical measure would we arrive
at if we did this? Think of this in terms of the units we use to
measure position and velocity. If we were to measure the
car's position from its starting point in miles, then we would
probably express its velocity in units of miles per hour
(dx/dt). If we were to differentiate the velocity (measured in
miles per hour) with respect to time, we would end up witha
unit of miles per hour per hour. Introductory physics classes
teach students about the behavior of falling objects,
measuring position in meters, velocity in meters per second,
and change in velocity over time in meters per second, per
second. This final measure is called acceleration: the rate of
change of velocity over time:
@Baar==
—
Position Velocity Acceleration
dx dx
dt dt
Tine —~ Tine —~ Tine —>~
—_> —_>
Differentiation Differentiation
The expression a’x/dt? is called the second derivative of
position (x) with regard to time (f). If we were to connect a
second differentiator circuit to the output of the first, the last
voltmeter would register acceleration:
Differentiator Differentiator
—
Deriving velocity from position, and acceleration from
velocity, we see the principle of differentiation very clearly
illustrated. These are not the only physical measurements
related to each other in this way, but they are, perhaps, the
most common. Another example of calculus in action is the
relationship between liquid flow (q) and liquid volume (v)
accumulated in a vessel over time:
vy = volume
A "Level Transmitter" device mounted on a water storage
tank provides a signal directly proportional to water level in
the tank, which -- if the tank is of constant cross-sectional
area throughout its height -- directly equates water volume
stored. If we were to take this volume signal and differentiate
it with respect to time (dv/dt), we would obtain a signal
proportional to the water flow rate through the pipe carrying
water to the tank. A differentiator circuit connected in such a
way as to receive this volume signal would produce an
output signal proportional to flow, possibly substituting for a
flow-measurement device ("Flow Transmitter") installed in
the pipe.
Returning to the car experiment, suppose that our
hypothetical car were equipped with a tachogenerator on
one of the wheels, producing a voltage signal directly
proportional to velocity. We could differentiate the signal to
obtain acceleration with one circuit, like this:
Differentiator
ome: > o> mo)
fo |
By its very nature, the tachogenerator differentiates the car's
position with respect to time, generating a voltage
proportional to how rapidly the wheel's angular position
changes over time. This provides us with a raw signal already
representative of velocity, with only a single step of
differentiation needed to obtain an acceleration signal. A
tachogenerator measuring velocity, of course, is a far more
practical example of automobile instrumentation than a giant
potentiometer measuring its physical position, but what we
gain in practicality we lose in position measurement. No
matter how many times we differentiate, we can never infer
the car's position from a velocity signal. If the process of
differentiation brought us from position to velocity to
acceleration, then somehow we need to perform the
"reverse" process of differentiation to go from velocity to
position. Such a mathematical process does exist, and it is
called integration. The "integrator" circuit may be used to
perform this function of integration with respect to time:
Integrator
Position
Differentiator
dv d°x
dt — dt
Acceleration +
—
Recall from the last chapter that an integrator circuit outputs
a voltage whose rate-of-change over time is proportional to
the input voltage's magnitude. Thus, given a constant input
voltage, the output voltage will change at a constant rate. If
the car travels at a constant velocity (constant voltage input
to the integrator circuit from the tachogenerator), then its
distance traveled will increase steadily as time progresses,
and the integrator will output a steadily changing voltage
proportional to that distance. If the car's velocity is not
constant, then neither will the rate-of-change over time be of
the integrator circuit's output, but the output voltage wil/
faithfully represent the amount of distance traveled by the
car at any given point in time.
The symbol for integration looks something like a very
narrow, cursive letter "S" (J). The equation utilizing this
symbol (fv dt = x) tells us that we are integrating velocity (v)
with respect to time (dt), and obtaining position (x) asa
result.
So, we may express three measures of the car's motion
(position, velocity, and acceleration) in terms of velocity (Vv)
just as easily as we could in terms of position (x):
—
Position Velocity Acceleration
Time —> Time — Time —
ener pitaantste
If we had an accelerometer attached to the car, generating a
signal proportional to the rate of acceleration or deceleration,
we could (hypothetically) obtain a velocity signal with one
step of integration, and a position signal with a second step
of integration:
Integrator
+
(v) lladt=x
Position
Integrator
a) V) lads
Velocity +
Acceleration [| _
Accel.
emo: —> =>
—
Thus, all three measures of the car's motion (position,
velocity, and acceleration) may be expressed in terms of
acceleration:
—
Position Velocity Acceleration
Jadt oe | | ladt J
Time — Time —> Time —
adie aoe
As you might have suspected, the process of integration may
be illustrated in, and applied to, other physical systems as
well. Take for example the water storage tank and flow
example shown earlier. If flow rate is the derivative of tank
volume with respect to time (q = dv/dt), then we could also
say that volume is the /ntegral of flow rate with respect to
time:
f= flow
Water
supply
| fdt = volume
If we were to use a "Flow Transmitter" device to measure
water flow, then by time-integration we could calculate the
volume of water accumulated in the tank over time. Although
it is theoretically possible to use a capacitive op-amp
integrator circuit to derive a volume signal from a flow signal,
mechanical and digital electronic "integrator" devices are
more suitable for integration over long periods of time, and
find frequent use in the water treatment and distribution
industries.
Just as there are symbolic techniques for differentiation,
there are also symbolic techniques for integration, although
they tend to be more complex and varied. Applying symbolic
integration to a real-world problem like the acceleration of a
car, though, is still contingent on the availability of an
equation precisely describing the measured signal -- often a
difficult or impossible thing to derive from measured data.
However, electronic integrator circuits perform this
mathematical function continuously, in real time, and for any
input signal profile, thus providing a powerful tool for
scientists and engineers.
Having said this, there are caveats to the using calculus
techniques to derive one type of measurement from another.
Differentiation has the undesirable tendency of amplifying
"noise" found in the measured variable, since the noise will
typically appear as frequencies much higher than the
measured variable, and high frequencies by their very nature
possess high rates-of-change over time.
To illustrate this problem, Suppose we were deriving a
measurement of car acceleration from the velocity signal
obtained from a tachogenerator with worn brushes or
commutator bars. Points of poor contact between brush and
commutator will produce momentary "dips" in the
tachogenerator's output voltage, and the differentiator
circuit connected to it will interpret these dips as very rapid
changes in velocity. For a car moving at constant speed --
neither accelerating nor decelerating -- the acceleration
signal should be 0 volts, but "noise" in the velocity signal
caused by a faulty tachogenerator will cause the
differentiated (acceleration) signal to contain "spikes,"
falsely indicating brief periods of high acceleration and
deceleration:
Differentiator
Noise voltage present in a signal to be differentiated need
not be of significant amplitude to cause trouble: all that is
required is that the noise profile have fast rise or fall times. In
other words, any electrical noise with a high dv/dt
component will be problematic when differentiated, even if it
is of low amplitude.
It should be noted that this problem is not an artifact (an
idiosyncratic error of the measuring/computing instrument)
of the analog circuitry; rather, it is inherent to the process of
differentiation. No matter how we might perform the
differentiation, "noise" in the velocity signal will invariably
corrupt the output signal. Of course, if we were
differentiating a signal twice, as we did to obtain both
velocity and acceleration from a position signal, the
amplified noise signal output by the first differentiator circuit
will be amplified again by the next differentiator, thus
compounding the problem:
more noise even more noise!
little noise
a H&
oe
® im Differentiator
—
Integration does not suffer from this problem, because
integrators act as low-pass filters, attenuating high-
frequency input signals. In effect, all the high and low peaks
resulting from noise on the signal become averaged together
over time, for a diminished net result. One might suppose,
then, that we could avoid all trouble by measuring
acceleration directly and integrating that signal to obtain
velocity; in effect, calculating in "reverse" from the way
shown previously:
Integrator
la dt=y
Velocity
—
Unfortunately, following this methodology might lead us into
other difficulties, one being a common artifact of analog
integrator circuits known as drift. All oop-amps have some
amount of input bias current, and this current will tend to
cause a Charge to accumulate on the capacitor in addition to
whatever charge accumulates as a result of the input voltage
signal. In other words, all analog integrator circuits suffer
from the tendency of having their output voltage "drift" or
“creep” even when there is absolutely no voltage input,
accumulating error over time as a result. Also, imperfect
Capacitors will tend to lose their stored charge over time due
to internal resistance, resulting in "drift" toward zero output
voltage. These problems are artifacts of the analog circuitry,
and may be eliminated through the use of digital
computation.
Circuit artifacts notwithstanding, possible errors may result
from the integration of one measurement (Such as
acceleration) to obtain another (such as velocity) simply
because of the way integration works. If the "Zero" calibration
point of the raw signal sensor is not perfect, it will output a
Slight positive or negative signal even in conditions when it
should output nothing. Consider a car with an imperfectly
calibrated accelerometer, or one that is influenced by gravity
to detect a slight acceleration unrelated to car motion. Even
with a perfect integrating computer, this sensor error will
cause the integrator to accumulate error, resulting in an
output signal indicating a change of velocity when the car is
neither accelerating nor decelerating.
Integrator
(slight positive
la dt=y
Velocity
(small rate
-— of change
a
As with differentiation, this error will also compound itself if
the integrated signal is passed on to another integrator
circuit, since the "drifting" output of the first integrator will
very soon present a significant positive or negative signal for
the next integrator to integrate. Therefore, care should be
taken when integrating sensor signals: if the "Zero"
adjustment of the sensor is not perfect, the integrated result
will drift, even if the integrator circuit itself is perfect.
So far, the only integration errors discussed have been
artificial in nature: originating from imperfections in the
circuitry and sensors. There also exists a source of error
inherent to the process of integration itself, and that is the
unknown constant problem. Beginning calculus students
learn that whenever a function is integrated, there exists an
unknown constant (usually represented as the variable C)
added to the result. This uncertainty is easiest to understand
by comparing the derivatives of several functions differing
only by the addition of a constant value:
d 2
<<. 3 = 6)
= x +4 x
— 3x’ = 6x
d 2
3x -6=6.
ax °* 7
Note how each of the parabolic curves (y = 3x? + C) share
the exact same shape, differing from each other in regard to
their vertical offset. However, they all share the exact same
derivative function: y’ = (d/dx)( 3x? + C) = 6x, because they
all share identical rates of change (slopes) at corresponding
points along the x axis. While this seems quite natural and
expected from the perspective of differentiation (different
equations sharing a common derivative), it usually strikes
beginning students as odd from the perspective of
integration, because there are multiple correct answers for
the integral of a function. Going from an equation to its
derivative, there is only one answer, but going from that
derivative back to the original equation leads us to a range
of correct solutions. In honor of this uncertainty, the symbolic
function of integration is called the indefinite integral.
When an integrator performs live signal integration with
respect to time, the output is the sum of the integrated input
Signal over time and an initial value of arbitrary magnitude,
representing the integrator's pre-existing output at the time
integration began. For example, if | integrate the velocity of a
car driving in a straight line away from a city, calculating
that a constant velocity of 50 miles per hour over a time of 2
hours will produce a distance (fv dt) of 100 miles, that does
not necessarily mean the car will be 100 miles away from the
city after 2 hours. All it tells us is that the car will be 100
miles further away from the city after 2 hours of driving. The
actual distance from the city after 2 hours of driving depends
on how far the car was from the city when integration began.
If we do not know this initial value for distance, we cannot
determine the car's exact distance from the city after 2 hours
of driving.
This same problem appears when we integrate acceleration
with respect to time to obtain velocity:
Integrator
jadt=v+ Vp
Where,
v, = Initial velocity
—
In this integrator system, the calculated velocity of the car
will only be valid if the integrator circuit is initialized to an
output value of zero when the car is stationary (v= 0).
Otherwise, the integrator could very well be outputting a
non-zero signal for velocity (vg) when the car is stationary,
for the accelerometer cannot tell the difference between a
stationary state (0 miles per hour) and a state of constant
velocity (say, 60 miles per hour, unchanging). This
uncertainty in integrator output is inherent to the process of
integration, and not an artifact of the circuitry or of the
sensor.
In summary, if maximum accuracy is desired for any physical
measurement, it is best to measure that variable directly
rather than compute it from other measurements. This is not
to say that computation is worthless. Quite to the contrary,
often it is the only practical means of obtaining a desired
measurement. However, the limits of computation must be
understood and respected in order that precise
measurements be obtained.
Measurement circuits -- INCOMPLETE
Figure below shows a photodiode amplifier for measuring low
levels of light. Best sensitivity and bandwidth are obtained
with a transimpedance amplifier, a current to voltage
amplifier, instead of a conventional operational amplifier. The
photodiode remains reverse biased for lowest diode
Capacitance, hence wider bandwidth, and lower noise. The
feedback resistor sets the “gain”, the current to voltage
amplification factor. Typical values are 1 to 10 Meg Q. Higher
values yield higher gain. A capacitor of a few pF may be
required to compensate for photodiode capacitance, and
prevents instability at the high gain. The wiring at the
summing node must be as compact as possible. This point is
sensitive to circuit board contaminants and must be
thoroughly cleaned. The most sensitive amplifiers contain
the photodiode and amplifier within a hybrid microcircuit
package or single die.
WV
\ Vo
Photodiode amplifier.
Control circuits -- PENDING
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Warren Young (August 2002): Initial idea and text for
“Power supply circuits" section. Paragraphs modified by Tony
Kuphaldt (changes in vocabulary, plus inclusion of additional
concepts).
Bill Marsden (April 2008) Author of “ElectroStatic
Discharge” section.
Bibliography
1. [LCC]Chin-Leong Lim, Lim Yeam Ch'ng, Goh Swee Chye,
“Diode Quad Is Foundation For PIN Diode Attenuator,”
Microwaves & RF, May 2006, at
http://www.mwrf.com/Articles/Index.cfm?
Ad=16ArticlelD=12523
2. [MUL]“Transistor Audio and Radio Circuits,” TP1399, 2nd
Ed., pp 39-40, Mullard, London, 1972.
3. [SIG]“AM Receiver Circuit TCA440,” Analog Data Manual,
2nd Ed., pp 14-20 to 14-26, Signetics, 1982.
4.[SNE]Sony “8-pin Single-Chip AM Radio with Builot-in
Power Amplifier,” pp 5, at
http://www.datasheetcatalog.com/datasheets_pdf/C/X/A/
1/CXA1600.shtml
5. [TX1]Texas Instruments “Solid State Communications,”
pp 318, McGraw-Hill, N.Y, 1966.
6. [TX2]Texas Instruments “Transistor Circuit Design,” pp
290, McGraw-Hill, N.Y., 1963.
7. [NXP] “Datasheet TDA7021T”, STR-NXP Wireless, at
http://www.nxp.com/acrobat_download/datasheets/TDA7
021T CNV_2.pdf
8.[DLS]Donald L. Stoner, L. A. Earnshaw, “The Transistor
Radio Handbook,” pp 76, Editors and Eenineers,
Sumerland, CA, 1963.
9. [CW1],Charles Wenzel, “Crystal Radio Circuits,” at
http://www.techlib.com/electronics/crystal. html.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
— 4 —>
—/ | 4]
Lessons In Electric Circuits
-- Volume lll
Chapter 10
ACTIVE FILTERS
e« Two pole active filters
ek PENDING **
Two pole active filters
Figure below
R
1
OK
Output C1
0.02uF Output
C2 Input
470pF
(a) (b)
Low Pass High pass
(a) 10Khz Low-pass filter. (b) 100Hz cutoff high-pass filter
110K
Test
c1-A1+R2 0 gy. R14 Re
V2R1I R20, V3R1IR20,
a, a ee
(R1 + R2)@ (R1 + R2)@.
Butterworth Linear Phase
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] 4]\—
—| | +4/l—
Lessons In Electric Circuits
-- Volume Ill
Chapter 11
DC MOTOR DRIVES
e Pulse Width Modulation
e Contributors
«& INCOMPLETE ***
Pulse Width Modulation
Pulse Width Modulation (PWM) uses digital signals to control
power applications, as well as being fairly easy to convert
back to analog with a minimum of hardware.
Analog systems, such as linear power supplies, tend to
generate a lot of heat since they are basically variable
resistors carrying a lot of current. Digital systems don't
generally generate as much heat. Almost all the heat
generated by a switching device is during the transition
(which is done quickly), while the device is neither on nor off,
but in between. This is because power follows the following
formula:
P = El, or Watts = Voltage X Current
If either voltage or current is near zero then power will be
near zero. PWM takes full advantage of this fact.
PWM can have many of the characteristics of an analog
control system, in that the digital signal can be free
wheeling. PWM does not have to capture data, although
there are exceptions to this with higher end controllers.
One of the parameters of any square wave is duty cycle. Most
square waves are 50%, this is the norm when discussing
them, but they don't have to be symmetrical. The ON time
can be varied completely between signal being off to being
fully on, O% to 100%, and all ranges between.
Shown below are examples of a 10%, 50%, and 90% duty
cycle. While the frequency is the same for each, this is not a
requirement.
ca 10% 50% 90%
Examples of PWM Waveforms
The reason PWM is popular is simple. Many loads, such as
resistors, integrate the power into a number matching the
percentage. Conversion into its analog equivalent value is
straightforward. LEDs are very nonlinear in their response to
current, give an LED half its rated current you you still get
more than half the light the LED can produce. With PWM the
light level produced by the LED is very linear. Motors, which
will be covered later, are alSo very responsive to PWM.
One of several ways PWM can be produced is by using a
sawtooth waveform and a comparator. As shown below the
sawtooth (or triangle wave) need not be symmetrical, but
linearity of the waveform is important. The frequency of the
sawtooth waveform is the sampling rate for the signal.
a Maat
| aegegege
pf Pee
PWM Modulator Why Ramp Symmetry Doesn't Matter
If there isn't any computation involved PWM can be fast. The
limiting factor is the comparators frequency response. This
may not be an issue since quite a few of the uses are fairly
low speed. Some microcontrollers have PWM built in, and can
record or create signals on demand.
Uses for PWM vary widely. It is the heart of Class D audio
amplifiers, by increasing the voltages you increase the
maximum output, and by selecting a frequency beyond
human hearing (typically 44Khz) PWM can be used. The
speakers do not respond to the high frequency, but
duplicates the low frequency, which is the audio signal.
Higher sampling rates can be used for even better fidelity,
and 100Khz or much higher is not unheard of.
ANAAANAARAAAAAAAAAAAAAAAL
VVVVVVVVVVVVVVVVVVVVVVVYV YY
How an Audio Signal is modulated with PWM
Another popular application is motor speed control. Motors as
a class require very high currents to operate. Being able to
vary their speed with PWM increases the efficiency of the
total system by quite a bit. PWM is more effective at
controlling motor speeds at low RPM than linear methods.
PWM is often used in conjunction with an H-Bridge. This
configuration is so named because it resembles the letter H,
and allows the effective voltage across the load to be
doubled, since the power supply can be switched across both
sides of the load. In the case of inductive loads, such as
motors, diodes are used to suppress inductive spikes, which
may damage the transistors. The inductance in a motor also
tends to reject the high frequency component of the
waveform. This configuration can also be used with speakers
for Class D audio amps.
While basically accurate, this schematic of an H-Bridge has
one serious flaw, it is possible while transitioning between
the MOSFETs that both transistors on top and bottom will be
on simultaneously, and will take the full brunt of what the
power supply can provide. This condition is referred to as
shoot through, and can happen with any type of transistor
used in a H-Bridge. If the power supply is powerful enough
the transistors will not survive. It is handled by using drivers
in front of the transistors that allow one to turn off before
allowing the other to turn on.
fc) 7 (J
it) ——
PWM ; PWM
S s SG
y
A simplified H Bridge
+
“w oTo w
Switching Mode Power Supplies (SMPS) can also use PWM,
although other methods also exist. Adding topologies that
use the stored power in both inductors and capacitors after
the main switching components can boost the efficiencies for
these devices quite high, exceeding 90% in some cases.
Below is an example of such a configuration.
Unregulated Voltage o—4 oRegulated Voltage Out
Voltage
Reference
Linear Comparator
Op 4mp
Example of SMPS using PWM
Efficiency in this case is measured as wattage. If you have a
SMPS with 90% efficiency, and it converts 12VDC to 5VDC at
10 Amps, the 12V side will be pulling approximately 4.6
Amps. The 10% (5 watts) not accounted for will show up as
waste heat. While being slightly noisier, this type of regulator
will run much cooler than its linear counterpart.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Bill Marsden (February 2010) Author of “Pulse Width
Modulation” section.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
Next
a
nts
zo
Co
joe
—/ | 4]
Lessons In Electric Circuits
-- Volume lll
Chapter 12
INVERTERS AND AC
MOTOR DRIVES
ek PENDING **
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] +]
—/ | 4]
Lessons In Electric Circuits
-- Volume lll
Chapter 13
ELECTRON TUBES
Introduction
Early tube history
e The triode
The tetrode
Beam power tubes
The pentode
Combination tubes
Tube parameters
lonization (gas-filled) tubes
Display tubes
Microwave tubes
Tubes versus Semiconductors
Introduction
An often neglected area of study in modern electronics is
that of tubes, more precisely known as vacuum tubes or
electron tubes. Almost completely overshadowed by
semiconductor, or "solid-state" components in most modern
applications, tube technology once dominated electronic
circuit design.
In fact, the historical transition from "electric" to "electronic"
circuits really began with tubes, for it was with tubes that we
entered into a whole new realm of circuit function: a way of
controlling the flow of electrons (current) in a circuit by
means of another electric signal (in the case of most tubes,
the controlling signal is a small voltage). The semiconductor
counterpart to the tube, of course, is the transistor.
Transistors perform much the same function as tubes:
controlling the flow of electrons in a circuit by means of
another flow of electrons in the case of the bipolar transistor,
and controlling the flow of electrons by means of a voltage
in the case of the field-effect transistor. In either case, a
relatively small electric signal controls a relatively large
electric current. This is the essence of the word "electronic,"
So as to distinguish it from "electric," which has more to do
with how electron flow is regulated by Ohm's Law and the
physical attributes of wire and components.
Though tubes are now obsolete for all but a few specialized
applications, they are still a worthy area of study. If nothing
else, it is fascinating to explore "the way things used to be
done" in order to better appreciate modern technology.
Early tube history
Thomas Edison, that prolific American inventor, is often
credited with the invention of the incandescent lamp. More
accurately, it could be said that Edison was the man who
perfected the incandescent lamp. Edison's successful design
of 1879 was actually preceded by 77 years by the British
scientist Sir Humphry Davy, who first demonstrated the
principle of using electric current to heat a thin strip of
metal (called a "filament") to the point of incandescence
(glowing white hot).
Edison was able to achieve his success by placing his
filament (made of carbonized sewing thread) inside of a
clear glass bulb from which the air had been forcibly
removed. In this vacuum, the filament could glow at white-
hot temperatures without being consumed by combustion:
clear, glass bulb
air removed
filament
In the course of his experimentation (sometime around
1883), Edison placed a strip of metal inside of an evacuated
(vacuum) glass bulb along with the filament. Between this
metal strip and one of the filament connections he attached
a sensitive ammeter. What he found was that electrons
would flow through the meter whenever the filament was
hot, but ceased when the filament cooled down:
metal strip
The white-hot filament in Edison's lamp was liberating free
electrons into the vacuum of the lamp, those electrons
finding their way to the metal strip, through the
galvanometer, and back to the filament. His curiosity
piqued, Edison then connected a fairly high-voltage battery
in the galvanometer circuit to aid the small current:
more
current |
Sure enough, the presence of the battery created a much
larger current from the filament to the metal strip. However,
when the battery was turned around, there was little to no
Current at all!
In effect, what Edison had stumbled upon was a diode!
Unfortunately, he saw no practical use for such a device and
proceeded with further refinements in his lamp design.
The one-way electron flow of this device (Known as the
Edison Effect) remained a curiosity until J. A. Fleming
experimented with its use in 1895. Fleming marketed his
device as a "valve," initiating a whole new area of study in
electric circuits. Vacuum tube diodes -- Fleming's "valves"
being no exception -- are not able to handle large amounts
of current, and so Fleming's invention was impractical for
any application in AC power, only for small electric signals.
Then in 1906, another inventor by the name of Lee De Forest
started playing around with the "Edison Effect," seeing what
more could be gained from the phenomenon. In doing so, he
made a Startling discovery: by placing a metal screen
between the glowing filament and the metal strip (which by
now had taken the form of a plate for greater surface area),
the stream of electrons flowing from filament to plate could
be regulated by the application of a small voltage between
the metal screen and the filament:
The DeForest "Audion" tube
—
“plate”
"grid"
"filament"
control
voltage
plate current can be controlled by the
application of a small control voltage
between the grid and filament!
De Forest called this metal screen between filament and
plate a grid. It wasn't just the amount of voltage between
grid and filament that controlled current from filament to
plate, it was the polarity as well. A negative voltage applied
to the grid with respect to the filament would tend to choke
off the natural flow of electrons, whereas a positive voltage
would tend to enhance the flow. Although there was some
amount of current through the grid, it was very small; much
smaller than the current through the plate.
Perhaps most importantly was his discovery that the small
amounts of grid voltage and grid current were having large
effects on the amount of plate voltage (with respect to the
filament) and plate current. In adding the grid to Fleming's
"valve," De Forest had made the valve adjustable: it now
functioned as an amplifying device, whereby a small
electrical signal could take control over a larger electrical
quantity.
The closest semiconductor equivalent to the Audion tube,
and to all of its more modern tube equivalents, is an n-
channel D-type MOSFET. It is a voltage-controlled device
with a large current gain.
Calling his invention the "Audion," he vigorously applied it
to the development of communications technology. In 1912
he sold the rights to his Audion tube as a telephone signal
amplifier to the American Telephone and Telegraph
Company (AT and T), which made long-distance telephone
communication practical. In the following year he
demonstrated the use of an Audion tube for generating
radio-frequency AC signals. In 1915 he achieved the
remarkable feat of broadcasting voice signals via radio from
Arlington, Virginia to Paris, and in 1916 inaugurated the first
radio news broadcast. Such accomplishments earned De
Forest the title "Father of Radio" in America.
SINGLE-TUBE RADIO
The triode
De Forest's Audion tube came to be known as the triode
tube, because it had three elements: filament, grid, and
plate (just as the "di" in the name diode refers to two
elements, filament and plate). Later developments in diode
tube technology led to the refinement of the electron
emitter: instead of using the filament directly as the
emissive element, another metal strip called the cathode
could be heated by the filament.
This refinement was necessary in order to avoid some
undesired effects of an incandescent filament as an electron
emitter. First, a filament experiences a voltage drop along its
length, as current overcomes the resistance of the filament
material and dissipates heat energy. This meant that the
voltage potential between different points along the length
of the filament wire and other elements in the tube would
not be constant. For this and similar reasons, alternating
current used as a power source for heating the filament wire
would tend to introduce unwanted AC "noise" in the rest of
the tube circuit. Furthermore, the surface area of a thin
filament was limited at best, and limited surface area on the
electron emitting element tends to place a corresponding
limit on the tube's current-carrying capacity.
The cathode was a thin metal cylinder fitting snugly over
the twisted wire of the filament. The cathode cylinder would
be heated by the filament wire enough to freely emit
electrons, without the undesirable side effects of actually
carrying the heating current as the filament wire had to. The
tube symbol for a triode with an indirectly-heated cathode
looks like this:
plate
grid
(
cathode filament
Since the filament is necessary for all but a few types of
vacuum tubes, it is often omitted in the symbol for
simplicity, or it may be included in the drawing but with no
power connections drawn to it:
no filament
shown ata no connections shown
to filament wires
A simple triode circuit is shown to illustrate its basic
operation as an amplifier:
Triode amplifier circuit
“plate supply”
— DC power
— source
input
voltage
The low-voltage AC signal connected between the grid and
cathode alternately suppresses, then enhances the electron
flow between cathode and plate. This causes a change in
voltage on the output of the circuit (between plate and
cathode). The AC voltage and current magnitudes on the
tube's grid are generally quite small compared with the
variation of voltage and current in the plate circuit. Thus,
the triode functions as an amplifier of the incoming AC
signal (taking high-voltage, high-current DC power supplied
from the large DC source on the right and "throttling" it by
means of the tube's controlled conductivity).
In the triode, the amount of current from cathode to plate
(the "controlled" current is a function both of grid-to-
cathode voltage (the controlling signal) and the plate-to-
cathode voltage (the electromotive force available to push
electrons through the vacuum). Unfortunately, neither of
these independent variables have a purely linear effect on
the amount of current through the device (often referred to
simply as the "plate current"). That is, triode current does
not necessarily respond in a direct, proportional manner to
the voltages applied.
In this particular amplifier circuit the nonlinearities are
compounded, as plate voltage (with respect to cathode)
changes along with the grid voltage (also with respect to
cathode) as plate current is throttled by the tube. The result
will be an output voltage waveform that doesn't precisely
resemble the waveform of the input voltage. In other words,
the quirkiness of the triode tube and the dynamics of this
particular circuit will distort the waveshape. If we really
wanted to get complex about how we stated this, we could
say that the tube introduces harmonics by failing to exactly
reproduce the input waveform.
Another problem with triode behavior is that of stray
Capacitance. Remember that any time we have two
conductive surfaces separated by an insulating medium, a
capacitor will be formed. Any voltage between those two
conductive surfaces will generate an electric field within
that insulating region, potentially storing energy and
introducing reactance into a circuit. Such is the case with
the triode, most problematically between the grid and the
plate. It is as if there were tiny capacitors connected
between the pairs of elements in the tube:
C
erid-plate
~plate-cathod
(
erid-cathod
Now, this stray capacitance is quite small, and the reactive
impedances usually high. Usually, that is, unless radio
frequencies are being dealt with. As we saw with De Forest's
Audion tube, radio was probably the prime application for
this new technology, so these "tiny" capacitances became
more than just a potential problem. Another refinement in
tube technology was necessary to overcome the limitations
of the triode.
The tetrode
As the name suggests, the tetrode tube contained four
elements: cathode (with the implicit filament, or "heater"),
grid, plate, and a new element called the screen. Similar in
construction to the grid, the screen was a wire mesh or coil
positioned between the grid and plate, connected toa
source of positive DC potential (with respect to the cathode,
as uSual) equal to a fraction of the plate voltage. When
connected to ground through an external capacitor, the
screen had the effect of electrostatically shielding the grid
from the plate. Without the screen, the capacitive linking
between the plate and the grid could cause significant
signal feedback at high frequencies, resulting in unwanted
oscillations.
The screen, being of less surface area and lower positive
potential than the plate, didn't attract many of the electrons
passing through the grid from the cathode, so the vast
majority of electrons in the tube still flew by the screen to be
collected by the plate:
Tetrode amplifier circuit
“plate supply”
DC power
source
With a constant DC screen voltage, electron flow from
cathode to plate became almost exclusively dependent
upon grid voltage, meaning the plate voltage could vary
over a wide range with little effect on plate current. This
made for more stable gains in amplifier circuits, and better
linearity for more accurate reproduction of the input signal
waveform.
Despite the advantages realized by the addition of a screen,
there were some disadvantages as well. The most significant
disadvantage was related to something known as secondary
emission. When electrons from the cathode strike the plate
at high velocity, they can cause free electrons to be jarred
loose from atoms in the metal of the plate. These electrons,
knocked off the plate by the impact of the cathode
electrons, are said to be "secondarily emitted." In a triode
tube, secondary emission is not that great a problem, but in
a tetrode with a positively-charged screen grid in close
proximity, these secondary electrons will be attracted to the
screen rather than the plate from which they came, resulting
in a loss of plate current. Less plate current means less gain
for the amplifier, which is not good.
Two different strategies were developed to address this
problem of the tetrode tube: beam power tubes and
pentodes. Both solutions resulted in new tube designs with
approximately the same electrical characteristics.
Beam power tubes
In the beam power tube, the basic four-element structure of
the tetrode was maintained, but the grid and screen wires
were carefully arranged along with a pair of auxiliary plates
to create an interesting effect: focused beams or "sheets" of
electrons traveling from cathode to plate. These electron
beams formed a stationary "cloud" of electrons between the
screen and plate (called a "space charge") which acted to
repel secondary electrons emitted from the plate back to the
plate. A set of "beam-forming" plates, each connected to the
cathode, were added to help maintain proper electron beam
focus. Grid and screen wire coils were arranged in such a
way that each turn or wrap of the screen fell directly behind
a wrap of the grid, which placed the screen wires in the
"shadow" formed by the grid. This precise alignment
enabled the screen to still perform its shielding function with
minimal interference to the passage of electrons from
cathode to plate.
rid
gridwires— beam-forming plates
(cross-sectional view)
(2)
“space charge"
cathode
electron beams
iy
screen, wires
(cross-sectional view)
This resulted in lower screen current (and more plate
current!) than an ordinary tetrode tube, with little added
expense to the construction of the tube.
Beam power tetrodes were often distinguished from their
non-beam counterparts by a different schematic symbol,
showing the beam-forming plates:
The "Beam power" tetrode tube
plate
grid screen
C)
cathode
The pentode
Another strategy for addressing the problem of secondary
electrons being attracted by the screen was the addition of a
fifth wire element to the tube structure: a suppressor. These
five-element tubes were naturally called pentodes.
The pentode tube
plate
suppressor
screen
grid
is
cathode
The suppressor was another wire coil or mesh situated
between the screen and the plate, usually connected
directly to ground potential. In some pentode tube designs,
the suppressor was internally connected to the cathode so
as to minimize the number of connection pins having to
penetrate the tube envelope:
plate
suppressor internall
Piacoa to cathode)
screen
grid
a
cathode
The suppressor's job was to repel any secondarily emitted
electrons back to the plate: a structural equivalent of the
beam power tube's space charge. This, of course, increased
plate current and decreased screen current, resulting in
better gain and overall performance. In some instances it
allowed for greater operating plate voltage as well.
Combination tubes
Similar in thought to the idea of the integrated circuit, tube
designers tried integrating different tube functions into
single tube envelopes to reduce space requirements in more
modern tube-type electronic equipment. A common
combination seen within a single glass shell was two either
diodes or two triodes. The idea of fitting pairs of diodes
inside a single envelope makes a lot of sense in light of
power supply full-wave rectifier designs, always requiring
multiple diodes.
Of course, it would have been quite impossible to combine
thousands of tube elements into a single tube envelope the
way that thousands of transistors can be etched onto a
single piece of silicon, but engineers still did their best to
push the limits of tube miniaturization and consolidation.
Some of these tubes, whimsically called compactrons, held
four or more complete tube elements within a single
envelope.
Sometimes the functions of two different tubes could be
integrated into a single, combination tube in a way that
simply worked more elegantly than two tubes ever could. An
example of this was the pentagrid converter, more generally
called a heptode, used in some superheterodyne radio
designs. These tubes contained seven elements: 5 grids plus
a cathode and a plate. Two of the grids were normally
reserved for signal input, the other three relegated to
screening and suppression (performance-enhancing)
functions. Combining the superheterodyne functions of
oscillator and signal mixer together in one tube, the signal
coupling between these two stages was intrinsic. Rather
than having separate oscillator and mixer circuits, the
oscillator creating an AC voltage and the mixer "mixing" that
voltage with another signal, the pentagrid converter's
oscillator section created an electron stream that oscillated
in intensity which then directly passed through another grid
for "mixing" with another signal.
This same tube was sometimes used in a different way: by
applying a DC voltage to one of the control grids, the gain of
the tube could be changed for a signal impressed on the
other control grid. This was known as variab/e-mu operation,
because the "mu" (yu) of the tube (its amplification factor,
measured as a ratio of plate-to-cathode voltage change over
grid-to-cathode voltage change with a constant plate
current) could be altered at will by a DC control voltage
signal.
Enterprising electronics engineers also discovered ways to
exploit such multi-variable capabilities of "lesser" tubes such
as tetrodes and pentodes. One such way was the so-called
ultralinear audio power amplifier, invented by a pair of
engineers named Hafler and Keroes, utilizing a tetrode tube
in combination with a "tapped" output transformer to
provide substantial improvements in amplifier linearity
(decreases in distortion levels). Consider a "single-ended"
triode tube amplifier with an output transformer coupling
power to the speaker:
Speaker
input
voltage
If we substitute a tetrode for a triode in this circuit, we will
see improvements in circuit gain resulting from the
electrostatic shielding offered by the screen, preventing
unwanted feedback between the plate and control grid:
Standard
configuration
of tetrode tube
in a single-ended
audio amplifier
input
voltage
However, the tetrode's screen may be used for functions
other than merely shielding the grid from the plate. It can
also be used as another control element, like the grid itself.
If a "tap" is made on the transformer's primary winding, and
this tap connected to the screen, the screen will receive a
voltage that varies with the signal being amplified
(feedback). More specifically, the feedback signal is
proportional to the rate-of-change of magnetic flux in the
transformer core (d®/dt), thus improving the amplifier's
ability to reproduce the input signal waveform at the
speaker terminals and not just in the primary winding of the
transformer:
"Ultralinear” Speaker
configuration
of tetrode tube
in a single-ended
audio amplifier
input
voltage
This signal feedback results in significant improvements in
amplifier linearity (and consequently, distortion), so long as
precautions are taken against "overpowering" the screen
with too great a positive voltage with respect to the cathode.
As aconcept, the ultralinear (screen-feedback) design
demonstrates the flexibility of operation granted by multiple
grid-elements inside a single tube: a capability rarely
matched by semiconductor components.
Some tube designs combined multiple tube functions in a
most economic way: dual plates with a single cathode, the
currents for each of the plates controlled by separate sets of
control grids. Common examples of these tubes were triode-
heptode and triode-hexode tubes (a hexode tube is a tube
with four grids, one cathode, and one plate).
Other tube designs simply incorporated separate tube
structures inside a single glass envelope for greater
economy. Dual diode (rectifier) tubes were quite common, as
were dual triode tubes, especially when the power
dissipation of each tube was relatively low.
Dual triode tube
=.
The 12AX7 and 12AU7 models are common examples of
dual-triode tubes, both of low-power rating. The 12AX7 is
especially common as a preamplifier tube in electric guitar
amplifier circuits.
Tube parameters
For bipolar junction transistors, the fundamental measure of
amplification is the Beta ratio (8), defined as the ratio of
collector current to base current (I¢/Ip). Other transistor
characteristics such as junction resistance, which in some
amplifier circuits may impact performance as much as 8, are
quantified for the benefit of circuit analysis. Electron tubes
are no different, their performance characteristics having
been explored and quantified long ago by electrical
engineers.
Before we can speak meaningfully on these characteristics,
we must define several mathematical variables used for
expressing common voltage, current, and resistance
measurements as well as some of the more complex
quantities:
ut = amplification factor, pronounced "mu"
(unitless)
g,, = Mutual conductance, in siemens
E,, = plate-to-cathode voltage
E, = grid-to-cathode voltage
I, = plate current
I, = cathode current
E, = input signal voltage
r, = dynamic plate resistance, in ohms
A= delta, the Greek symbol for change
The two most basic measures of an amplifying tube's
characteristics are its amplification factor (u) and its mutual
conductance (g,,), also Known as transconductance.
Transconductance is defined here just the same as it is for
field-effect transistors, another category of voltage-
controlled devices. Here are the two equations defining each
of these performance characteristics:
AE,
[=
AE
with constant I, (plate current)
AI,
jo :
om ~~
with constant E, (plate voltage)
Another important, though more abstract, measure of tube
performance is its plate resistance. This is the measurement
of plate voltage change over plate current change for a
constant value of grid voltage. In other words, this is an
expression of how much the tube acts like a resistor for any
given amount of grid voltage, analogous to the operation of
a JFET in its ohmic mode:
AE,,
>= with constant E, (grid voltage)
p
The astute reader will notice that plate resistance may be
determined by dividing the amplification factor by the
transconductance:
bn 9 =?
AE, “AR,
... dividing u by g,,...
AE,,
AE,
7 :
P
Al,
AE,
AE, AE,
i= — ——
P AE, AL
ee AF,
3 P
These three performance measures of tubes are subject to
change from tube to tube (just as B ratios between two
"identical" bipolar transistors are never precisely the same)
and between different operating conditions. This variability
is due partly to the unavoidable nonlinearities of electron
tubes and partly due to how they are defined. Even
supposing the existence of a perfectly linear tube, it will be
impossible for all three of these measures to be constant
over the allowable ranges of operation. Consider a tube that
perfectly regulates current at any given amount of grid
voltage (like a bipolar transistor with an absolutely constant
B): that tube's plate resistance must vary with plate voltage,
because plate current will not change even though plate
voltage does.
Nevertheless, tubes were (and are) rated by these values at
given operating conditions, and may have their
characteristic curves published just like transistors.
lonization (gas-filled) tubes
So far, we've explored tubes which are totally "evacuated" of
all gas and vapor inside their glass envelopes, properly
known as vacuum tubes. With the addition of certain gases
or vapors, however, tubes take on significantly different
characteristics, and are able to fulfill certain special roles in
electronic circuits.
When a high enough voltage is applied across a distance
occupied by a gas or vapor, or when that gas or vapor is
heated sufficiently, the electrons of those gas molecules will
be stripped away from their respective nuclei, creating a
condition of /onization. Having freed the electrons from their
electrostatic bonds to the atoms' nuclei, they are free to
migrate in the form of a current, making the ionized gas a
relatively good conductor of electricity. In this state, the gas
is more properly referred to as a plasma.
lonized gas is not a perfect conductor. As such, the flow of
electrons through ionized gas will tend to dissipate energy
in the form of heat, thereby helping to keep the gas ina
state of ionization. The result of this is a tube that will begin
to conduct under certain conditions, then tend to stay ina
state of conduction until the applied voltage across the gas
and/or the heat-generating current drops to a minimum
level.
The astute observer will note that this is precisely the kind
of behavior exhibited by a class of semiconductor devices
called "thyristors," which tend to stay "on" once turned "on"
and tend to stay "off" once turned "off." Gas-filled tubes, it
can be said, manifest this same property of hysteresis.
Unlike their vacuum counterparts, ionization tubes were
often manufactured with no filament (heater) at all. These
were called co/d-cathode tubes, with the heated versions
designated as hot-cathode tubes. Whether or not the tube
contained a source of heat obviously impacted the
characteristics of a gas-filled tube, but not to the extent that
lack of heat would impact the performance of a hard-
vacuum tube.
The simplest type of ionization device is not necessarily a
tube at all; rather, it is constructed of two electrodes
separated by a gas-filled gap. Simply called a spark gap, the
gap between the electrodes may be occupied by ambient
air, other times a special gas, in which case the device must
have a sealed envelope of some kind.
Spark gap
a
——_@ @
enclosure (optional)
electrodes
A prime application for spark gaps is in overvoltage
protection. Engineered not to ionize, or "break down" (begin
conducting), with normal system voltage applied across the
electrodes, the spark gap's function is to conduct in the
event of a significant increase in voltage. Once conducting,
it will act as a heavy load, holding the system voltage down
through its large current draw and subsequent voltage drop
along conductors and other series impedances. In a properly
engineered system, the spark gap will stop conducting
("extinguish") when the system voltage decreases to a
normal level, well below the voltage required to initiate
conduction.
One major caveat of spark gaps is their significantly finite
life. The discharge generated by such a device can be quite
violent, and as such will tend to deteriorate the surfaces of
the electrodes through pitting and/or melting.
Spark gaps can be made to conduct on command by placing
a third electrode (usually with a sharp edge or point)
between the other two and applying a high voltage pulse
between that electrode and one of the other electrodes. The
pulse will create a small spark between the two electrodes,
ionizing part of the pathway between the two large
electrodes, and enabling conduction between them if the
applied voltage is high enough:
Triggered spark gap
main ;
(high voltage,
ee high current)
Load
spark gap
iy third electrode
.
triggering Wels source
(high voltage, low current)
Spark gaps of both the triggered and untriggered variety
can be built to handle huge amounts of current, some even
into the range of mega-amps (millions of amps)! Physical
size is the primary limiting factor to the amount of current a
Spark gap can Safely and reliably handle.
When the two main electrodes are placed in a sealed tube
filled with a special gas, a discharge tube is formed. The
most common type of discharge tube is the neon light, used
popularly as a source of colorful illumination, the color of the
light emitted being dependent on the type of gas filling the
tube.
Construction of neon lamps closely resembles that of spark
gaps, but the operational characteristics are quite different:
high voltage power supply (AC or DC)
NEON LAMP electrode
electrode
current through the tube
causes the neon gas to glow
glass tube
_ small neon
indicator lamp
Neon lamp schematic symbol
By controlling the spacing of the electrodes and the type of
gas in the tube, neon lights can be made to conduct without
drawing the excessive currents that spark gaps do. They still
exhibit hysteresis in that it takes a higher voltage to initiate
conduction than it does to make them "extinguish," and
their resistance is definitely nonlinear (the more voltage
applied across the tube, the more current, thus more heat,
thus lower resistance). Given this nonlinear tendency, the
voltage across a neon tube must not be allowed to exceed a
certain limit, lest the tube be damaged by excessive
temperatures.
This nonlinear tendency gives the neon tube an application
other than colorful illumination: it can act somewhat like a
zener diode, "clamping" the voltage across it by drawing
more and more current if the voltage decreases. When used
in this fashion, the tube is known as a g/ow tube, or voltage-
regulator tube, and was a popular means of voltage
regulation in the days of electron tube circuit design.
voltage across load
held relative constant
with variations of voltage
source and load resistance
oe
Please take note of the black dot found in the tube symbol
shown above (and in the neon lamp symbol shown before
that). That marker indicates the tube is gas-filled. It is a
common marker used in all gas-filled tube symbols.
R
load
One example of a glow tube designed for voltage regulation
was the VR-150, with a nominal regulating voltage of 150
volts. Its resistance throughout the allowable limits of
current could vary from 5 kQ to 30 kQ, a 6:1 span. Like zener
diode regulator circuits of today, glow tube regulators could
be coupled to amplifying tubes for better voltage regulation
and higher load current ranges.
If a regular triode was filled with gas instead of a hard
vacuum, it would manifest all the hysteresis and
nonlinearity of other gas tubes with one major advantage:
the amount of voltage applied between grid and cathode
would determine the minimum plate-to cathode voltage
necessary to initiate conduction. In essence, this tube was
the equivalent of the semiconductor SCR (Silicon-Controlled
Rectifier), and was called the thyratron.
high voltage
AC source
control
voltage
It should be noted that the schematic shown above is
greatly simplified for most purposes and thyratron tube
designs. Some thyratrons, for instance, required that the
grid voltage switch polarity between their "on" and "off"
states in order to properly work. Also, some thyratrons had
more than one grid!
Thyratrons found use in much the same way as SCR's find
use today: controlling rectified AC to large loads such as
motors. Thyratron tubes have been manufactured with
different types of gas fillings for different characteristics:
inert (chemically non-reactive) gas, hydrogen gas, and
mercury (vaporized into a gas form when activated).
Deuterium, a rare isotope of hydrogen, was used in some
special applications requiring the switching of high voltages.
Display tubes
In addition to performing tasks of amplification and
switching, tubes can be designed to serve as display
devices.
Perhaps the best-known display tube is the cathode ray
tube, or CRT. Originally invented as an instrument to study
the behavior of "cathode rays" (electrons) in a vacuum,
these tubes developed into instruments useful in detecting
voltage, then later as video projection devices with the
advent of television. The main difference between CRTs used
in oscilloscopes and CRTs used in televisions is that the
oscilloscope variety exclusively use electrostatic (plate)
deflection, while televisions use electromagnetic (coil)
deflection. Plates function much better than coils over a
wider range of signal frequencies, which is great for
oscilloscopes but irrelevant for televisions, since a television
electron beam sweeps vertically and horizontally at fixed
frequencies. Electromagnetic deflection coils are much
preferred in television CRT construction because they do not
have to penetrate the glass envelope of the tube, thus
decreasing the production costs and increasing tube
reliability.
An interesting "cousin" to the CRT is the Cat-Eye or Magic-
Eye indicator tube. Essentially, this tube is a voltage-
measuring device with a display resembling a glowing green
ring. Electrons emitted by the cathode of this tube impinge
on a fluorescent screen, causing the green-colored light to
be emitted. The shape of the glow produced by the
fluorescent screen varies as the amount of voltage applied
to a grid changes:
"Cat-Eye" indicator tube displays
large shadow slight shadow minimal shadow
The width of the shadow is directly determined by the
potential difference between the control electrode and the
fluorescent screen. The control electrode is a narrow rod
placed between the cathode and the fluorescent screen. If
that control electrode (rod) is significantly more negative
than the fluorescent screen, it will deflect some electrons
away from the that area of the screen. The area of the screen
"shadowed" by the control electrode will appear darker when
there is a significant voltage difference between the two.
When the control electrode and fluorescent screen are at
equal potential (zero voltage between them), the shadowing
effect will be minimal and the screen will be equally
illuminated.
The schematic symbol for a "cat-eye" tube looks something
like this:
"Cat-Eye" or "Magic-Eye"
indicator tube
fluorescent
plate screen
control
electrode
amplifie (\
grid
cathode
Here is a photograph of a cat-eye tube, showing the circular
display region as well as the glass envelope, socket (black,
at far end of tube), and some of its internal structure:
Normally, only the end of the tube would protrude from a
hole in an instrument panel, so the user could view the
circular, fluorescent screen.
In its simplest usage, a "cat-eye” tube could be operated
without the use of the amplifier grid. However, in order to
make it more sensitive, the amplifier grid /s used, and it is
used like this:
"Cat-Eye” indicator tube circuit
As the signal voltage increases, current through
the tube is choked off. This decreases the ae
between the plate and the fluorescent screen,
lessening the shadow effect (shadow narrows).
The cathode, amplifier grid, and plate act as a triode to
create large changes in plate-to-cathode voltage for small
changes in grid-to-cathode voltage. Because the control
electrode is internally connected to the plate, it is
electrically common to it and therefore possesses the same
amount of voltage with respect to the cathode that the plate
does. Thus, the large voltage changes induced on the plate
due to small voltage changes on the amplifier grid end up
causing large changes in the width of the shadow seen by
whoever is viewing the tube.
Control electrode negative with No voltage between control
respect to the fluorescent screen. electrode and flourescent screen.
This is caused by a positive This is caused by a negative
amplifier grid voltage (with amplifier grid voliage (with
respect to the cathode). respect to the cathode).
"Cat-eye" tubes were never accurate enough to be equipped
with a graduated scale as is the case with CRT's and
electromechanical meter movements, but they served well
as null detectors in bridge circuits, and as signal strength
indicators in radio tuning circuits. An unfortunate limitation
to the "cat-eye" tube as a null detector was the fact that it
was not directly capable of voltage indication in both
polarities.
Microwave tubes
For extremely high-frequency applications (above 1 GHz),
the interelectrode capacitances and transit-time delays of
standard electron tube construction become prohibitive.
However, there seems to be no end to the creative ways in
which tubes may be constructed, and several high-
frequency electron tube designs have been made to
overcome these challenges.
It was discovered in 1939 that a toroidal cavity made of
conductive material called a cavity resonator surrounding
an electron beam of oscillating intensity could extract power
from the beam without actually intercepting the beam itself.
The oscillating electric and magnetic fields associated with
the beam "echoed" inside the cavity, in a manner similar to
the sounds of traveling automobiles echoing in a roadside
canyon, allowing radio-frequency energy to be transferred
from the beam to a waveguide or coaxial cable connected to
the resonator with a coupling loop. The tube was called an
inductive output tube, or /OT:
The inductive output tube (IOT)
coaxial
output
cable
RF power
I output
— toroidal
cavity
nf
DC supply
Two of the researchers instrumental in the initial
development of the IOT, a pair of brothers named Sigurd and
Russell Varian, added a second cavity resonator for signal
input to the inductive output tube. This input resonator
acted as a pair of inductive grids to alternately "bunch" and
release packets of electrons down the drift space of the tube,
so the electron beam would be composed of electrons
traveling at different velocities. This "velocity modulation" of
the beam translated into the same sort of amplitude
variation at the output resonator, where energy was
extracted from the beam. The Varian brothers called their
invention a k/ystron.
The klystron tube
coaxial
signal output
input cable
RF power
l-— output
Beam —
contro] ——
TTT TEED TEE TUT PEED ETE EEE
electron beam
DC supply
Another invention of the Varian brothers was the reflex
klystron tube. In this tube, electrons emitted from the
heated cathode travel through the cavity grids toward the
repeller plate, then are repelled and returned back the way
they came (hence the name reflex) through the cavity grids.
Self-sustaining oscillations would develop in this tube, the
frequency of which could be changed by adjusting the
repeller voltage. Hence, this tube operated as a voltage-
controlled oscillator.
The reflex klystron tube
cavity repeller
grids
RF output
~~ cavity
control grid
cathode
As a voltage-controlled oscillator, reflex klystron tubes
served commonly as "local oscillators" for radar equipment
and microwave receivers:
Reflex klystron tube used as
a voltagé-controlled oscillator
Initially developed as low-power devices whose output
required further amplification for radio transmitter use,
reflex klystron design was refined to the point where the
tubes could serve as power devices in their own right. Reflex
klystrons have since been superseded by semiconductor
devices in the application of local oscillators, but
amplification klystrons continue to find use in high-power,
high-frequency radio transmitters and in scientific research
applications.
One microwave tube performs its task so well and so cost-
effectively that it continues to reign supreme in the
competitive realm of consumer electronics: the magnetron
tube. This device forms the heart of every microwave oven,
generating several hundred watts of microwave RF energy
used to heat food and beverages, and doing so under the
most grueling conditions for a tube: powered on and off at
random times and for random durations.
Magnetron tubes are representative of an entirely different
kind of tube than the IOT and klystron. Whereas the latter
tubes use a linear electron beam, the magnetron directs its
electron beam in a circular pattern by means of a strong
magnetic field:
The magnetron tube
cavit
resonators
RF output
Once again, cavity resonators are used as microwave-
frequency "tank circuits," extracting energy from the
passing electron beam inductively. Like all microwave-
frequency devices using a cavity resonator, at least one of
the resonator cavities is tapped with a coupling loop: a loop
of wire magnetically coupling the coaxial cable to the
resonant structure of the cavity, allowing RF power to be
directed out of the tube to a load. In the case of the
microwave oven, the output power is directed through a
waveguide to the food or drink to be heated, the water
molecules within acting as tiny load resistors, dissipating the
electrical energy in the form of heat.
The magnet required for magnetron operation is not shown
in the diagram. Magnetic flux runs perpendicular to the
plane of the circular electron path. In other words, from the
view of the tube shown in the diagram, you are looking
straight at one of the magnetic poles.
Tubes versus Semiconductors
Devoting a whole chapter in a modern electronics text to the
design and function of electron tubes may seem a bit
strange, seeing as how semiconductor technology has all
but obsoleted tubes in almost every application. However,
there is merit in exploring tubes not just for historical
purposes, but also for those niche applications that
necessitate the qualifying phrase "a/most every application"
in regard to semiconductor supremacy.
In some applications, electron tubes not only continue to see
practical use, but perform their respective tasks better than
any solid-state device yet invented. In some cases the
performance and reliability of electron tube technology is far
superior.
In the fields of high-power, high-speed circuit switching,
specialized tubes such as hydrogen thyratrons and krytrons
are able to switch far larger amounts of current, far faster
than any semiconductor device designed to date. The
thermal and temporal limits of semiconductor physics place
limitations on switching ability that tubes -- which do not
operate on the same principles -- are exempt from.
In high-power microwave transmitter applications, the
excellent thermal tolerance of tubes alone secures their
dominance over semiconductors. Electron conduction
through semiconducting materials is greatly impacted by
temperature. Electron conduction through a vacuum is not.
As a consequence, the practical thermal limits of
semiconductor devices are rather low compared to that of
tubes. Being able to operate tubes at far greater
temperatures than equivalent semiconductor devices allows
tubes to dissipate more thermal energy for a given amount
of dissipation area, which makes them smaller and lighter in
continuous high power applications.
Another decided advantage of tubes over semiconductor
components in high-power applications is their
rebuildability. When a large tube fails, it may be
disassembled and repaired at far lower cost than the
purchase price of a new tube. When a semiconductor
component fails, large or small, there is generally no means
of repair.
The following photograph shows the front panel of a 1960's
vintage 5 kW AM radio transmitter. One of two "Eimac"
brand power tubes can be seen in a recessed area, behind
the glass door. According to the station engineer who gave
the facility tour, the rebuild cost for such a tube is only
$800: quite inexpensive compared to the cost of a new tube,
and still quite reasonable in contrast to the price of a new,
comparable semiconductor component!
Tubes, being less complex in their manufacture than
semiconductor components, are potentially cheaper to
produce as well, although the huge volume of
semiconductor device production in the world greatly offsets
this theoretical advantage. Semiconductor manufacture is
quite complex, involving many dangerous chemical
substances and necessitating super-clean assembly
environments. Tubes are essentially nothing more than glass
and metal, with a vacuum seal. Physical tolerances are
"loose" enough to permit hand-assembly of vacuum tubes,
and the assembly work need not be done in a "clean room"
environment as is necessary for semiconductor manufacture.
One modern area where electron tubes enjoy supremacy
over semiconductor components is in the professional and
high-end audio amplifier markets, although this is partially
due to musical culture. Many professional guitar players, for
example, prefer tube amplifiers over transistor amplifiers
because of the specific distortion produced by tube circuits.
An electric guitar amplifier is designed to produce distortion
rather than avoid distortion as is the case with audio-
reproduction amplifiers (this is why an electric guitar sounds
so much different than an acoustical guitar), and the type of
distortion produced by an amplifier is as much a matter of
personal taste as it is technical measurement. Since rock
music in particular was born with guitarists playing tube-
amplifier equipment, there is a significant level of "tube
appeal" inherent to the genre itself, and this appeal shows
itself in the continuing demand for "tubed" guitar amplifiers
among rock guitarists.
As an illustration of the attitude among some guitarists,
consider the following quote taken from the technical
glossary page of a tube-amplifier website which will remain
nameless:
Solid State: A component that has been specifically
designed to make a guitar amplifier sound bad.
Compared to tubes, these devices can have a very long
lifespan, which guarantees that your amplifier will retain
its thin, lifeless, and buzzy sound for a long time to
come.
In the area of audio reproduction amplifiers (music studio
amplifiers and home entertainment amplifiers), it is best for
an amplifier to reproduce the musical signal with as //tt/e
distortion as possible. Paradoxically, in contrast to the guitar
amplifier market where distortion is a design goal, high-end
audio is another area where tube amplifiers enjoy continuing
consumer demand. Though one might suppose the
objective, technical requirement of low distortion would
eliminate any subjective bias on the part of audiophiles, one
would be very wrong. The market for high-end "tubed"
amplifier equipment is quite volatile, changing rapidly with
trends and fads, driven by highly subjective claims of
“magical" sound from audio system reviewers and
salespeople. As in the electric guitar world, there is no small
measure of cult-like devotion to tube amplifiers among some
quarters of the audiophile world. As an example of this
irrationality, consider the design of many ultra-high-end
amplifiers, with chassis built to display the working tubes
openly, even though this physical exposure of the tubes
obviously enhances the undesirable effect of microphonics
(changes in tube performance as a result of sound waves
vibrating the tube structure).
Having said this, though, there is a wealth of technical
literature contrasting tubes against semiconductors for
audio power amplifier use, especially in the area of
distortion analysis. More than a few competent electrical
engineers prefer tube amplifier designs over transistors, and
are able to produce experimental evidence in support of
their choice. The primary difficulty in quantifying audio
system performance is the uncertain response of human
hearing. A// amplifiers distort their input signal to some
degree, especially when overloaded, so the question is
which type of amplifier design distorts the least. However,
since human hearing is very nonlinear, people do not
interpret all types of acoustic distortion equally, and so
some amplifiers will sound "better" than others even if a
quantitative distortion analysis with electronic instruments
indicates similar distortion levels. To determine what type of
audio amplifier will distort a musical signal "the least," we
must regard the human ear and brain as part of the whole
acoustical system. Since no complete model yet exists for
human auditory response, objective assessment is difficult
at best. However, some research indicates that the
characteristic distortion of tube amplifier circuits (especially
when overloaded) is less objectionable than distortion
produced by transistors.
Tubes also possess the distinct advantage of low "drift" over
a wide range of operating conditions. Unlike semiconductor
components, whose barrier voltages, B ratios, bulk
resistances, and junction capacitances may change
substantially with changes in device temperature and/or
other operating conditions, the fundamental characteristics
of a vacuum tube remain nearly constant over a wide range
in operating conditions, because those characteristics are
determined primarily by the physical dimensions of the
tube's structural elements (cathode, grid(s), and plate)
rather than the interactions of subatomic particles ina
crystalline lattice.
This is one of the major reasons solid-state amplifier
designers typically engineer their circuits to maximize
power-efficiency even when it compromises distortion
performance, because a power-inefficient amplifier
dissipates a lot of energy in the form of waste heat, and
transistor characteristics tend to change substantially with
temperature. Temperature-induced "drift" makes it difficult
to stabilize "Q" points and other important performance-
related measures in an amplifier circuit. Unfortunately,
power efficiency and low distortion seem to be mutually
exclusive design goals.
For example, class A audio amplifier circuits typically exhibit
very low distortion levels, but are very wasteful of power,
meaning that it would be difficult to engineer a solid-state
class A amplifier of any substantial power rating due to the
consequent drift of transistor characteristics. Thus, most
solid-state audio amplifier designers choose class B circuit
configurations for greater efficiency, even though class B
designs are notorious for producing a type of distortion
known as crossover distortion. However, with tubes it is easy
to design a stable class A audio amplifier circuit because
tubes are not as adversely affected by the changes in
temperature experienced in a such a power-inefficient circuit
configuration.
Tube performance parameters, though, tend to "drift" more
than semiconductor devices when measured over long
periods of time (years). One major mechanism of tube
"aging" appears to be vacuum leaks: when air enters the
inside of a vacuum tube, its electrical characteristics
become irreversibly altered. This same phenomenon is a
major cause of tube mortality, or why tubes typically do not
last as long as their respective solid-state counterparts.
When tube vacuum is maintained at a high level, though,
excellent performance and life is possible. An example of
this is a klystron tube (used to produce the high-frequency
radio waves used in a radar system) that lasted for 240,000
hours of operation (cited by Robert S. Symons of Litton
Electron Devices Division in his informative paper, "Tubes:
Still vital after all these years," printed in the April 1998
issue of /EEE Spectrum magazine).
If nothing else, the tension between audiophiles over tubes
versus semiconductors has spurred a remarkable degree of
experimentation and technical innovation, serving as an
excellent resource for those wishing to educate themselves
on amplifier theory. Taking a wider view, the versatility of
electron tube technology (different physical configurations,
multiple control grids) hints at the potential for circuit
designs of far greater variety than is possible using
semiconductors. For this and other reasons, electron tubes
will never be "obsolete," but will continue to serve in niche
roles, and to foster innovation for those electronics
engineers, inventors, and hobbyists who are unwilling to let
their minds by stifled by convention.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] 4]\—
— 4 —
Appendix 1
ABOUT THIS BOOK
Purpose
They say that necessity is the mother of invention. At least
in the case of this book, that adage is true. As an industrial
electronics instructor, | was forced to use a sub-standard
textbook during my first year of teaching. My students were
daily frustrated with the many typographical errors and
obscure explanations in this book, having spent much time
at home struggling to comprehend the material within.
Worse yet were the many incorrect answers in the back of
the book to selected problems. Adding insult to injury was
the $100+ price.
Contacting the publisher proved to be an exercise in futility.
Even though the particular text | was using had been in
print and in popular use for a couple of years, they claimed
my complaint was the first they'd ever heard. My request to
review the draft for the next edition of their book was met
with disinterest on their part, and | resolved to find an
alternative text.
Finding a Suitable alternative was more difficult than | had
imagined. Sure, there were plenty of texts in print, but the
really good books seemed a bit too heavy on the math and
the less intimidating books omitted a lot of information | felt
was important. Some of the best books were out of print, and
those that were still being printed were quite expensive.
It was out of frustration that | compiled Lessons in Electric
Circuits from notes and ideas | had been collecting for years.
My primary goal was to put readable, high-quality
information into the hands of my students, but a secondary
goal was to make the book as affordable as possible. Over
the years, | had experienced the benefit of receiving free
instruction and encouragement in my pursuit of learning
electronics from many people, including several teachers of
mine in elementary and high school. Their selfless
assistance played a key role in my own studies, paving the
way for a rewarding career and fascinating hobby. If only |
could extend the gift of their help by giving to other people
what they gavetome...
So, | decided to make the book freely available. More than
that, | decided to make it "open," following the same
development model used in the making of free software
(most notably the various UNIX utilities released by the Free
Software Foundation, and the Linux operating system,
whose fame Is growing even as | write). The goal was to
copyright the text -- so as to protect my authorship -- but
expressly allow anyone to distribute and/or modify the text
to suit their own needs with a minimum of legal
encumbrance. This willful and formal revoking of standard
distribution limitations under copyright is whimsically
termed copyleft. Anyone can "copyleft" their creative work
simply by appending a notice to that effect on their work,
but several Licenses already exist, covering the fine legal
points in great detail.
The first such License | applied to my work was the GPL --
General Public License -- of the Free Software Foundation
(GNU). The GPL, however, is intended to copyleft works of
computer software, and although its introductory language
is broad enough to cover works of text, its wording is not as
clear as it could be for that application. When other, less
specific copyleft Licenses began appearing within the free
software community, | chose one of them (the Design
Science License, or DSL) as the official notice for my project.
In "copylefting" this text, | guaranteed that no instructor
would be limited by a text insufficient for their needs, as |
had been with error-ridden textbooks from major publishers.
I'm sure this book in its initial form will not satisfy everyone,
but anyone has the freedom to change it, leveraging my
efforts to suit variant and individual requirements. For the
beginning student of electronics, learn what you can from
this book, editing it as you feel necessary if you come across
a useful piece of information. Then, if you pass it on to
someone else, you will be giving them something better
than what you received. For the instructor or electronics
professional, feel free to use this as a reference manual,
adding or editing to your heart's content. The only "catch" is
this: if you plan to distribute your modified version of this
text, you must give credit where credit is due (to me, the
Original author, and anyone else whose modifications are
contained in your version), and you must ensure that
whoever you give the text to is aware of their freedom to
similarly share and edit the text. The next chapter covers
this process in more detail.
It must be mentioned that although | strive to maintain
technical accuracy in all of this book's content, the subject
matter is broad and harbors many potential dangers.
Electricity maims and kills without provocation, and
deserves the utmost respect. | strongly encourage
experimentation on the part of the reader, but only with
circuits powered by small batteries where there is no risk of
electric shock, fire, explosion, etc. High-power electric
circuits should be left to the care of trained professionals!
The Design Science License clearly states that neither | nor
any contributors to this book bear any liability for what is
done with its contents.
The use of SPICE
One of the best ways to learn how things work is to follow
the inductive approach: to observe specific instances of
things working and derive general conclusions from those
observations. In science education, labwork is the
traditionally accepted venue for this type of learning,
although in many cases labs are designed by educators to
reinforce principles previously learned through lecture or
textbook reading, rather than to allow the student to learn
on their own through a truly exploratory process.
Having taught myself most of the electronics that | know, |
appreciate the sense of frustration students may have in
teaching themselves from books. Although electronic
components are typically inexpensive, not everyone has the
means or opportunity to set up a laboratory in their own
homes, and when things go wrong there's no one to ask for
help. Most textbooks seem to approach the task of education
from a deductive perspective: tell the student how things
are supposed to work, then apply those principles to specific
instances that the student may or may not be able to
explore by themselves. The inductive approach, as useful as
it is, is hard to find in the pages of a book.
However, textbooks don't have to be this way. | discovered
this when | started to learn a computer program called
SPICE. It is a text-based piece of software intended to model
circuits and provide analyses of voltage, current, frequency,
etc. Although nothing is quite as good as building real
circuits to gain knowledge in electronics, computer
simulation is an excellent alternative. In learning how to use
this powerful tool, | made a discovery: SPICE could be used
within a textbook to present circuit simulations to allow
students to "observe" the phenomena for themselves. This
way, the readers could learn the concepts inductively (by
interpreting SPICE's output) as well as deductively (by
interpreting my explanations). Furthermore, in seeing SPICE
used over and over again, they should be able to
understand how to use it themselves, providing a perfectly
safe means of experimentation on their own computers with
circuit simulations of their own design.
Another advantage to including computer analyses in a
textbook is the empirical verification it adds to the concepts
presented. Without demonstrations, the reader is left to take
the author's statements on faith, trusting that what has
been written is indeed accurate. The problem with faith, of
course, is that it is only as good as the authority in which it
is placed and the accuracy of interpretation through which it
is understood. Authors, like all human beings, are liable to
err and/or communicate poorly. With demonstrations,
however, the reader can immediately see for themselves
that what the author describes is indeed true.
Demonstrations also serve to clarify the meaning of the text
with concrete examples.
SPICE is introduced early in volume | (DC) of this book
series, and hopefully in a gentle enough way that it doesn't
create confusion. For those wishing to learn more, a chapter
in the Reference volume (volume V) contains an overview of
SPICE with many example circuits. There may be more flashy
(graphic) circuit simulation programs in existence, but SPICE
is free, a virtue complementing the charitable philosophy of
this book very nicely.
Acknowledgements
First, | wish to thank my wife, whose patience during those
many and long evenings (and weekends!) of typing has
been extraordinary.
| also wish to thank those whose open-source software
development efforts have made this endeavor all the more
affordable and pleasurable. The following is a list of various
free computer software used to make this book, and the
respective programmers:
e GNU/Linux Operating System -- Linus Torvalds, Richard
Stallman, and a host of others too numerous to mention.
e Vim text editor -- Bram Moolenaar and others.
Xcircuit drafting program -- Tim Edwards.
SPICE circuit simulation program -- too many
contributors to mention.
e T-X text processing system -- Donald Knuth and others.
e Texinfo document formatting system -- Free Software
Foundation.
¢ LATEX document formatting system -- Leslie Lamport and
others.
e Gimp image manipulation program -- too many
contributors to mention.
Appreciation is also extended to Robert L. Boylestad, whose
first edition of Introductory Circuit Analysis taught me more
about electric circuits than any other book. Other important
texts in my electronics studies include the 1939 edition of
The "Radio" Handbook, Bernard Grob's second edition of
Introduction to Electronics I, and Forrest Mims' original
Engineer's Notebook.
Thanks to the staff of the Bellingham Antique Radio
Museum, who were generous enough to let me terrorize their
establishment with my camera and flash unit.
| wish to specifically thank Jeffrey Elkner and all those at
Yorktown High School for being willing to host my book as
part of their Open Book Project, and to make the first effort
in contributing to its form and content. Thanks also to David
Sweet (website: [*]) and Ben Crowell (website: [*]) for
providing encouragement, constructive criticism, and a
wider audience for the online version of this book.
Thanks to Michael Stutz for drafting his Design Science
License, and to Richard Stallman for pioneering the concept
of copyleft.
Last but certainly not least, many thanks to my parents and
those teachers of mine who saw in me a desire to learn
about electricity, and who kindled that flame into a passion
for discovery and intellectual adventure. | honor you by
helping others as you have helped me.
Tony Kuphaldt, July 2001
"A candle loses nothing of its light when lighting
another"
Kahlil Gibran
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—||4]l_—
—| | +]
Appendix 2
CONTRIBUTOR LIST
How to contribute to this book
As a copylefted work, this book is open to revision and expansion by
any interested parties. The only "catch" is that credit must be given
where credit is due. This /s a copyrighted work: it is notin the public
domain!
If you wish to cite portions of this book in a work of your own, you
must follow the same guidelines as for any other copyrighted work.
Here is a Sample from the Design Science License:
The Work is copyright the Author. All rights to the Work are reserved
by the Author, except as specifically described below. This License
describes the terms and conditions under which the Author permits you
to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright Law.
Your right to operate, perform, read or otherwise interpret and/or
execute the Work is unrestricted; however, you do so at your own risk,
because the Work comes WITHOUT ANY WARRANTY -- see Section 7 ("NO
WARRANTY") below.
If you wish to modify this book in any way, you must document the
nature of those modifications in the "Credits" section along with your
name, and ideally, information concerning how you may be
contacted. Again, the Design Science License:
Permission is granted to modify or sample from a copy of the Work,
producing a derivative work, and to distribute the derivative work
under the terms described in the section for distribution above,
provided that the following terms are met:
(a) The new, derivative work is published under the terms of this
License.
(b) The derivative work is given a new name, so that its name or
title can not be confused with the Work, or with a version of
the Work, in any way.
(c) Appropriate authorship credit is given: for the differences
between the Work and the new derivative work, authorship is
attributed to you, while the material sampled or used from
the Work remains attributed to the original Author; appropriate
notice must be included with the new work indicating the nature
and the dates of any modifications of the Work made by you.
Given the complexities and security issues surrounding the
maintenance of files comprising this book, it is recommended that
you submit any revisions or expansions to the original author (Tony R.
Kuphaldt). You are, of course, welcome to modify this book directly by
editing your own personal copy, but we would all stand to benefit
from your contributions if your ideas were incorporated into the
online “master copy” where all the world can see it.
Credits
All entries arranged in alphabetical order of surname. Major
contributions are listed by individual name with some detail on the
nature of the contribution(s), date, contact info, etc. Minor
contributions (typo corrections, etc.) are listed by name only for
reasons of brevity. Please understand that when | classify a
contribution as “minor,” it is in no way inferior to the effort or value of
a “major” contribution, just smaller in the sense of less text changed.
Any and all contributions are gratefully accepted. | am indebted to all
those who have given freely of their own knowledge, time, and
resources to make this a better book!
Tony R. Kuphaldt
« Date(s) of contribution(s): 1996 to present
¢ Nature of contribution: Original author.
¢ Contact at: Liec0@lycos.com
Dennis Crunkilton
Date(s) of contribution(s): July 2004 to present
Nature of contribution: Mini table of contents, all chapters
except appendicies; html, latex, ps, pdf; See Devel/tutorial.html;
01/2006.
¢ Nature of contribution: Completed Ch4 Bipolar junction
transistors, CH7 Thyristors; Ch9 Practical anlog ckts, a few
additions; Ch8 Opamps, minor; 04/2009
¢ Contact at: dcrunkilton(at)att(dot)net
Bill Marsden
« Date(s) of contribution(s): May 2003 - present
¢ Nature of contribution: Update to LED subsection, Diodes Ch 3
, Nov 2003.
¢ Nature of contribution: Original author: “ElectroStatic
Discharge” Section, Chapter 9, May 2008.
Nature of contribution: Chapter 3, LED's update, photodiode
update, Feburary 2009.
Nature of contribution: Chapter 11, Section author: "Pulse
Width Modulation", Feburary 2010.
Nature of contribution: Chapter 9, Section author: Derek
Payne "Power Supply Introduction", "Linear power supplies",
Feburary 2020.
Contact at: bill _marsden2(at) hotmail (dot) com
John Anhalt
Date(s) of contribution(s): June 2011
Nature of contribution: Updated Si SP3 electron hybridization,
Ch 2
Contact at: jpa@anhalt.org
Derek Payne
Date(s) of contribution(s):February 2020
Nature of contribution: Chapter 9, Section author: Derek
Payne "Power Supply Introduction", "Linear power supplies",
Feburary 2020.
Contact at:
Typo corrections and other “minor” contributions
Line-allaboutcircuits.com (June 2005) Typographical error
correction in Volumes 1,2,3,5, various chapters ,(:S/visa-
versa/vice versa/).
Colin Creitz (May 2007) Chapters: several, s/it's/its.
Dennis Crunkilton (October 2005) Typographical capitlization
correction to sectiontitles, chapter 9.
Jeff DeFreitas (March 2006)Improve appearance: replace “/" and "/"
Chapters: Al, A2.
Paul Stokes, Program Chair, Computer and Electronics Engineering
Technology, ITT Technical Institute, Houston, Tx (October 2004)
Change (1001, = - 819 + 710 = -149) to (1001, aan - 819 + lio = -lio9),
CH2, Binary Arithmetic
Paul Stokes, Program Chair Computer and Electronics Engineering
Technology, ITT Technical Institute, Houston, Tx (October 2004)
Near "Fold up the corners" change Out=B'C' to Qut=B'D', 14118.eps
Same change, Karnaugh Mapping
The students of Bellingham Technical College's Instrumentation
program,
Roger Hollingsworth (May 2003) Suggested a way to make the PLC
motor control system fail-safe.
Jan-Willem Rensman (May 2002) Suggested the inclusion of Schmitt
triggers and gate hysteresis to the "Logic Gates" chapter.
Don Stalkowski (June 2002) Technical help with PostScript-to-PDF
file format conversion.
Joseph Teichman (June 2002) Suggestion and technical help
regarding use of PNG images instead of JPEG.
Unregistered@allaboutcircuits.com (November 2007) “Boolean
algebra”, images 14019.pes 14021.eps output of gates incorrect
S/0/A S/1/A .
Dan Simon (February 2008) “Numeration Systems”, After BINARY TO
OCTAL CONVERSION, position of decimal point ---.
Timothy Kingman (March 2008) Changed default roman font to
newcent.
Imranullah Syed (March 2008) Suggested centering of uncaptioned
schematics.
Chris01720@allaboutcircuits.com (March 2008) Ch 15, Inaccuracy
involving CD-ROM production.
studiot@allaboutcircuits.com (March 2008) Ch 15, s/disk/disc/ in
CDROM .
Keith@allaboutcircuits.com (April 2008) Ch 12, s/laralel-
out/parallel-out/ .
Ken Braswell (May 2008) Ch 3, s/drips/drops/.
Guest@allaboutcircuits.com (Oct 2008) Ch 2, s/are in close/are
close/.
Radoslav@allaboutcircuits.com (Oct 2008) Ch 8, s/that 1 mA of/that
6 mA/.
Scanman@allaboutcircuits.com (Dec 2008) Ch 2, s/shells are
hold/shells hold/.
dgeorge@allaboutcircuits.com (Dec 2008) Ch 7, image 03320.png,
Swapped anode and anode gate. left diagram.
Unregistered Guest@allaboutcircuits.com (Feb 2009) Ch 2 s/than
FET's/than JFET's.
Unregistered Guest@allaboutcircuits.com (March 2009) Ch 8,
13061.png, change formula for inverting gain to include "-" .
dezurtrat@allaboutcircuits.com (March 2009) Ch 3, 03443.png, s/p-
p/peak.
Bill Marsden@allaboutcircuits.com (April 2009) Ch 3, s/I would/It
would/
Peter O@allaboutcircuits.com (April 2009) Ch 1, closing
parenthesis, above replaced with reference to figure.
Nanophotonics@allaboutcircuits.com (April 2009) Ch 9, image
53009.jpg s/courtisy/courtesy.
Bill Marsden@allaboutcircuits.com (April 2009) Ch 8, images
2001.png, 2002.png appearance.
D Crunkilton (April 2009) Ch 4, images 23006.png, 23007.png
updated.
Unregistered Guest@allaboutcircuits.com (June 2009) Ch 7, s/SCR
schematic symboLl/TRIAC schematic symbol .
Peter O'Dette (June 2009) Ch 1, s/is 1 watts/is 1 Watt , s/10
watt/10 Watts , s/ watt/ Watt
Unregistered Guest@allaboutcircuits.com (June 2009) Ch 3,
s/being/begin , near "voltage at which they" . s/is/in near "The
diodes must be".
regrehan@allaboutcircuits.com (June 2009) Ch 4, s/r1 12 1/rl1 1 2
1k in common-emitter amplifier SPICE list.
Unregistered Guest@allaboutcircuits.com (July 2009) Ch 3, s/Note
polarity change on coil changed/Note polarity change on coil.
Unregistered Guest@allaboutcircuits.com (August 2009) Ch 4, Swap
PNP & NPN at (b) & (c), caption of 03075.png
Unregistered Guest@allaboutcircuits.com (August 2009) equation
typos 03077.png 03479.png
Peter O'Dette@allaboutcircuits.com (August 2009) Ch 2, Numerous
changes, and 03409.png
Bill Marsden@allaboutcircuits.com (November 2009) Ch 4, Beta
formula, "Transistor atings and Packages".
Unregistered Guest@allaboutcircuits.com (November 2009) Ch 3,
Image 03288.eps changed polarized capacitor to non-polarized.
Unregistered Guest@allaboutcircuits.com (November 2009) Ch 4
s/hasre/share/ s/common=emitter/common-emitter/
Uisge@allaboutcircuits.com (November 2009) Ch 3, s/once every
half-cycle/one half of every full cycle/ , s/much/half/ .
Unregistered Guest@allaboutcircuits.com (November 2009) Ch 4 s/To
maintaining/To maintain
Unregistered Guest@allaboutcircuits.com (November 2009) Ch 3
s/[ model] /[ modeLlname] / .
gareththegeek@allaboutcircuits.com (November 2009) Ch 2 numerous
typos, omissions
Dcrunkilton@allaboutcircuits.com (November 2009) Ch 2 minor chages
to text and image 03392.eps
waynerr@allaboutcircuits.com (December 2009) Ch 4 equations 4 and
7 of image 03488.eps .
jkenny@allaboutcircuits.com (January 2010) Ch 7 s/will will/will/
BHijazi@allaboutcircuits.com (February 2010) Ch 1, Clarification
of text between images 03378.png and 03379.png
SgtWookiei@allaboutcircuits.com (March 2010) Ch 4, image
03375.png, flipped pnp and battery .
Bill_Marsden@allaboutcircuits.com (March 2010) Ch 9, Changes to
ESD section.
SgtWookiei@allaboutcircuits.com (April 2010) Ch 4, image
03078.png, added resistors.
silv3rm00n@allaboutcircuits.com (April 2010) Ch 4, typo in SPICE
listing near image 20004.png.
optomistl@allaboutcircuits.com (July 2010) Ch 2, typo
s/campared/compared/.
Bill_Marsden@allaboutcircuits.com (July 2010) Ch 11, change [I] to
italic tags in dcdrive.sml
Unregistered guest @allaboutcircuits.com (August 2010) Ch 2, s/The
bopolar transistor/The bipolar junction transistor/ .
Unregistered guest @allaboutcircuits.com (August 2010) Ch 4,
D Crunkilton (Sept 2010) Ch 2 s/minuscule/minuscule; Ch 3 ,4 ,5,
7, S/useable/usable.
beenthere@allaboutcircuits.com (Oct 2010) Ch 3, AC line powered
LED material removed.
mulebones@allaboutcircuits.com (Feb 2011) Ch 3, s/5 Vptp/10 Vptp/
Skfir@allaboutcircuits.com (Feb 2011) Ch 1, s/ ource/source/
Skfir@allaboutcircuits.com (Feb 2011) Ch 2, 4, A3 s/the the/the/
Skfir@allaboutcircuits.com (Feb 2011) Ch 2, s/insulator
insulator/insulator/
Skfir@allaboutcircuits.com (Feb 2011) Ch 3, s/a approximately/at
approximately/ , s/frequency my/frequency may/ , S/application
a/appliation is as/ , s/been produce/been produced/; Ch4
s/approximage/approximate/ s/resistor is a short/capacitor is a
short/ ; s/Iis it/Is it/ s/The the/The/ s/the these/these/,
s/distortion distortion/distortion/
D. Crunkilton (June 2011) hi.latex, header file; updated link to
openbookproject.net
SamAtOz@allaboutcircuits.com (May 2012) Ch 2 s/occurr/occur
s/repells/repels/ , s/is increases/increases , at (c) changed to
full reference, .
john207@allaboutcircuits.com (May 2012) Ch 4, various
Bill_Marsden@allaboutcircuits.com (May 2012) Ch 4, Clarification
of text near: Bipolar transistors are contructed. . ..
kintzlr@allaboutcircuits.com (January 2013) Ch 4,image 03495.eps
corrected. Added Ohm symbol to 0.26, above 2600 Ohm.
sby64@allaboutcircuits.com (January 2013) Ch 4, caption image
03495.png s/resistance Vth/resistance Rth.
keithostertag@allaboutcircuits.com (January 2013) Ch 4, caption
image 03495.png s/resistance Vth/resistance Rth.
Eugene Smirnoff (January 2013) Ch 2, near "A SQUID'" s/is an/is a/
s/Superconduction/Superconducting.
mrchen@allaboutcircuits.com (February 2014) Ch 3, s/inversely
proportional/iverted/ in Common Emitter section .
slidercrank@allaboutcircuits.com (February 2014) Ch 1,symbol for
neper s/n/Np/. Ch 2, s/Dimitri/Dmitri/, s/always"risky"/always
"risky"/
triffid_hunter@allaboutcircuits.com (February 2014) Ch 3,
s/common-base/common-emitter , caption and image 03502.eps in
Cascode section .
mnada@allaboutcircuits.com (February 2014) Ch 4, s/RB/RE in table
near image 03488.png and in image 03488.png
adam555@allaboutcircuits.com (February 2014) Ch 4, change b to
Beta in image 03488.png; above 13074.png s/base resistor/emitter
resistor. After internal resistance: s/RE/REE. s+(Beta)REE/IE+
(Beta) REE+.
LvW@allaboutcircuits.com (February 2014) Ch 4, change 22 instance
of REE to ree in text; same for images: 03489.eps 03494.eps
03495.eps 03497.eps 13062.eps
georacer@allaboutcircuits.com (February 2014) Ch 4, insert
bigspcace tag above Bypass Capacitor for R.
peek65408@allaboutcircuits.com (February 2014) Ch 4, s/Small
emitter base current controls large collector emitter current
flowing against emitter arrow/Small Base-Emitter current controls
large Collector-Emitter current flowing against emitter arrow/.
image:13048.eps changed Euler's constant to Euler's Number.
¢ tshuck@allaboutcircuits.com (February 2014) Ch 3, insert missing
image 03300.png into diode.sml.
¢ Roman Kaluzniacki (October 2014) Ch 2, s/principle/principal with
respect to "principal quantum number".
¢ va-Ssawyer (August 2015) Ch 4, s/@.26mV/26m/ s/Re'/rEE/.
e DC (Feb 2020) Ch 4, broken reference s/bjt.tbl/bjt6. tbl
S/>023014.png/23014.png.
e DC (Feb 2020) Ch 2, broken reference Figure 03302.png (c)
S/0a3462/03462/ . s/>0396.png/>03296.png/.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt,
under the terms and conditions of the Design Science License.
—| | +4]
—/ | 4]
Appendix 3
DESIGN SCIENCE LICENSE
Copyright © 1999-2000 Michael Stutz stutz@dsl.org
Verbatim copying of this document is permitted, in any
medium.
0. Preamble
Copyright law gives certain exclusive rights to the author of
a work, including the rights to copy, modify and distribute
the work (the "reproductive," "adaptative," and
"distribution" rights).
The idea of "copyleft" is to willfully revoke the exclusivity of
those rights under certain terms and conditions, so that
anyone can copy and distribute the work or properly
attributed derivative works, while all copies remain under
the same terms and conditions as the original.
The intent of this license is to be a general "copyleft" that
can be applied to any kind of work that has protection under
copyright. This license states those certain conditions under
which a work published under its terms may be copied,
distributed, and modified.
Whereas "design science" is a strategy for the development
of artifacts as a way to reform the environment (not people)
and subsequently improve the universal standard of living,
this Design Science License was written and deployed as a
strategy for promoting the progress of science and art
through reform of the environment.
1. Definitions
"License" shall mean this Design Science License. The
License applies to any work which contains a notice placed
by the work's copyright holder stating that it is published
under the terms of this Design Science License.
"Work" shall mean such an aforementioned work. The
License also applies to the output of the Work, only if said
output constitutes a "derivative work" of the licensed Work
as defined by copyright law.
“Object Form" shall mean an executable or performable form
of the Work, being an embodiment of the Work in some
tangible medium.
"Source Data" shall mean the origin of the Object Form,
being the entire, machine-readable, preferred form of the
Work for copying and for human modification (usually the
language, encoding or format in which composed or
recorded by the Author); plus any accompanying files,
scripts or other data necessary for installation, configuration
or compilation of the Work.
(Examples of "Source Data" include, but are not limited to,
the following: if the Work is an image file composed and
edited in 'PNG' format, then the original PNG source file is
the Source Data; if the Work is an MPEG 1.0 layer 3 digital
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"Author" shall mean the copyright holder(s) of the Work.
The individual licensees are referred to as "you."
2. Rights and copyright
The Work is copyright the Author. All rights to the Work are
reserved by the Author, except as specifically described
below. This License describes the terms and conditions
under which the Author permits you to copy, distribute and
modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright law.
Your right to operate, perform, read or otherwise interpret
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Permission is granted to modify or sample from a copy of the
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is attributed to you, while the material sampled or used from
the Work remains attributed to the original Author;
appropriate notice must be included with the new work
indicating the nature and the dates of any modifications of
the Work made by you.
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Copying, distributing or modifying the Work (including but
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[ $Id: dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]
— + —
Lessons In Electric Circuits
Copyright (C) 2000-2020, Tony R.
Kuphaldt
See the Design Science License (Appendix 3)
for details regarding copying and distribution
Revised October 18, 2006
Master Index
Chapter 1: BASIC CONCEPTS OF ELECTRICITY
Chapter 2: OHM'S LAW
Chapter 3: ELECTRICAL SAFETY
Chapter 4: SCIENTIFIC NOTATION AND METRIC PREFIXES
Chapter 5: SERIES AND PARALLEL CIRCUITS
Chapter 6: DIVIDER CIRCUITS AND KIRCHHOFF'S LAWS
Chapter 7: SERIES-PARALLEL COMBINATION CIRCUITS
Chapter 8: DC METERING CIRCUITS
Chapter 9: ELECTRICAL INSTRUMENTATION SIGNALS
Chapter 10: DC NETWORK ANALYSIS
Chapter 11: BATTERIES AND POWER SYSTEMS
Chapter 12: THE PHYSICS OF CONDUCTORS AND
INSULATORS
Chapter 13: CAPACITORS
Chapter 14: MAGNETISM AND ELECTROMAGNETISM
Chapter 15: INDUCTORS
Chapter 16: RC AND L/R TIME CONSTANTS
Appendix 1: ABOUT THIS BOOK
Appendix 2: CONTRIBUTOR LIST
Appendix 3: DESIGN SCIENCE LICENSE
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Back to Master Index
—| | +4/l—
Lessons In Electric Circuits
-- Volume |
Chapter 1
BASIC CONCEPTS OF
ELECTRICITY
Static electricity
Conductors, insulators, and electron flow
Electric circuits
Voltage and current
Resistance
Voltage and current in a practical circuit
Conventional versus electron flow
Contributors
Static electricity
It was discovered centuries ago that certain types of
materials would mysteriously attract one another after being
rubbed together. For example: after rubbing a piece of silk
against a piece of glass, the silk and glass would tend to stick
together. Indeed, there was an attractive force that could be
demonstrated even when the two materials were separated:
—P> —+_
attraction
Glass rod Silk cloth
Glass and silk aren't the only materials known to behave like
this. Anyone who has ever brushed up against a latex balloon
only to find that it tries to stick to them has experienced this
Same phenomenon. Paraffin wax and wool cloth are another
pair of materials early experimenters recognized as
manifesting attractive forces after being rubbed together:
—— ——
attraction
Wax
Wool cloth
This phenomenon became even more interesting when it was
discovered that identical materials, after having been rubbed
with their respective cloths, always repelled each other:
a See oll
repulsion
Glass rod Glass rod
———— ——
repulsion
Wax Wax
It was alSo noted that when a piece of glass rubbed with silk
was exposed to a piece of wax rubbed with wool, the two
materials would attract one another:
— <———
attraction
Wax
Glass rod
Furthermore, it was found that any material demonstrating
properties of attraction or repulsion after being rubbed could
be classed into one of two distinct categories: attracted to
glass and repelled by wax, or repelled by glass and attracted
to wax. It was either one or the other: there were no materials
found that would be attracted to or repelled by both glass
and wax, or that reacted to one without reacting to the other.
More attention was directed toward the pieces of cloth used
to do the rubbing. It was discovered that after rubbing two
pieces of glass with two pieces of silk cloth, not only did the
glass pieces repel each other, but so did the cloths. The same
phenomenon held for the pieces of wool used to rub the wax:
—_ —_—P
repulsion
Silk cloth Silk cloth
at —
repulsion
Wool cloth Wool cloth
Now, this was really strange to witness. After all, none of
these objects were visibly altered by the rubbing, yet they
definitely behaved differently than before they were rubbed.
Whatever change took place to make these materials attract
or repel one another was invisible.
Some experimenters speculated that invisible "fluids" were
being transferred from one object to another during the
process of rubbing, and that these "fluids" were able to effect
a physical force over a distance. Charles Dufay was one of
the early experimenters who demonstrated that there were
definitely two different types of changes wrought by rubbing
certain pairs of objects together. The fact that there was more
than one type of change manifested in these materials was
evident by the fact that there were two types of forces
produced: attraction and repulsion. The hypothetical fluid
transfer became known as a charge.
One pioneering researcher, Benjamin Franklin, came to the
conclusion that there was only one fluid exchanged between
rubbed objects, and that the two different "charges" were
nothing more than either an excess or a deficiency of that
one fluid. After experimenting with wax and wool, Franklin
suggested that the coarse wool removed some of this
invisible fluid from the smooth wax, causing an excess of
fluid on the wool and a deficiency of fluid on the wax. The
resulting disparity in fluid content between the wool and wax
would then cause an attractive force, as the fluid tried to
regain its former balance between the two materials.
Postulating the existence of a single "fluid" that was either
gained or lost through rubbing accounted best for the
observed behavior: that all these materials fell neatly into
one of two categories when rubbed, and most importantly,
that the two active materials rubbed against each other
always fell into opposing categories as evidenced by their
invariable attraction to one another. In other words, there
was never a time where two materials rubbed against each
other both became either positive or negative.
Following Franklin's speculation of the wool rubbing
something off of the wax, the type of charge that was
associated with rubbed wax became known as "negative"
(because it was supposed to have a deficiency of fluid) while
the type of charge associated with the rubbing wool became
known as "positive" (because it was supposed to have an
excess of fluid). Little did he know that his innocent
conjecture would cause much confusion for students of
electricity in the future!
Precise measurements of electrical charge were carried out
by the French physicist Charles Coulomb in the 1780's using
a device called a torsional balance measuring the force
generated between two electrically charged objects. The
results of Coulomb's work led to the development of a unit of
electrical charge named in his honor, the coulomb. |f two
"point" objects (hypothetical objects having no appreciable
surface area) were equally charged to a measure of 1
coulomb, and placed 1 meter (approximately 1 yard) apart,
they would generate a force of about 9 billion newtons
(approximately 2 billion pounds), either attracting or
repelling depending on the types of charges involved. The
operational definition of a coulomb as the unit of electrical
charge (in terms of force generated between point charges)
was found to be equal to an excess or deficiency of about
6,250,000,000,000,000,000 electrons. Or, stated in reverse
terms, one electron has a charge of about
0.00000000000000000016 coulombs. Being that one
electron is the smallest known carrier of electric charge, this
last figure of charge for the electron is defined as the
elementary charge.
It was discovered much later that this "fluid" was actually
composed of extremely small bits of matter called e/ectrons,
SO Named in honor of the ancient Greek word for amber:
another material exhibiting charged properties when rubbed
with cloth. Experimentation has since revealed that all
objects are composed of extremely small "building-blocks"
known as atoms, and that these atoms are in turn composed
of smaller components known as particles. The three
fundamental particles comprising most atoms are called
protons, neutrons and electrons. Whilst the majority of atoms
have a combination of protons, neutrons, and electrons, not
all atoms have neutrons; an example is the protium isotope
(,H*?) of hydrogen (Hydrogen-1) which is the lightest and
most common form of hydrogen which only has one proton
and one electron. Atoms are far too small to be seen, but if
we could look at one, it might appear something like this:
© © = electron
®) = proton
) = neutron
Even though each atom in a piece of material tends to hold
together as a unit, there's actually a lot of empty space
between the electrons and the cluster of protons and
neutrons residing in the middle.
This crude model is that of the element carbon, with six
protons, six neutrons, and six electrons. In any atom, the
protons and neutrons are very tightly bound together, which
IS an important quality. The tightly-bound clump of protons
and neutrons in the center of the atom is called the nuc/eus,
and the number of protons in an atom's nucleus determines
its elemental identity: change the number of protons in an
atom's nucleus, and you change the type of atom that it is. In
fact, if you could remove three protons from the nucleus of
an atom of lead, you will have achieved the old alchemists'
dream of producing an atom of gold! The tight binding of
protons in the nucleus is responsible for the stable identity of
chemical elements, and the failure of alchemists to achieve
their dream.
Neutrons are much less influential on the chemical character
and identity of an atom than protons, although they are just
as hard to add to or remove from the nucleus, being so
tightly bound. If neutrons are added or gained, the atom will
still retain the same chemical identity, but its mass will
change slightly and it may acquire strange nuclear
properties such as radioactivity.
However, electrons have significantly more freedom to move
around in an atom than either protons or neutrons. In fact,
they can be knocked out of their respective positions (even
leaving the atom entirely!) by far less energy than what it
takes to dislodge particles in the nucleus. If this happens, the
atom still retains its chemical identity, but an important
imbalance occurs. Electrons and protons are unique in the
fact that they are attracted to one another over a distance. It
is this attraction over distance which causes the attraction
between rubbed objects, where electrons are moved away
from their original atoms to reside around atoms of another
object.
Electrons tend to repel other electrons over a distance, as do
protons with other protons. The only reason protons bind
together in the nucleus of an atom is because of a much
stronger force called the strong nuclear force which has
effect only under very short distances. Because of this
attraction/repulsion behavior between individual particles,
electrons and protons are said to have opposite electric
charges. That is, each electron has a negative charge, and
each proton a positive charge. In equal numbers within an
atom, they counteract each other's presence so that the net
charge within the atom is zero. This is why the picture of a
carbon atom had six electrons: to balance out the electric
charge of the six protons in the nucleus. If electrons leave or
extra electrons arrive, the atom's net electric charge will be
imbalanced, leaving the atom "charged" as a whole, causing
it to interact with charged particles and other charged atoms
nearby. Neutrons are neither attracted to or repelled by
electrons, protons, or even other neutrons, and are
consequently categorized as having no charge at all.
The process of electrons arriving or leaving is exactly what
happens when certain combinations of materials are rubbed
together: electrons from the atoms of one material are forced
by the rubbing to leave their respective atoms and transfer
over to the atoms of the other material. In other words,
electrons comprise the "fluid" hypothesized by Benjamin
Franklin.
The result of an imbalance of this "fluid" (electrons) between
objects is called static electricity. It is called "static" because
the displaced electrons tend to remain stationary after being
moved from one insulating material to another. In the case of
wax and wool, it was determined through further
experimentation that electrons in the wool actually
transferred to the atoms in the wax, which is exactly opposite
of Franklin's conjecture! In honor of Franklin's designation of
the wax's charge being "negative" and the wool's charge
being "positive," electrons are said to have a "negative"
charging influence. Thus, an object whose atoms have
received a surplus of electrons is said to be negatively
charged, while an object whose atoms are lacking electrons
iS Said to be positively charged, as confusing as these
designations may seem. By the time the true nature of
electric "fluid" was discovered, Franklin's nomenclature of
electric charge was too well established to be easily changed,
and so it remains to this day.
Michael Faraday proved (1832) that static electricity was the
same as that produced by a battery or a generator. Static
electricity is, for the most part, a nuisance. Black powder and
smokeless powder have graphite added to prevent ignition
due to static electricity. It causes damage to sensitive
semiconductor circuitry. While it is possible to produce
motors powered by high voltage and low current
characteristic of static electricity, this is not economic. The
few practical applications of static electricity include
xerographic printing, the electrostatic air filter, and the high
voltage Van de Graaff generator.
e REVIEW:
e All materials are made up of tiny “building blocks" known
as atoms.
e All naturally occurring atoms contain particles called
electrons, protons, and neutrons, with the exception of
the protium isotope (,H?) of hydrogen.
e Electrons have a negative (-) electric charge.
e Protons have a positive (+) electric charge.
e Neutrons have no electric charge.
e Electrons can be dislodged from atoms much easier than
protons or neutrons.
e The number of protons in an atom's nucleus determines
its identity as a unique element.
Conductors, insulators, and electron
flow
The electrons of different types of atoms have different
degrees of freedom to move around. With some types of
materials, such as metals, the outermost electrons in the
atoms are so loosely bound that they chaotically move in the
Space between the atoms of that material by nothing more
than the influence of room-temperature heat energy. Because
these virtually unbound electrons are free to leave their
respective atoms and float around in the space between
adjacent atoms, they are often called free e/ectrons.
In other types of materials such as glass, the atoms' electrons
have very little freedom to move around. While external
forces such as physical rubbing can force some of these
electrons to leave their respective atoms and transfer to the
atoms of another material, they do not move between atoms
within that material very easily.
This relative mobility of electrons within a material is known
as electric conductivity. Conductivity is determined by the
types of atoms in a material (the number of protons in each
atom's nucleus, determining its chemical identity) and how
the atoms are linked together with one another. Materials
with high electron mobility (many free electrons) are called
conductors, while materials with low electron mobility (few or
no free electrons) are called insulators.
Here are a few common examples of conductors and
insulators:
e Conductors:
e silver
e copper
e gold
e aluminum
e iron
e steel
e brass
e bronze
e mercury
e graphite
e dirty water
e concrete
e Insulators:
e glass
e rubber
e oll
e asphalt
e fiberglass
e porcelain
e ceramic
e quartz
e (dry) cotton
¢ (dry) paper
e (dry) wood
e plastic
e air
e diamond
e pure water
It must be understood that not all conductive materials have
the same level of conductivity, and not all insulators are
equally resistant to electron motion. Electrical conductivity is
analogous to the transparency of certain materials to light:
materials that easily "conduct" light are called "transparent,"
while those that don't are called "opaque." However, not all
transparent materials are equally conductive to light.
Window glass is better than most plastics, and certainly
better than "clear" fiberglass. So it is with electrical
conductors, some being better than others.
For instance, silver is the best conductor in the "conductors"
list, offering easier passage for electrons than any other
material cited. Dirty water and concrete are also listed as
conductors, but these materials are substantially less
conductive than any metal.
It should also be understood that some materials experience
changes in their electrical properties under different
conditions. Glass, for instance, is a very good insulator at
room temperature, but becomes a conductor when heated to
a very high temperature. Gases such as air, normally
insulating materials, also become conductive if heated to
very high temperatures. Most metals become poorer
conductors when heated, and better conductors when
cooled. Many conductive materials become perfectly
conductive (this is called superconductivity) at extremely low
temperatures.
While the normal motion of "free" electrons in a conductor is
random, with no particular direction or speed, electrons can
be influenced to move in a coordinated fashion through a
conductive material. This uniform motion of electrons is what
we Call e/ectricity, or electric current. To be more precise, it
could be called dynamic electricity in contrast to static
electricity, which is an unmoving accumulation of electric
charge. Just like water flowing through the emptiness of a
pipe, electrons are able to move within the empty space
within and between the atoms of a conductor. The conductor
may appear to be solid to our eyes, but any material
composed of atoms is mostly empty space! The liquid-flow
analogy is so fitting that the motion of electrons through a
conductor is often referred to as a "flow."
A noteworthy observation may be made here. As each
electron moves uniformly through a conductor, it pushes on
the one ahead of it, such that all the electrons move together
as a group. The starting and stopping of electron flow
through the length of a conductive path is virtually
instantaneous from one end of a conductor to the other, even
though the motion of each electron may be very slow. An
approximate analogy is that of a tube filled end-to-end with
marbles:
Tube
@ - ©0000000000000008 -0@
Marble Marble
The tube is full of marbles, just as a conductor is full of free
electrons ready to be moved by an outside influence. Ifa
single marble is suddenly inserted into this full tube on the
left-hand side, another marble will immediately try to exit the
tube on the right. Even though each marble only traveled a
short distance, the transfer of motion through the tube is
virtually instantaneous from the left end to the right end, no
matter how long the tube is. With electricity, the overall
effect from one end of a conductor to the other happens at
the speed of light: a swift 186,000 miles per second!!! Each
individual electron, though, travels through the conductor at
a much slower pace.
If we want electrons to flow in a certain direction to a certain
place, we must provide the proper path for them to move,
just as a plumber must install piping to get water to flow
where he or she wants it to flow. To facilitate this, wires are
made of highly conductive metals such as copper or
aluminum in a wide variety of sizes.
Remember that electrons can flow only when they have the
opportunity to move in the space between the atoms of a
material. This means that there can be electric current only
where there exists a continuous path of conductive material
providing a conduit for electrons to travel through. In the
marble analogy, marbles can flow into the left-hand side of
the tube (and, consequently, through the tube) if and only if
the tube is open on the right-hand side for marbles to flow
out. If the tube is blocked on the right-hand side, the marbles
will just "pile up" inside the tube, and marble "flow" will not
occur. The same holds true for electric current: the
continuous flow of electrons requires there be an unbroken
path to permit that flow. Let's look at a diagram to illustrate
how this works:
A thin, solid line (as shown above) is the conventional
symbol for a continuous piece of wire. Since the wire is made
of a conductive material, such as copper, its constituent
atoms have many free electrons which can easily move
through the wire. However, there will never be a continuous
or uniform flow of electrons within this wire unless they have
a place to come from and a place to go. Let's adda
hypothetical electron "Source" and "Destination:"
Electron ee ei a a= —— Electron
Source _ Destination
Now, with the Electron Source pushing new electrons into the
wire on the left-hand side, electron flow through the wire can
occur (as indicated by the arrows pointing from left to right).
However, the flow will be interrupted if the conductive path
formed by the wire is broken:
Electron no flow! no flow! Electron
Source (break) Destination
Since air is an insulating material, and an air gap separates
the two pieces of wire, the once-continuous path has now
been broken, and electrons cannot flow from Source to
Destination. This is like cutting a water pipe in two and
Capping off the broken ends of the pipe: water can't flow if
there's no exit out of the pipe. In electrical terms, we had a
condition of electrical continuity when the wire was in one
piece, and now that continuity is broken with the wire cut
and separated.
If we were to take another piece of wire leading to the
Destination and simply make physical contact with the wire
leading to the Source, we would once again have a
continuous path for electrons to flow. The two dots in the
diagram indicate physical (metal-to-metal) contact between
the wire pieces:
Electron —— no flow! —. = Electron
Source (break) Destination
a ~_ _
Now, we have continuity from the Source, to the newly-made
connection, down, to the right, and up to the Destination.
This is analogous to putting a "tee" fitting in one of the
capped-off pipes and directing water through a new segment
of pipe to its destination. Please take note that the broken
segment of wire on the right hand side has no electrons
flowing through it, because it is no longer part of a complete
path from Source to Destination.
It is interesting to note that no "wear" occurs within wires
due to this electric current, unlike water-carrying pipes which
are eventually corroded and worn by prolonged flows.
Electrons do encounter some degree of friction as they move,
however, and this friction can generate heat in a conductor.
This is a topic we'll explore in much greater detail later.
e REVIEW:
e In conductive materials, the outer electrons in each atom
can easily come or go, and are called free e/ectrons.
e In insulating materials, the outer electrons are not so free
to move.
e All metals are electrically conductive.
e Dynamic electricity, or electric current, is the uniform
motion of electrons through a conductor.
e Static electricity is an unmoving (if on an insulator),
accumulated charge formed by either an excess or
deficiency of electrons in an object. It is typically formed
by charge separation by contact and separation of
dissimilar materials.
e For electrons to flow continuously (indefinitely) through a
conductor, there must be a complete, unbroken path for
them to move both into and out of that conductor.
Electric circuits
You might have been wondering how electrons can
continuously flow in a uniform direction through wires
without the benefit of these hypothetical electron Sources
and Destinations. In order for the Source-and-Destination
scheme to work, both would have to have an infinite capacity
for electrons in order to sustain a continuous flow! Using the
marble-and-tube analogy, the marble source and marble
destination buckets would have to be infinitely large to
contain enough marble capacity for a "flow" of marbles to be
sustained.
The answer to this paradox is found in the concept of a
circuit: a never-ending looped pathway for electrons. If we
take a wire, or many wires joined end-to-end, and loop it
around so that it forms a continuous pathway, we have the
means to support a uniform flow of electrons without having
to resort to infinite Sources and Destinations:
electrons can flaw |
in a path without A marble-and-
beginning or end, hula-hoop "circuit"
| continuing forever! |
Each electron advancing clockwise in this circuit pushes on
the one in front of it, which pushes on the one in front of it,
and so on, and so on, just like a hula-hoop filled with
marbles. Now, we have the capability of supporting a
continuous flow of electrons indefinitely without the need for
infinite electron supplies and dumps. All we need to maintain
this flow is a continuous means of motivation for those
electrons, which we'll address in the next section of this
chapter.
It must be realized that continuity is just as important ina
circuit as it is in a straight piece of wire. Just as in the
example with the straight piece of wire between the electron
Source and Destination, any break in this circuit will prevent
electrons from flowing through it:
no flow!
continuous
electron flow cannot
occur anywhere
in a "broken" circuit!
no flow!
no flow!
An important principle to realize here is that /t doesn't
matter where the break occurs. Any discontinuity in the
circuit will prevent electron flow throughout the entire circuit.
Unless there is a continuous, unbroken loop of conductive
material for electrons to flow through, a sustained flow
simply cannot be maintained.
no flow!
continuous
electron flow cannot
occur anywhere
ina “broken” circuit!
no flow! (break)
no flow!
¢ REVIEW:
e A circuit is an unbroken loop of conductive material that
allows electrons to flow through continuously without
beginning or end.
e If a circuit is "broken," that means its conductive
elements no longer form a complete path, and
continuous electron flow cannot occur in it.
e The location of a break in a circuit is irrelevant to its
inability to sustain continuous electron flow. Any break,
anywhere in a circuit prevents electron flow throughout
the circuit.
Voltage and current
As was previously mentioned, we need more than just a
continuous path (circuit) before a continuous flow of
electrons will occur: we also need some means to push these
electrons around the circuit. Just like marbles in a tube or
water in a pipe, it takes some kind of influencing force to
initiate flow. With electrons, this force is the same force at
work in static electricity: the force produced by an imbalance
of electric charge.
If we take the examples of wax and wool which have been
rubbed together, we find that the surplus of electrons in the
wax (negative charge) and the deficit of electrons in the wool
(positive charge) creates an imbalance of charge between
them. This imbalance manifests itself as an attractive force
between the two objects:
oo ~t
attraction
Wax
Wool cloth
If a conductive wire is placed between the charged wax and
wool, electrons will flow through it, as some of the excess
electrons in the wax rush through the wire to get back to the
wool, filling the deficiency of electrons there:
Wool cloth
The imbalance of electrons between the atoms in the wax
and the atoms in the wool creates a force between the two
materials. With no path for electrons to flow from the wax to
the wool, all this force can do Is attract the two objects
together. Now that a conductor bridges the insulating gap,
however, the force will provoke electrons to flow in a uniform
direction through the wire, if only momentarily, until the
charge in that area neutralizes and the force between the
wax and wool diminishes.
The electric charge formed between these two materials by
rubbing them together serves to store a certain amount of
energy. This energy is not unlike the energy stored in a high
reservoir of water that has been pumped from a lower-level
pond:
Energy stored
|
Water flow
The influence of gravity on the water in the reservoir creates
a force that attempts to move the water down to the lower
level again. If a suitable pipe is run from the reservoir back to
the pond, water will flow under the influence of gravity down
from the reservoir, through the pipe:
|
Energy released
It takes energy to pump that water from the low-level pond to
the high-level reservoir, and the movement of water through
the piping back down to its original level constitutes a
releasing of energy stored from previous pumping.
If the water is pumped to an even higher level, it will take
even more energy to do so, thus more energy will be stored,
and more energy released if the water is allowed to flow
through a pipe back down again:
Energy stored
Energy released
More energy released
!
!
!
!
Electrons are not much different. If we rub wax and wool
together, we "pump" electrons away from their normal
"levels," creating a condition where a force exists between
the wax and wool, as the electrons seek to re-establish their
former positions (and balance within their respective atoms).
The force attracting electrons back to their original positions
around the positive nuclei of their atoms is analogous to the
force gravity exerts on water in the reservoir, trying to draw it
down to its former level.
Just as the pumping of water to a higher level results in
energy being stored, "pumping" electrons to create an
electric charge imbalance results in a certain amount of
energy being stored in that imbalance. And, just as providing
a way for water to flow back down from the heights of the
reservoir results in a release of that stored energy, providing
a way for electrons to flow back to their original "levels"
results in a release of stored energy.
When the electrons are poised in that static condition (just
like water sitting still, high in a reservoir), the energy stored
there is called potential energy, because it has the possibility
(potential) of release that has not been fully realized yet.
When you scuff your rubber-soled shoes against a fabric
carpet on a dry day, you create an imbalance of electric
charge between yourself and the carpet. The action of
scuffing your feet stores energy in the form of an imbalance
of electrons forced from their original locations. This charge
(static electricity) is stationary, and you won't realize that
energy is being stored at all. However, once you place your
hand against a metal doorknob (with lots of electron mobility
to neutralize your electric charge), that stored energy will be
released in the form of a sudden flow of electrons through
your hand, and you will perceive it as an electric shock!
This potential energy, stored in the form of an electric charge
imbalance and capable of provoking electrons to flow
through a conductor, can be expressed as a term called
voltage, which technically is a measure of potential energy
per unit charge of electrons, or something a physicist would
call specific potential energy. Defined in the context of static
electricity, voltage is the measure of work required to move a
unit charge from one location to another, against the force
which tries to keep electric charges balanced. In the context
of electrical power sources, voltage is the amount of potential
energy available (work to be done) per unit charge, to move
electrons through a conductor.
Because voltage is an expression of potential energy,
representing the possibility or potential for energy release as
the electrons move from one "level" to another, it is always
referenced between two points. Consider the water reservoir
analogy:
Location #1
Drop
Location #2
Because of the difference in the height of the drop, there's
potential for much more energy to be released from the
reservoir through the piping to location 2 than to location 1.
The principle can be intuitively understood in dropping a
rock: which results in a more violent impact, a rock dropped
from a height of one foot, or the same rock dropped from a
height of one mile? Obviously, the drop of greater height
results in greater energy released (a more violent impact).
We cannot assess the amount of stored energy in a water
reservoir simply by measuring the volume of water any more
than we can predict the severity of a falling rock's impact
simply from knowing the weight of the rock: in both cases we
must also consider how far these masses will drop from their
initial height. The amount of energy released by allowing a
mass to drop is relative to the distance between its starting
and ending points. Likewise, the potential energy available
for moving electrons from one point to another is relative to
those two points. Therefore, voltage is always expressed as a
quantity between two points. Interestingly enough, the
analogy of a mass potentially "dropping" from one height to
another is such an apt model that voltage between two
points is sometimes called a vo/tage drop.
Voltage can be generated by means other than rubbing
certain types of materials against each other. Chemical
reactions, radiant energy, and the influence of magnetism on
conductors are a few ways in which voltage may be
produced. Respective examples of these three sources of
voltage are batteries, solar cells, and generators (Such as the
"alternator" unit under the hood of your automobile). For
now, we won't go into detail as to how each of these voltage
sources works -- more important is that we understand how
voltage sources can be applied to create electron flow in a
circuit.
Let's take the symbol for a chemical battery and build a
circuit step by step:
Any source of voltage, including batteries, have two points
for electrical contact. In this case, we have point 1 and point
2 in the above diagram. The horizontal lines of varying
length indicate that this is a battery, and they further
indicate the direction which this battery's voltage will try to
push electrons through a circuit. The fact that the horizontal
lines in the battery symbol appear separated (and thus
unable to serve as a path for electrons to move) is no cause
for concern: in real life, those horizontal lines represent
metallic plates immersed in a liquid or semi-solid material
that not only conducts electrons, but also generates the
voltage to push them along by interacting with the plates.
Notice the little "+" and "-" signs to the immediate left of the
battery symbol. The negative (-) end of the battery is always
the end with the shortest dash, and the positive (+) end of
the battery is always the end with the longest dash. Since we
have decided to call electrons "negatively" charged (thanks,
Ben!), the negative end of a battery is that end which tries to
push electrons out of it. Likewise, the positive end is that end
which tries to attract electrons.
With the "+" and "-" ends of the battery not connected to
anything, there will be voltage between those two points, but
there will be no flow of electrons through the battery,
because there is no continuous path for the electrons to
move.
Water analogy
Electric Battery
|
No flow “— Battery
|
The same principle holds true for the water reservoir and
pump analogy: without a return pipe back to the pond, stored
energy in the reservoir cannot be released in the form of
water flow. Once the reservoir is completely filled up, no flow
can occur, no matter how much pressure the pump may
generate. There needs to be a complete path (circuit) for
water to flow from the pond, to the reservoir, and back to the
pond in order for continuous flow to occur.
No flow (once the
reservoir has been
completely filled)
We can provide such a path for the battery by connecting a
piece of wire from one end of the battery to the other.
Forming a circuit with a loop of wire, we will initiate a
continuous flow of electrons in a clockwise direction:
Electric Circuit
= ~<_+
electron flow!
Water analogy
|
water flow!
water flow!
|
So long as the battery continues to produce voltage and the
continuity of the electrical path isn't broken, electrons will
continue to flow in the circuit. Following the metaphor of
water moving through a pipe, this continuous, uniform flow
of electrons through the circuit is called a current. So long as
the voltage source keeps "pushing" in the same direction, the
electron flow will continue to move in the same direction in
the circuit. This single-direction flow of electrons is called a
Direct Current, or DC. In the second volume of this book
series, electric circuits are explored where the direction of
current switches back and forth: A/ternating Current, or AC.
But for now, we'll just concern ourselves with DC circuits.
Because electric current is composed of individual electrons
flowing in unison through a conductor by moving along and
pushing on the electrons ahead, just like marbles through a
tube or water through a pipe, the amount of flow throughout
a single circuit will be the same at any point. If we were to
monitor a cross-section of the wire in a single circuit,
counting the electrons flowing by, we would notice the exact
Same quantity per unit of time as in any other part of the
circuit, regardless of conductor length or conductor diameter.
If we break the circuit's continuity at any point, the electric
current will cease in the entire loop, and the full voltage
produced by the battery will be manifested across the break,
between the wire ends that used to be connected:
no flow!
— Batter voltage
y (break) drop
+ ‘ a
2
no flow!
Notice the "+" and "-" signs drawn at the ends of the break in
the circuit, and how they correspond to the "+" and "-" signs
next to the battery's terminals. These markers indicate the
direction that the voltage attempts to push electron flow,
that potential direction commonly referred to as polarity.
Remember that voltage is always relative between two
points. Because of this fact, the polarity of a voltage drop is
also relative between two points: whether a point in a circuit
gets labeled with a "+" or a "-" depends on the other point to
which it is referenced. Take a look at the following circuit,
where each corner of the loop is marked with a number for
reference:
no flow!
1 a 2
aes Battery (break)
f
+
4 3
no flow!
With the circuit's continuity broken between points 2 and 3,
the polarity of the voltage dropped between points 2 and 3 is
"-" for point 2 and "+" for point 3. The battery's polarity (1 "-"
and 4 '"+") is trying to push electrons through the loop
clockwise from 1 to 2 to 3 to 4 and back to 1 again.
Now let's see what happens if we connect points 2 and 3
back together again, but place a break in the circuit between
points 3 and 4:
no flow!
no flow!
(break)
With the break between 3 and 4, the polarity of the voltage
drop between those two points is "+" for 4 and "-" for 3. Take
special note of the fact that point 3's "sign" is opposite of
that in the first example, where the break was between
points 2 and 3 (where point 3 was labeled "+"). It is
impossible for us to say that point 3 in this circuit will always
be either "+" or "-", because polarity, like voltage itself, is not
specific to a single point, but is always relative between two
points!
REVIEW:
Electrons can be motivated to flow through a conductor
by the same force manifested in static electricity.
Voltage is the measure of specific potential energy
(potential energy per unit charge) between two locations.
In layman's terms, it is the measure of "push" available to
motivate electrons.
Voltage, as an expression of potential energy, is always
relative between two locations, or points. Sometimes it is
called a voltage "drop."
When a voltage source is connected to a circuit, the
voltage will cause a uniform flow of electrons through
that circuit called a current.
In a single (one loop) circuit, the amount of current at
any point is the same as the amount of current at any
other point.
e If a circuit containing a voltage source is broken, the full
voltage of that source will appear across the points of the
break.
e The +/- orientation of a voltage drop is called the
polarity. It is also relative between two points.
Resistance
The circuit in the previous section is not a very practical one.
In fact, it can be quite dangerous to build (directly
connecting the poles of a voltage source together with a
single piece of wire). The reason it is dangerous is because
the magnitude of electric current may be very large in such a
short circuit, and the release of energy very dramatic (usually
in the form of heat). Usually, electric circuits are constructed
in such a way as to make practical use of that released
energy, in as safe a manner as possible.
One practical and popular use of electric current is for the
operation of electric lighting. The simplest form of electric
lamp is a tiny metal "filament" inside of a clear glass bulb,
which glows white-hot ("incandesces") with heat energy
when sufficient electric current passes through it. Like the
battery, it has two conductive connection points, one for
electrons to enter and the other for electrons to exit.
Connected to a source of voltage, an electric lamp circuit
looks something like this:
electron flow
electron flow
As the electrons work their way through the thin metal
filament of the lamp, they encounter more opposition to
motion than they typically would in a thick piece of wire. This
opposition to electric current depends on the type of
material, its cross-sectional area, and its temperature. It is
technically known as resistance. (It can be said that
conductors have low resistance and insulators have very high
resistance.) This resistance serves to limit the amount of
current through the circuit with a given amount of voltage
supplied by the battery, as compared with the "short circuit"
where we had nothing but a wire joining one end of the
voltage source (battery) to the other.
When electrons move against the opposition of resistance,
"friction" is generated. Just like mechanical friction, the
friction produced by electrons flowing against a resistance
manifests itself in the form of heat. The concentrated
resistance of a lamp's filament results in a relatively large
amount of heat energy dissipated at that filament. This heat
energy is enough to cause the filament to glow white-hot,
producing light, whereas the wires connecting the lamp to
the battery (which have much lower resistance) hardly even
get warm while conducting the same amount of current.
As in the case of the short circuit, if the continuity of the
circuit is broken at any point, electron flow stops throughout
the entire circuit. With a lamp in place, this means that it will
stop glowing:
no flow! no flow!
(break)
- +
: aes |
drop
Battery ——
Electric lamp
(not glowing)
no flow!
As before, with no flow of electrons, the entire potential
(voltage) of the battery is available across the break, waiting
for the opportunity of a connection to bridge across that
break and permit electron flow again. This condition is known
as an open circuit, where a break in the continuity of the
circuit prevents current throughout. All it takes is a single
break in continuity to "open" a circuit. Once any breaks have
been connected once again and the continuity of the circuit
re-established, it is known as a Closed circuit.
What we see here is the basis for switching lamps on and off
by remote switches. Because any break in a circuit's
continuity results in current stopping throughout the entire
circuit, we can use a device designed to intentionally break
that continuity (called a switch), mounted at any convenient
location that we can run wires to, to control the flow of
electrons in the circuit:
switch
It doesn’t matter how twisted or
convoluted a route the wires take
conducting current, so long as they
form a complete, uninterrupted
loop (circuit).
This is how a switch mounted on the wall of a house can
control a lamp that is mounted down a long hallway, or even
in another room, far away from the switch. The switch itself is
constructed of a pair of conductive contacts (usually made of
some kind of metal) forced together by a mechanical lever
actuator or pushbutton. When the contacts touch each other,
electrons are able to flow from one to the other and the
circuit's continuity is established; when the contacts are
separated, electron flow from one to the other is prevented
by the insulation of the air between, and the circuit's
continuity is broken.
Perhaps the best kind of switch to show for illustration of the
basic principle is the "knife" switch:
A knife switch is nothing more than a conductive lever, free
to pivot on a hinge, coming into physical contact with one or
more stationary contact points which are also conductive.
The switch shown in the above illustration is constructed on
a porcelain base (an excellent insulating material), using
copper (an excellent conductor) for the "blade" and contact
points. The handle is plastic to insulate the operator's hand
from the conductive blade of the switch when opening or
closing it.
Here is another type of knife switch, with two stationary
contacts instead of one:
The particular knife switch shown here has one "blade" but
two stationary contacts, meaning that it can make or break
more than one circuit. For now this is not terribly important
to be aware of, just the basic concept of what a switch is and
how it works.
Knife switches are great for illustrating the basic principle of
how a switch works, but they present distinct safety
problems when used in high-power electric circuits. The
exposed conductors in a knife switch make accidental
contact with the circuit a distinct possibility, and any
sparking that may occur between the moving blade and the
stationary contact is free to ignite any nearby flammable
materials. Most modern switch designs have their moving
conductors and contact points sealed inside an insulating
case in order to mitigate these hazards. A photograph of a
few modern switch types show how the switching
mechanisms are much more concealed than with the knife
design:
Toggle switch
Multiposition rotary
selector switch
In keeping with the "open" and "closed" terminology of
circuits, a switch that is making contact from one connection
terminal to the other (example: a knife switch with the blade
fully touching the stationary contact point) provides
continuity for electrons to flow through, and is called a closed
switch. Conversely, a switch that is breaking continuity
(example: a knife switch with the blade not touching the
stationary contact point) won't allow electrons to pass
through and is called an open switch. This terminology is
often confusing to the new student of electronics, because
the words "open" and "closed" are commonly understood in
the context of a door, where "open" is equated with free
passage and "closed" with blockage. With electrical switches,
these terms have opposite meaning: "open" means no flow
while "closed" means free passage of electrons.
¢ REVIEW:
e Resistance is the measure of opposition to electric
current.
e A short circuit is an electric circuit offering little or no
resistance to the flow of electrons. Short circuits are
dangerous with high voltage power sources because the
high currents encountered can cause large amounts of
heat energy to be released.
e An open circuit is one where the continuity has been
broken by an interruption in the path for electrons to
flow.
e A closed circuit is one that is complete, with good
continuity throughout.
e A device designed to open or close a circuit under
controlled conditions is called a switch.
e The terms “open" and "closed" refer to switches as well
as entire circuits. An open switch is one without
continuity: electrons cannot flow through it. A closed
switch is one that provides a direct (low resistance) path
for electrons to flow through.
Voltage and current in a practical
circuit
Because it takes energy to force electrons to flow against the
opposition of a resistance, there will be voltage manifested
(or "dropped") between any points in a circuit with resistance
between them. It is important to note that although the
amount of current (the quantity of electrons moving past a
given point every second) is uniform in a simple circuit, the
amount of voltage (potential energy per unit charge)
between different sets of points in a single circuit may vary
considerably:
same rate of current...
... at all points in this circuit
Take this circuit as an example. If we label four points in this
circuit with the numbers 1, 2, 3, and 4, we will find that the
amount of current conducted through the wire between
points 1 and 2 is exactly the same as the amount of current
conducted through the lamp (between points 2 and 3). This
same quantity of current passes through the wire between
points 3 and 4, and through the battery (between points 1
and 4).
However, we will find the voltage appearing between any two
of these points to be directly proportional to the resistance
within the conductive path between those two points, given
that the amount of current along any part of the circuit's
path is the same (which, for this simple circuit, it is). In a
normal lamp circuit, the resistance of a lamp will be much
greater than the resistance of the connecting wires, so we
should expect to see a substantial amount of voltage
between points 2 and 3, with very little between points 1 and
2, or between 3 and 4. The voltage between points 1 and 4,
of course, will be the full amount of "force" offered by the
battery, which will be only slightly greater than the voltage
across the lamp (between points 2 and 3).
This, again, is analogous to the water reservoir system:
|
(energy stored)
Waterwheel
(energy released)
Pump
| 3
Between points 2 and 3, where the falling water is releasing
energy at the water-wheel, there is a difference of pressure
between the two points, reflecting the opposition to the flow
of water through the water-wheel. From point 1 to point 2, or
from point 3 to point 4, where water is flowing freely through
reservoirs with little opposition, there is little or no difference
of pressure (no potential energy). However, the rate of water
flow in this continuous system is the same everywhere
(assuming the water levels in both pond and reservoir are
unchanging): through the pump, through the water-wheel,
and through all the pipes. So it is with simple electric
circuits: the rate of electron flow is the same at every point in
the circuit, although voltages may differ between different
sets of points.
Conventional versus electron flow
"The nice thing about standards Is that there are so
many of them to choose from."
Andrew S. Tanenbaum, computer science
professor
When Benjamin Franklin made his conjecture regarding the
direction of charge flow (from the smooth wax to the rough
wool), he set a precedent for electrical notation that exists to
this day, despite the fact that we know electrons are the
constituent units of charge, and that they are displaced from
the wool to the wax -- not from the wax to the wool -- when
those two substances are rubbed together. This is why
electrons are said to have a negative charge: because
Franklin assumed electric charge moved in the opposite
direction that it actually does, and so objects he called
"negative" (representing a deficiency of charge) actually
have a surplus of electrons.
By the time the true direction of electron flow was
discovered, the nomenclature of "positive" and "negative"
had already been so well established in the scientific
community that no effort was made to change it, although
calling electrons "positive" would make more sense in
referring to "excess" charge. You see, the terms "positive"
and "negative" are human inventions, and as such have no
absolute meaning beyond our own conventions of language
and scientific description. Franklin could have just as easily
referred to a surplus of charge as "black" and a deficiency as
"white," in which case scientists would speak of electrons
having a "white" charge (assuming the same incorrect
conjecture of charge position between wax and wool).
However, because we tend to associate the word "positive"
with "surplus" and "negative" with "deficiency," the standard
label for electron charge does seem backward. Because of
this, many engineers decided to retain the old concept of
electricity with "positive" referring to a surplus of charge, and
label charge flow (current) accordingly. This became known
as conventional flow notation:
Conventional flow notation
| SE el oo EE eel
Electric charge moves
from the positive (surplus)
side of the battery to the
negative (deficiency) side.
Others chose to designate charge flow according to the
actual motion of electrons in a circuit. This form of symbology
became known as e/ectron flow notation:
Electron flow notation
{ ~~ ~$ ~t
Electric charge moves
from the negative (surplus)
side of the battery to the
positive (deficiency) side.
In conventional flow notation, we show the motion of charge
according to the (technically incorrect) labels of + and -. This
way the labels make sense, but the direction of charge flow is
incorrect. In electron flow notation, we follow the actual
motion of electrons in the circuit, but the + and - labels seem
backward. Does it matter, really, how we designate charge
flow in a circuit? Not really, so long as we're consistent in the
use of our symbols. You may follow an imagined direction of
current (conventional flow) or the actual (electron flow) with
equal success insofar as circuit analysis is concerned.
Concepts of voltage, current, resistance, continuity, and even
mathematical treatments such as Ohm's Law (chapter 2) and
Kirchhoff's Laws (chapter 6) remain just as valid with either
style of notation.
You will find conventional flow notation followed by most
electrical engineers, and illustrated in most engineering
textbooks. Electron flow is most often seen in introductory
textbooks (this one included) and in the writings of
professional scientists, especially solid-state physicists who
are concerned with the actual motion of electrons in
substances. These preferences are cultural, in the sense that
certain groups of people have found it advantageous to
envision electric current motion in certain ways. Being that
most analyses of electric circuits do not depend ona
technically accurate depiction of charge flow, the choice
between conventional flow notation and electron flow
notation is arbitrary ... almost.
Many electrical devices tolerate real currents of either
direction with no difference in operation. Incandescent lamps
(the type utilizing a thin metal filament that glows white-hot
with sufficient current), for example, produce light with equal
efficiency regardless of current direction. They even function
well on alternating current (AC), where the direction changes
rapidly over time. Conductors and switches operate
irrespective of current direction, as well. The technical term
for this irrelevance of charge flow is nonpolarization. We
could say then, that incandescent lamps, switches, and wires
are nonpolarized components. Conversely, any device that
functions differently on currents of different direction would
be called a polarized device.
There are many such polarized devices used in electric
circuits. Most of them are made of so-called semiconductor
Substances, and as such aren't examined in detail until the
third volume of this book series. Like switches, lamps, and
batteries, each of these devices is represented in a schematic
diagram by a unique symbol. As one might guess, polarized
device symbols typically contain an arrow within them,
somewhere, to designate a preferred or exclusive direction of
current. This is where the competing notations of
conventional and electron flow really matter. Because
engineers from long ago have settled on conventional flow as
their "culture's" standard notation, and because engineers
are the same people who invent electrical devices and the
symbols representing them, the arrows used in these devices'
symbols a// point in the direction of conventional flow, not
electron flow. That is to say, all of these devices’ symbols
have arrow marks that point aga/nst the actual flow of
electrons through them.
Perhaps the best example of a polarized device is the diode.
A diode is a one-way "valve" for electric current, analogous to
a check valve for those familiar with plumbing and hydraulic
systems. Ideally, a diode provides unimpeded flow for current
in one direction (little or no resistance), but prevents flow in
the other direction (infinite resistance). Its schematic symbol
looks like this:
Diode
—>-
Placed within a battery/lamp circuit, its operation is as such:
Diode operation
Current permitted Current prohibited
When the diode is facing in the proper direction to permit
current, the lamp glows. Otherwise, the diode blocks all
electron flow just like a break in the circuit, and the lamp will
not glow.
If we label the circuit current using conventional flow
notation, the arrow symbol of the diode makes perfect sense:
the triangular arrowhead points in the direction of charge
flow, from positive to negative:
Current shown using
conventional flow notation
On the other hand, if we use electron flow notation to show
the true direction of electron travel around the circuit, the
diode's arrow symbology seems backward:
Current shown using
electron flow notation
For this reason alone, many people choose to make
conventional flow their notation of choice when drawing the
direction of charge motion in a circuit. If for no other reason,
the symbols associated with semiconductor components like
diodes make more sense this way. However, others choose to
show the true direction of electron travel so as to avoid
having to tell themselves, "just remember the electrons are
actually moving the other way" whenever the true direction
of electron motion becomes an issue.
In this series of textbooks, | have committed to using electron
flow notation. Ironically, this was not my first choice. | found
it much easier when | was first learning electronics to use
conventional flow notation, primarily because of the
directions of semiconductor device symbol arrows. Later,
when | began my first formal training in electronics, my
instructor insisted on using electron flow notation in his
lectures. In fact, he asked that we take our textbooks (which
were illustrated using conventional flow notation) and use
our pens to change the directions of all the current arrows so
as to point the "correct" way! His preference was not
arbitrary, though. In his 20-year career as a U.S. Navy
electronics technician, he worked on a lot of vacuum-tube
equipment. Before the advent of semiconductor components
like transistors, devices known as vacuum tubes or electron
tubes were used to amplify small electrical signals. These
devices work on the phenomenon of electrons hurtling
through a vacuum, their rate of flow controlled by voltages
applied between metal plates and grids placed within their
path, and are best understood when visualized using
electron flow notation.
When | graduated from that training program, | went back to
my old habit of conventional flow notation, primarily for the
sake of minimizing confusion with component symbols, since
vacuum tubes are all but obsolete except in special
applications. Collecting notes for the writing of this book, |
had full intention of illustrating it using conventional flow.
Years later, when | became a teacher of electronics, the
curriculum for the program | was going to teach had already
been established around the notation of electron flow. Oddly
enough, this was due in part to the legacy of my first
electronics instructor (the 20-year Navy veteran), but that's
another story entirely! Not wanting to confuse students by
teaching "differently" from the other instructors, | had to
overcome my habit and get used to visualizing electron flow
instead of conventional. Because | wanted my book to bea
useful resource for my students, | begrudgingly changed
plans and illustrated it with all the arrows pointing the
"correct" way. Oh well, sometimes you just can't win!
On a positive note (no pun intended), | have subsequently
discovered that some students prefer electron flow notation
when first learning about the behavior of semiconductive
substances. Also, the habit of visualizing electrons flowing
against the arrows of polarized device symbols isn't that
difficult to learn, and in the end I've found that | can follow
the operation of a circuit equally well using either mode of
notation. Still, | sometimes wonder if it would all be much
easier if we went back to the source of the confusion -- Ben
Franklin's errant conjecture -- and fixed the problem there,
calling electrons "positive" and protons "negative."
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Bill Heath (September 2002): Pointed out error in
illustration of carbon atom -- the nucleus was shown with
seven protons instead of six.
Ben Crowell, Ph.D. (January 13, 2001): suggestions on
improving the technical accuracy of vo/tage and charge
definitions.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
— & —*
Previous Contents Next
— 4 —>
Lessons In Electric Circuits --
Volume |
Chapter 2
OHM's LAW
e« How voltage, current, and resistance relate
An analogy for Ohm's Law
Power in electric circuits
Calculating electric power
Resistors
Nonlinear conduction
Circuit wiring
Polarity of voltage drops
Computer simulation of electric circuits
e Contributors
“One microampere flowing in one ohm causes a one microvolt potential
drop."
Georg Simon Ohm
How voltage, current, and resistance relate
An electric circuit is formed when a conductive path is created to allow free
electrons to continuously move. This continuous movement of free electrons
through the conductors of a circuit is called a current, and it is often referred to
in terms of "flow," just like the flow of a liquid through a hollow pipe.
The force motivating electrons to "flow" in a circuit is called vo/tage. Voltage is
a specific measure of potential energy that is always relative between two
points. When we speak of a certain amount of voltage being present in a circuit,
we are referring to the measurement of how much potentia/ energy exists to
move electrons from one particular point in that circuit to another particular
point. Without reference to two particular points, the term "voltage" has no
meaning.
Free electrons tend to move through conductors with some degree of friction, or
opposition to motion. This opposition to motion is more properly called
resistance. The amount of current in a circuit depends on the amount of voltage
available to motivate the electrons, and also the amount of resistance in the
circuit to oppose electron flow. Just like voltage, resistance is a quantity relative
between two points. For this reason, the quantities of voltage and resistance are
often stated as being "between" or "across" two points in a Circuit.
To be able to make meaningful statements about these quantities in circuits, we
need to be able to describe their quantities in the same way that we might
quantify mass, temperature, volume, length, or any other kind of physical
quantity. For mass we might use the units of "kilogram" or "gram." For
temperature we might use degrees Fahrenheit or degrees Celsius. Here are the
standard units of measurement for electrical current, voltage, and resistance:
PResistance| ek | Ohm | a
The "symbol" given for each quantity is the standard alphabetical letter used to
represent that quantity in an algebraic equation. Standardized letters like these
are common in the disciplines of physics and engineering, and are
internationally recognized. The "unit abbreviation" for each quantity represents
the alphabetical symbol used as a shorthand notation for its particular unit of
measurement. And, yes, that strange-looking "horseshoe" symbol is the capital
Greek letter Q, just a character in a foreign alphabet (apologies to any Greek
readers here).
Each unit of measurement is named after a famous experimenter in electricity:
The amp after the Frenchman Andre M. Ampere, the vo/t after the Italian
Alessandro Volta, and the ohm after the German Georg Simon Ohm.
The mathematical symbol for each quantity is meaningful as well. The "R" for
resistance and the "V" for voltage are both self-explanatory, whereas "I" for
current seems a bit weird. The "I" is thought to have been meant to represent
"Intensity" (of electron flow), and the other symbol for voltage, "E," stands for
“Electromotive force." From what research I've been able to do, there seems to
be some dispute over the meaning of "I." The symbols "E" and "V" are
interchangeable for the most part, although some texts reserve "E" to represent
voltage across a source (Such as a battery or generator) and "V" to represent
voltage across anything else.
All of these symbols are expressed using capital letters, except in cases where a
quantity (especially voltage or current) is described in terms of a brief period of
time (called an "instantaneous" value). For example, the voltage of a battery,
which is stable over a long period of time, will be symbolized with a capital
letter "E," while the voltage peak of a lightning strike at the very instant it hits
a power line would most likely be symbolized with a lower-case letter "e" (or
lower-case "v") to designate that value as being at a single moment in time.
This same lower-case convention holds true for current as well, the lower-case
letter "i" representing current at some instant in time. Most direct-current (DC)
measurements, however, being stable over time, will be symbolized with capital
letters.
One foundational unit of electrical measurement, often taught in the
beginnings of electronics courses but used infrequently afterwards, is the unit
of the coulomb, which is a measure of electric charge proportional to the
number of electrons in an imbalanced state. One coulomb of charge is equal to
6,250,000,000,000,000,000 electrons. The symbol for electric charge quantity
is the capital letter "Q," with the unit of coulombs abbreviated by the capital
letter "C." It so happens that the unit for electron flow, the amp, is equal to 1
coulomb of electrons passing by a given point in a circuit in 1 second of time.
Cast in these terms, current is the rate of electric charge motion through a
conductor.
As stated before, voltage is the measure of potential energy per unit charge
available to motivate electrons from one point to another. Before we can
precisely define what a "volt" is, we must understand how to measure this
quantity we call "potential energy." The general metric unit for energy of any
kind is the jou/e, equal to the amount of work performed by a force of 1 newton
exerted through a motion of 1 meter (in the same direction). In British units,
this is slightly less than 3/4 pound of force exerted over a distance of 1 foot. Put
in common terms, it takes about 1 joule of energy to lift a 3/4 pound weight 1
foot off the ground, or to drag something a distance of 1 foot using a parallel
pulling force of 3/4 pound. Defined in these scientific terms, 1 volt is equal to 1
joule of electric potential energy per (divided by) 1 coulomb of charge. Thus, a
9 volt battery releases 9 joules of energy for every coulomb of electrons moved
through a circuit.
These units and symbols for electrical quantities will become very important to
know as we begin to explore the relationships between them in circuits. The
first, and perhaps most important, relationship between current, voltage, and
resistance is called Ohm's Law, discovered by Georg Simon Ohm and published
in his 1827 paper, The Galvanic Circuit Investigated Mathematically. Ohm's
principal discovery was that the amount of electric current through a metal
conductor in a circuit is directly proportional to the voltage impressed across it,
for any given temperature. Ohm expressed his discovery in the form of a simple
equation, describing how voltage, current, and resistance interrelate:
E=1K
In this algebraic expression, voltage (E) is equal to current (1) multiplied by
resistance (R). Using algebra techniques, we can manipulate this equation into
two variations, solving for | and for R, respectively:
eee eos
R I
Let's see how these equations might work to help us analyze simple circuits:
electron flow
- Electric lamp (glowing)
\
electron flow
In the above circuit, there is only one source of voltage (the battery, on the left)
and only one source of resistance to current (the lamp, on the right). This makes
it very easy to apply Ohm's Law. If we know the values of any two of the three
quantities (voltage, current, and resistance) in this circuit, we can use Ohm's
Law to determine the third.
In this first example, we will calculate the amount of current (I) in a circuit,
given values of voltage (E) and resistance (R):
= 79?
= 797
What is the amount of current (I) in this circuit?
(Se Se SH
R 32
In this second example, we will calculate the amount of resistance (R) ina
circuit, given values of voltage (E) and current (I):
1=4A
What is the amount of resistance (R) offered by the lamp?
In the last example, we will calculate the amount of voltage supplied by a
battery, given values of current (1) and resistance (R):
1=2A
1=2A
What is the amount of voltage provided by the battery?
E = 1R = (2A\(7Q)=14V
Ohm's Law is a very simple and useful tool for analyzing electric circuits. It is
used so often in the study of electricity and electronics that it needs to be
committed to memory by the serious student. For those who are not yet
comfortable with algebra, there's a trick to remembering how to solve for any
one quantity, given the other two. First, arrange the letters E, l,and Rina
triangle like this:
/\
MES
If you Know E and I, and wish to determine R, just eliminate R from the picture
and see what's left:
a
>
>
If you Know E and R, and wish to determine I, eliminate | and see what's left:
p- =
R
2
pi
Lastly, if you know | and R, and wish to determine E, eliminate E and see what's
oO
ae
m
ll
wu
Eventually, you'll have to be familiar with algebra to seriously study electricity
and electronics, but this tip can make your first calculations a little easier to
remember. If you are comfortable with algebra, all you need to do is commit
E=IR to memory and derive the other two formulae from that when you need
them!
¢ REVIEW:
e Voltage measured in vo/ts, symbolized by the letters "E" or "V".
e Current measured in amps, symbolized by the letter "I".
e Resistance measured in ohms, symbolized by the letter "R".
e Ohm's Law: E = 1R;1=E/R; R= E/|
An analogy for Ohm's Law
Ohm's Law also makes intuitive sense if you apply it to the water-and-pipe
analogy. If we have a water pump that exerts pressure (voltage) to push water
around a "circuit" (current) through a restriction (resistance), we can model how
the three variables interrelate. If the resistance to water flow stays the same
and the pump pressure increases, the flow rate must also increase.
Pressure = increase Voltage = increase
Flow rate = increase Current = increase
Resistance= same Resistance= same
If the pressure stays the same and the resistance increases (making it more
difficult for the water to flow), then the flow rate must decrease:
Pressure = same Voltage = same
Flow rate = decrease Current = decrease
Resistance= increase Resistance= increase
If the flow rate were to stay the same while the resistance to flow decreased, the
required pressure from the pump would necessarily decrease:
Pressure = decrease Voltage = decrease
Flow rate = same Current = same
Resistance= decrease Resistance= decrease
E=IR
As odd as it may seem, the actual mathematical relationship between pressure,
flow, and resistance is actually more complex for fluids like water than it is for
electrons. If you pursue further studies in physics, you will discover this for
yourself. Thankfully for the electronics student, the mathematics of Ohm's Law
is very straightforward and simple.
¢ REVIEW:
e With resistance steady, current follows voltage (an increase in voltage
means an increase in current, and vice versa).
e With voltage steady, changes in current and resistance are opposite (an
increase in current means a decrease in resistance, and vice versa).
e With current steady, voltage follows resistance (an increase in resistance
means an increase in voltage).
Power in electric circuits
In addition to voltage and current, there is another measure of free electron
activity in a circuit: power. First, we need to understand just what power is
before we analyze it in any circuits.
Power is a measure of how much work can be performed in a given amount of
time. Work is generally defined in terms of the lifting of a weight against the
pull of gravity. The heavier the weight and/or the higher it is lifted, the more
work has been done. Power is a measure of how rapidly a standard amount of
work is done.
For American automobiles, engine power is rated in a unit called "horsepower,"
invented initially as a way for steam engine manufacturers to quantify the
working ability of their machines in terms of the most common power source of
their day: horses. One horsepower is defined in British units as 550 ft-lbs of
work per second of time. The power of a car's engine won't indicate how tall of a
hill it can climb or how much weight it can tow, but it will indicate how fast it
can climb a specific hill or tow a specific weight.
The power of a mechanical engine is a function of both the engine's speed and
its torque provided at the output shaft. Soeed of an engine's output shaft is
measured in revolutions per minute, or RPM. Torque is the amount of twisting
force produced by the engine, and it is usually measured in pound-feet, or |b-ft
(not to be confused with foot-pounds or ft-lbs, which is the unit for work).
Neither speed nor torque alone is a measure of an engine's power.
A 100 horsepower diesel tractor engine will turn relatively slowly, but provide
great amounts of torque. A 100 horsepower motorcycle engine will turn very
fast, but provide relatively little torque. Both will produce 100 horsepower, but
at different speeds and different torques. The equation for shaft horsepower is
simple:
20ST
Horsepower =
ad 33,000
Where,
S = shaft speed in r.p.m.
T = shaft torque in lb-ft.
Notice how there are only two variable terms on the right-hand side of the
equation, S and T. All the other terms on that side are constant: 2, pi, and
33,000 are all constants (they do not change in value). The horsepower varies
only with changes in speed and torque, nothing else. We can re-write the
equation to show this relationship:
Horsepower « S T
This symbol means
* "proportional to”
Because the unit of the "horsepower" doesn't coincide exactly with speed in
revolutions per minute multiplied by torque in pound-feet, we can't say that
horsepower equals ST. However, they are proportional to one another. As the
mathematical product of ST changes, the value for horsepower will change by
the same proportion.
In electric circuits, power is a function of both voltage and current. Not
surprisingly, this relationship bears striking resemblance to the "proportional"
horsepower formula above:
P=1E
In this case, however, power (P) is exactly equal to current (1) multiplied by
voltage (E), rather than merely being proportional to IE. When using this
formula, the unit of measurement for power is the watt, abbreviated with the
letter "W."
It must be understood that neither voltage nor current by themselves constitute
power. Rather, power is the combination of both voltage and current in a circuit.
Remember that voltage is the specific work (or potential energy) per unit
charge, while current is the rate at which electric charges move through a
conductor. Voltage (Specific work) is analogous to the work done in lifting a
weight against the pull of gravity. Current (rate) is analogous to the speed at
which that weight is lifted. Together as a product (multiplication), voltage
(work) and current (rate) constitute power.
Just as in the case of the diesel tractor engine and the motorcycle engine, a
circuit with high voltage and low current may be dissipating the same amount
of power as a circuit with low voltage and high current. Neither the amount of
voltage alone nor the amount of current alone indicates the amount of power in
an electric circuit.
In an open circuit, where voltage is present between the terminals of the source
and there is zero current, there is zero power dissipated, no matter how great
that voltage may be. Since P=IE and I=0 and anything multiplied by zero is
zero, the power dissipated in any open circuit must be zero. Likewise, if we were
to have a short circuit constructed of a loop of superconducting wire (absolutely
zero resistance), we could have a condition of current in the loop with zero
voltage, and likewise no power would be dissipated. Since P=IE and E=0 and
anything multiplied by zero is zero, the power dissipated in a superconducting
loop must be zero. (We'll be exploring the topic of superconductivity in a later
chapter).
Whether we measure power in the unit of "horsepower" or the unit of "watt,"
we're still talking about the same thing: how much work can be done in a given
amount of time. The two units are not numerically equal, but they express the
same kind of thing. In fact, European automobile manufacturers typically
advertise their engine power in terms of kilowatts (kW), or thousands of watts,
instead of horsepower! These two units of power are related to each other by a
simple conversion formula:
1 Horsepower = 745.7 Watts
So, our 100 horsepower diesel and motorcycle engines could also be rated as
"74570 watt" engines, or more properly, aS "74.57 kilowatt" engines. In
European engineering specifications, this rating would be the norm rather than
the exception.
e REVIEW:
e Power is the measure of how much work can be done in a given amount of
time.
e Mechanical power is commonly measured (in America) in "horsepower."
e Electrical power is almost always measured in "watts," and it can be
calculated by the formula P = IE.
e Electrical power is a product of both voltage and current, not either one
separately.
e Horsepower and watts are merely two different units for describing the
same kind of physical measurement, with 1 horsepower equaling 745.7
watts.
Calculating electric power
We've seen the formula for determining the power in an electric circuit: by
multiplying the voltage in "volts" by the current in "amps" we arrive at an
answer in "watts." Let's apply this to a circuit example:
1=7?79?
In the above circuit, we know we have a battery voltage of 18 volts and a lamp
resistance of 3 QO. Using Ohm's Law to determine current, we get:
E 13 V
ae ee
Now that we know the current, we can take that value and multiply it by the
voltage to determine power:
P=1E= (6 A)(18 V) = 108 W
Answer: the lamp is dissipating (releasing) 108 watts of power, most likely in
the form of both light and heat.
Let's try taking that same circuit and increasing the battery voltage to see what
happens. Intuition should tell us that the circuit current will increase as the
voltage increases and the lamp resistance stays the same. Likewise, the power
will increase as well:
1= 79?
—
1= 79?
Now, the battery voltage is 36 volts instead of 18 volts. The lamp is still
providing 3 Q of electrical resistance to the flow of electrons. The current is now:
E 36V ,
= Ro ae Steele
This stands to reason: if | = E/R, and we double E while R stays the same, the
current should double. Indeed, it has: we now have 12 amps of current instead
of 6. Now, what about power?
P=1E= (12 A)\(36 V)=432 W
Notice that the power has increased just as we might have suspected, but it
increased quite a bit more than the current. Why is this? Because power is a
function of voltage multiplied by current, and both voltage and current doubled
from their previous values, the power will increase by a factor of 2 x 2, or 4. You
can check this by dividing 432 watts by 108 watts and seeing that the ratio
between them is indeed 4.
Using algebra again to manipulate the formulae, we can take our original power
formula and modify it for applications where we don't know both voltage and
current:
If we only know voltage (E) and resistance (R):
i fess and P=lE
Then, P=—E or P=
ne
If we only know current (I) and resistance (R):
If, E=1R and P=1E
Then, P=1(1R) or P=IR
A historical note: it was James Prescott Joule, not Georg Simon Ohm, who first
discovered the mathematical relationship between power dissipation and
current through a resistance. This discovery, published in 1841, followed the
form of the last equation (P = |?R), and is properly known as Joule's Law.
However, these power equations are so commonly associated with the Ohm's
Law equations relating voltage, current, and resistance (E=IR ; I=E/R; and
R=E/I) that they are frequently credited to Ohm.
Power equations
e REVIEW:
e Power measured in watts, symbolized by the letter "W".
¢ Joule's Law: P = |?R; P=IE; P= E2/R
Resistors
Because the relationship between voltage, current, and resistance in any circuit
is So regular, we can reliably control any variable in a circuit simply by
controlling the other two. Perhaps the easiest variable in any circuit to control is
its resistance. This can be done by changing the material, size, and shape of its
conductive components (remember how the thin metal filament of a lamp
created more electrical resistance than a thick wire?).
Special components called resistors are made for the express purpose of
creating a precise quantity of resistance for insertion into a circuit. They are
typically constructed of metal wire or carbon, and engineered to maintain a
stable resistance value over a wide range of environmental conditions. Unlike
lamps, they do not produce light, but they do produce heat as electric power is
dissipated by them in a working circuit. Typically, though, the purpose of a
resistor is not to produce usable heat, but simply to provide a precise quantity
of electrical resistance.
The most common schematic symbol for a resistor is a zig-zag line:
WV
Resistor values in ohms are usually shown as an adjacent number, and if several
resistors are present in a circuit, they will be labeled with a unique identifier
number such as Rj, R>, R3, etc. As you can see, resistor symbols can be shown
either horizontally or vertically:
R, This is resistor "R,"
VW with a resistance value
150 of 150 ohms.
This is resistor "R2"
R, 225 with a resistance value
: of 25 ohms.
Real resistors look nothing like the zig-zag symbol. Instead, they look like small
tubes or cylinders with two wires protruding for connection to a circuit. Here isa
sampling of different kinds and sizes of resistors:
In keeping more with their physical appearance, an alternative schematic
symbol for a resistor looks like a small, rectangular box:
——
Resistors can also be shown to have varying rather than fixed resistances. This
might be for the purpose of describing an actual physical device designed for
the purpose of providing an adjustable resistance, or it could be to show some
component that just happens to have an unstable resistance:
variable
resistance
¥ 0. ff
In fact, any time you see a component symbol drawn with a diagonal arrow
through it, that component has a variable rather than a fixed value. This symbol
"modifier" (the diagonal arrow) is standard electronic symbol convention.
Variable resistors must have some physical means of adjustment, either a
rotating shaft or lever that can be moved to vary the amount of electrical
resistance. Here is a photograph showing some devices called potentiometers,
which can be used as variable resistors:
Because resistors dissipate heat energy as the electric currents through them
overcome the "friction" of their resistance, resistors are also rated in terms of
how much heat energy they can dissipate without overheating and sustaining
damage. Naturally, this power rating is specified in the physical unit of "watts."
Most resistors found in small electronic devices such as portable radios are
rated at 1/4 (0.25) watt or less. The power rating of any resistor is roughly
proportional to its physical size. Note in the first resistor photograph how the
power ratings relate with size: the bigger the resistor, the higher its power
dissipation rating. Also note how resistances (in ohms) have nothing to do with
size!
Although it may seem pointless now to have a device doing nothing but
resisting electric current, resistors are extremely useful devices in circuits.
Because they are simple and so commonly used throughout the world of
electricity and electronics, we'll spend a considerable amount of time analyzing
circuits composed of nothing but resistors and batteries.
For a practical illustration of resistors' usefulness, examine the photograph
below. It is a picture of a printed circuit board, or PCB: an assembly made of
sandwiched layers of insulating phenolic fiber-board and conductive copper
strips, into which components may be inserted and secured by a low-
temperature welding process called "soldering." The various components on
this circuit board are identified by printed labels. Resistors are denoted by any
label beginning with the letter "R".
: Ww
ZX6475
This particular circuit board is a computer accessory called a "modem," which
allows digital information transfer over telephone lines. There are at least a
dozen resistors (all rated at 1/4 watt power dissipation) that can be seen on this
modem's board. Every one of the black rectangles (called "integrated circuits"
or "chips") contain their own array of resistors for their internal functions, as
well.
Another circuit board example shows resistors packaged in even smaller units,
called "surface mount devices." This particular circuit board is the underside of
a personal computer hard disk drive, and once again the resistors soldered onto
it are designated with labels beginning with the letter "R":
=
PTET ce
There are over one hundred surface-mount resistors on this circuit board, and
this count of course does not include the number of resistors internal to the
black "chips." These two photographs should convince anyone that resistors --
devices that "merely" oppose the flow of electrons -- are very important
components in the realm of electronics!
In schematic diagrams, resistor symbols are sometimes used to illustrate any
general type of device in a circuit doing something useful with electrical energy.
Any non-specific electrical device is generally called a /oad, so if you see a
schematic diagram showing a resistor symbol labeled "load," especially in a
tutorial circuit diagram explaining some concept unrelated to the actual use of
electrical power, that symbol may just be a kind of shorthand representation of
something else more practical than a resistor.
To summarize what we've learned in this lesson, let's analyze the following
circuit, determining all that we can from the information given:
1=2A
Battery = R=777
at oe P=27?
All we've been given here to start with is the battery voltage (10 volts) and the
circuit current (2 amps). We don't know the resistor's resistance in ohms or the
power dissipated by it in watts. Surveying our array of Ohm's Law equations, we
find two equations that give us answers from known quantities of voltage and
current:
R=— and P=l1E
Inserting the known quantities of voltage (E) and current (I) into these two
equations, we can determine circuit resistance (R) and power dissipation (P):
R-_lOV _so
2A
P= (2 A)(10 V)=20 W
For the circuit conditions of 10 volts and 2 amps, the resistor's resistance must
be 5 Q. If we were designing a circuit to operate at these values, we would have
to specify a resistor with a minimum power rating of 20 watts, or else it would
overheat and fail.
e REVIEW:
e Devices called resistors are built to provide precise amounts of resistance in
electric circuits. Resistors are rated both in terms of their resistance (ohms)
and their ability to dissipate heat energy (watts).
e Resistor resistance ratings cannot be determined from the physical size of
the resistor(s) in question, although approximate power ratings can. The
larger the resistor is, the more power it can safely dissipate without
suffering damage.
e Any device that performs some useful task with electric power is generally
known as a /Joad. Sometimes resistor symbols are used in schematic
diagrams to designate a non-specific load, rather than an actual resistor.
Nonlinear conduction
"Advances are made by answering questions. Discoveries are made by
questioning answers."
Bernhard Haisch, Astrophysicist
Ohm's Law is a simple and powerful mathematical tool for helping us analyze
electric circuits, but it has limitations, and we must understand these
limitations in order to properly apply it to real circuits. For most conductors,
resistance is a rather stable property, largely unaffected by voltage or current.
For this reason we can regard the resistance of many circuit components as a
constant, with voltage and current being directly related to each other.
For instance, our previous circuit example with the 3 QO lamp, we calculated
current through the circuit by dividing voltage by resistance (I=E/R). With an
18 volt battery, our circuit current was 6 amps. Doubling the battery voltage to
36 volts resulted in a doubled current of 12 amps. All of this makes sense, of
course, so long as the lamp continues to provide exactly the same amount of
friction (resistance) to the flow of electrons through it: 3 Q.
1=6A
However, reality is not always this simple. One of the phenomena explored ina
later chapter is that of conductor resistance changing with temperature. In an
incandescent lamp (the kind employing the principle of electric current heating
a thin filament of wire to the point that it glows white-hot), the resistance of the
filament wire will increase dramatically as it warms from room temperature to
operating temperature. If we were to increase the supply voltage in a real lamp
circuit, the resulting increase in current would cause the filament to increase
temperature, which would in turn increase its resistance, thus preventing
further increases in current without further increases in battery voltage.
Consequently, voltage and current do not follow the simple equation "I=E/R"
(with R assumed to be equal to 3 Q) because an incandescent lamp's filament
resistance does not remain stable for different currents.
The phenomenon of resistance changing with variations in temperature is one
shared by almost all metals, of which most wires are made. For most
applications, these changes in resistance are small enough to be ignored. In the
application of metal lamp filaments, the change happens to be quite large.
This is just one example of "nonlinearity" in electric circuits. It is by no means
the only example. A "linear" function in mathematics is one that tracks a
straight line when plotted on a graph. The simplified version of the lamp circuit
with a constant filament resistance of 3 O generates a plot like this:
|
(current)
E
(voltage)
The straight-line plot of current over voltage indicates that resistance is a
stable, unchanging value for a wide range of circuit voltages and currents. In an
"ideal" situation, this is the case. Resistors, which are manufactured to provide
a definite, stable value of resistance, behave very much like the plot of values
seen above. A mathematician would call their behavior "linear."
A more realistic analysis of a lamp circuit, however, over several different values
of battery voltage would generate a plot of this shape:
|
(current)
E
(voltage)
The plot is no longer a straight line. It rises sharply on the left, as voltage
increases from zero to a low level. As it progresses to the right we see the line
flattening out, the circuit requiring greater and greater increases in voltage to
achieve equal increases in current.
If we try to apply Ohm's Law to find the resistance of this lamp circuit with the
voltage and current values plotted above, we arrive at several different values.
We could say that the resistance here is nonlinear, increasing with increasing
current and voltage. The nonlinearity is caused by the effects of high
temperature on the metal wire of the lamp filament.
Another example of nonlinear current conduction is through gases such as air.
At standard temperatures and pressures, air is an effective insulator. However, if
the voltage between two conductors separated by an air gap is increased
greatly enough, the air molecules between the gap will become "ionized,"
having their electrons stripped off by the force of the high voltage between the
wires. Once ionized, air (and other gases) become good conductors of
electricity, allowing electron flow where none could exist prior to ionization. If
we were to plot current over voltage on a graph as we did with the lamp circuit,
the effect of ionization would be clearly seen as nonlinear:
|
(current)
(voltage) |
ionization potential
The graph shown is approximate for a small air gap (less than one inch). A
larger air gap would yield a higher ionization potential, but the shape of the I/E
curve would be very similar: practically no current until the ionization potential
was reached, then substantial conduction after that.
Incidentally, this is the reason lightning bolts exist as momentary surges rather
than continuous flows of electrons. The voltage built up between the earth and
clouds (or between different sets of clouds) must increase to the point where it
overcomes the ionization potential of the air gap before the air ionizes enough
to support a substantial flow of electrons. Once it does, the current will continue
to conduct through the ionized air until the static charge between the two
points depletes. Once the charge depletes enough so that the voltage falls
below another threshold point, the air de-ionizes and returns to its normal state
of extremely high resistance.
Many solid insulating materials exhibit similar resistance properties: extremely
high resistance to electron flow below some critical threshold voltage, then a
much lower resistance at voltages beyond that threshold. Once a solid
insulating material has been compromised by high-voltage breakdown, as it is
called, it often does not return to its former insulating state, unlike most gases.
It may insulate once again at low voltages, but its breakdown threshold voltage
will have been decreased to some lower level, which may allow breakdown to
occur more easily in the future. This is a common mode of failure in high-
voltage wiring: insulation damage due to breakdown. Such failures may be
detected through the use of special resistance meters employing high voltage
(1000 volts or more).
There are circuit components specifically engineered to provide nonlinear
resistance curves, one of them being the varistor. Commonly manufactured
from compounds such as zinc oxide or silicon carbide, these devices maintain
high resistance across their terminals until a certain "firing" or "breakdown"
voltage (equivalent to the "ionization potential" of an air gap) is reached, at
which point their resistance decreases dramatically. Unlike the breakdown of an
insulator, varistor breakdown is repeatable: that is, it is designed to withstand
repeated breakdowns without failure. A picture of a varistor is shown here:
Sa
There are also special gas-filled tubes designed to do much the same thing,
exploiting the very same principle at work in the ionization of air by a lightning
bolt.
Other electrical components exhibit even stranger current/voltage curves than
this. Some devices actually experience a decrease in current as the applied
voltage /ncreases. Because the slope of the current/voltage for this
phenomenon is negative (angling down instead of up as it progresses from left
to right), it is known as negative resistance.
region of
| negative
resistance
(current) aes cme,
'
E
(voltage)
Most notably, high-vacuum electron tubes known as tetrodes and
semiconductor diodes known as Esaki or tunne! diodes exhibit negative
resistance for certain ranges of applied voltage.
Ohm's Law is not very useful for analyzing the behavior of components like
these where resistance varies with voltage and current. Some have even
suggested that "Ohm's Law" should be demoted from the status of a "Law"
because it is not universal. It might be more accurate to call the equation
(R=E/I) a definition of resistance, befitting of a certain class of materials under
a narrow range of conditions.
For the benefit of the student, however, we will assume that resistances
specified in example circuits are stable over a wide range of conditions unless
otherwise specified. | just wanted to expose you to a little bit of the complexity
of the real world, lest | give you the false impression that the whole of electrical
phenomena could be summarized in a few simple equations.
e REVIEW:
e« The resistance of most conductive materials is stable over a wide range of
conditions, but this is not true of all materials.
e Any function that can be plotted on a graph as a straight line is called a
linear function. For circuits with stable resistances, the plot of current over
voltage is linear (I=E/R).
e In circuits where resistance varies with changes in either voltage or current,
the plot of current over voltage will be nonlinear (not a straight line).
¢ A varistor is a component that changes resistance with the amount of
voltage impressed across it. With little voltage across it, its resistance is
high. Then, at a certain "breakdown" or "firing" voltage, its resistance
decreases dramatically.
e Negative resistance is where the current through a component actually
decreases as the applied voltage across it is increased. Some electron tubes
and semiconductor diodes (most notably, the tetrode tube and the Esaki, or
tunnel diode, respectively) exhibit negative resistance over a certain range
of voltages.
Circuit wiring
So far, we've been analyzing single-battery, single-resistor circuits with no
regard for the connecting wires between the components, so long as a complete
circuit is formed. Does the wire length or circuit "shape" matter to our
calculations? Let's look at a couple of circuit configurations and find out:
1 2
Battery — Resistor
10 V 5Q
4 3
1 2
Battery — Resistor
10 V 52
4 3
When we draw wires connecting points in a circuit, we usually assume those
wires have negligible resistance. As such, they contribute no appreciable effect
to the overall resistance of the circuit, and so the only resistance we have to
contend with is the resistance in the components. In the above circuits, the only
resistance comes from the 5 Q resistors, so that is all we will consider in our
calculations. In real life, metal wires actually do have resistance (and so do
power sources!), but those resistances are generally so much smaller than the
resistance present in the other circuit components that they can be safely
ignored. Exceptions to this rule exist in power system wiring, where even very
small amounts of conductor resistance can create significant voltage drops
given normal (high) levels of current.
If connecting wire resistance is very little or none, we can regard the connected
points in a circuit as being e/ectrically common. That is, points 1 and 2 in the
above circuits may be physically joined close together or far apart, and it
doesn't matter for any voltage or resistance measurements relative to those
points. The same goes for points 3 and 4. It is as if the ends of the resistor were
attached directly across the terminals of the battery, so far as our Ohm's Law
calculations and voltage measurements are concerned. This is useful to know,
because it means you can re-draw a circuit diagram or re-wire a circuit,
shortening or lengthening the wires as desired without appreciably impacting
the circuit's function. All that matters is that the components attach to each
other in the same sequence.
It also means that voltage measurements between sets of "electrically common"
points will be the same. That is, the voltage between points 1 and 6 (directly
across the battery) will be the same as the voltage between points 3 and 4
(directly across the resistor). Take a close look at the following circuit, and try to
determine which points are common to each other:
1 2
Resistor
52
6 5
Here, we only have 2 components excluding the wires: the battery and the
resistor. Though the connecting wires take a convoluted path in forming a
complete circuit, there are several electrically common points in the electrons’
path. Points 1, 2, and 3 are all common to each other, because they're directly
connected together by wire. The same goes for points 4, 5, and 6.
The voltage between points 1 and 6 is 10 volts, coming straight from the
battery. However, since points 5 and 4 are common to 6, and points 2 and 3
common to 1, that same 10 volts also exists between these other pairs of
points:
Between points 1 and 4 = 10 volts
Between points 2 and 4 = 10 volts
Between points 3 and 4 = 10 volts (directly across the resistor)
Between points 1 and 5 = 10 volts
Between points 2 and 5 = 10 volts
Between points 3 and 5 = 10 volts
Between points 1 and 6 = 10 volts (directly across the battery)
Between points 2 and 6 = 10 volts
Between points 3 and 6 = 10 volts
Since electrically common points are connected together by (zero resistance)
wire, there is no significant voltage drop between them regardless of the
amount of current conducted from one to the next through that connecting
wire. Thus, if we were to read voltages between common points, we should show
(practically) zero:
Between points 1 and 2 = 0 volts Points 1, 2, and 3 are
Between points 2 and 3 = 0 volts electrically common
Between points 1 and 3 = 0 volts
Between points 4 and 5 = 0 volts Points 4, 5, and 6 are
Between points 5 and 6 = 0 volts electrically common
Between points 4 and 6 = 0 volts
This makes sense mathematically, too. With a 10 volt battery and a 5 OQ resistor,
the circuit current will be 2 amps. With wire resistance being zero, the voltage
drop across any continuous stretch of wire can be determined through Ohm's
Law as such:
E=1R
E=(2 A)(OQ)
E=0V
It should be obvious that the calculated voltage drop across any uninterrupted
length of wire in a circuit where wire is assumed to have zero resistance will
always be zero, no matter what the magnitude of current, since zero multiplied
by anything equals zero.
Because common points in a circuit will exhibit the same relative voltage and
resistance measurements, wires connecting common points are often labeled
with the same designation. This is not to say that the termina/ connection
points are labeled the same, just the connecting wires. Take this circuit as an
example:
1 wire #2 2
wire #2
10 V
Resistor
wire #1 32
6
wire #1
wire #1
Points 1, 2, and 3 are all common to each other, so the wire connecting point 1
to 2 is labeled the same (wire 2) as the wire connecting point 2 to 3 (wire 2). In
a real circuit, the wire stretching from point 1 to 2 may not even be the same
color or size as the wire connecting point 2 to 3, but they should bear the exact
same label. The same goes for the wires connecting points 6, 5, and 4.
Knowing that electrically common points have zero voltage drop between them
is a valuable troubleshooting principle. If | measure for voltage between points
in a circuit that are supposed to be common to each other, | should read zero. If,
however, | read substantial voltage between those two points, then | know with
certainty that they cannot be directly connected together. If those points are
supposed to be electrically common but they register otherwise, then | know
that there is an "open failure" between those points.
One final note: for most practical purposes, wire conductors can be assumed to
possess zero resistance from end to end. In reality, however, there will always
be some small amount of resistance encountered along the length of a wire,
unless its a Superconducting wire. Knowing this, we need to bear in mind that
the principles learned here about electrically common points are all valid toa
large degree, but not to an abso/ute degree. That is, the rule that electrically
common points are guaranteed to have zero voltage between them is more
accurately stated as such: electrically common points will have very //ttle
voltage dropped between them. That small, virtually unavoidable trace of
resistance found in any piece of connecting wire is bound to create a small
voltage across the length of it as current is conducted through. So long as you
understand that these rules are based upon /dea/ conditions, you won't be
perplexed when you come across some condition appearing to be an exception
to the rule.
« REVIEW:
e Connecting wires in a circuit are assumed to have zero resistance unless
otherwise stated.
e Wires in a circuit can be shortened or lengthened without impacting the
circuit's function -- all that matters is that the components are attached to
one another in the same sequence.
e Points directly connected together in a circuit by zero resistance (wire) are
considered to be electrically common.
e Electrically common points, with zero resistance between them, will have
zero voltage dropped between them, regardless of the magnitude of current
(ideally).
e The voltage or resistance readings referenced between sets of electrically
common points will be the same.
e These rules apply to /dea/ conditions, where connecting wires are assumed
to possess absolutely zero resistance. In real life this will probably not be
the case, but wire resistances should be low enough so that the general
principles stated here still hold.
Polarity of voltage drops
We can trace the direction that electrons will flow in the same circuit by starting
at the negative (-) terminal and following through to the positive (+) terminal of
the battery, the only source of voltage in the circuit. From this we can see that
the electrons are moving counter-clockwise, from point 6 to 5 to 4 to 3 to2 tol
and back to 6 again.
As the current encounters the 5 QO resistance, voltage is dropped across the
resistor's ends. The polarity of this voltage drop is negative (-) at point 4 with
respect to positive (+) at point 3. We can mark the polarity of the resistor's
voltage drop with these negative and positive symbols, in accordance with the
direction of current (whichever end of the resistor the current is entering is
negative with respect to the end of the resistor it is ex/ting:
2
current
current
Sere
Resistor
52
6 5
We could make our table of voltages a little more complete by marking the
polarity of the voltage for each pair of points in this circuit:
Between points 1 (+) and 4 (-) = 10 volts
Between points 2 (+) and 4 (-) = 10 volts
Between points 3 (+) and 4 (-) = 10 volts
Between points 1 (+) and 5 (-) = 10 volts
Between points 2 (+) and 5 (-) = 10 volts
Between points 3 (+) and 5 (-) = 10 volts
Between points 1 (+) and 6 (-) = 10 volts
Between points 2 (+) and 6 (-) = 10 volts
Between points 3 (+) and 6 (-) = 10 volts
While it might seem a little silly to document polarity of voltage drop in this
circuit, it is an important concept to master. It will be critically important in the
analysis of more complex circuits involving multiple resistors and/or batteries.
It should be understood that polarity has nothing to do with Ohm's Law: there
will never be negative voltages, currents, or resistance entered into any Ohm's
Law equations! There are other mathematical principles of electricity that do
take polarity into account through the use of signs (+ or -), but not Ohm's Law.
e REVIEW:
e The polarity of the voltage drop across any resistive component is
determined by the direction of electron flow through it: negative entering,
and positive exiting.
Computer simulation of electric circuits
Computers can be powerful tools if used properly, especially in the realms of
science and engineering. Software exists for the simulation of electric circuits
by computer, and these programs can be very useful in helping circuit
designers test ideas before actually building real circuits, saving much time and
money.
These same programs can be fantastic aids to the beginning student of
electronics, allowing the exploration of ideas quickly and easily with no
assembly of real circuits required. Of course, there is no substitute for actually
building and testing real circuits, but computer simulations certainly assist in
the learning process by allowing the student to experiment with changes and
see the effects they have on circuits. Throughout this book, I'll be incorporating
computer printouts from circuit simulation frequently in order to illustrate
important concepts. By observing the results of a computer simulation, a
student can gain an intuitive grasp of circuit behavior without the intimidation
of abstract mathematical analysis.
To simulate circuits on computer, | make use of a particular program called
SPICE, which works by describing a circuit to the computer by means of a listing
of text. In essence, this listing is a kind of computer program in itself, and must
adhere to the syntactical rules of the SPICE language. The computer is then
used to process, or "run," the SPICE program, which interprets the text listing
describing the circuit and outputs the results of its detailed mathematical
analysis, also in text form. Many details of using SPICE are described in volume
5 ("Reference") of this book series for those wanting more information. Here, I'll
just introduce the basic concepts and then apply SPICE to the analysis of these
simple circuits we've been reading about.
First, we need to have SPICE installed on our computer. As a free program, it is
commonly available on the internet for download, and in formats appropriate
for many different operating systems. In this book, | use one of the earlier
versions of SPICE: version 2G6, for its simplicity of use.
Next, we need a circuit for SPICE to analyze. Let's try one of the circuits
illustrated earlier in the chapter. Here is its schematic diagram:
This simple circuit consists of a battery and a resistor connected directly
together. We know the voltage of the battery (10 volts) and the resistance of
the resistor (5 Q), but nothing else about the circuit. If we describe this circuit to
SPICE, it should be able to tell us (at the very least), how much current we have
in the circuit by using Ohm's Law (I=E/R).
SPICE cannot directly understand a schematic diagram or any other form of
graphical description. SPICE is a text-based computer program, and demands
that a circuit be described in terms of its constituent components and
connection points. Each unique connection point in a circuit is described for
SPICE by a "node" number. Points that are electrically common to each other in
the circuit to be simulated are designated as such by sharing the same number.
It might be helpful to think of these numbers as "wire" numbers rather than
"node" numbers, following the definition given in the previous section. This is
how the computer knows what's connected to what: by the sharing of common
wire, or node, numbers. In our example circuit, we only have two "nodes," the
top wire and the bottom wire. SPICE demands there be a node 0 somewhere in
the circuit, so we'll label our wires O and 1:
In the above illustration, I've shown multiple "1" and "0" labels around each
respective wire to emphasize the concept of common points sharing common
node numbers, but still this is a graphic image, not a text description. SPICE
needs to have the component values and node numbers given to it in text form
before any analysis may proceed.
Creating a text file in a computer involves the use of a program called a text
editor. Similar to a word processor, a text editor allows you to type text and
record what you've typed in the form of a file stored on the computer's hard
disk. Text editors lack the formatting ability of word processors (no italic, bold,
or underlined characters), and this is a good thing, since programs such as
SPICE wouldn't know what to do with this extra information. If we want to create
a plain-text file, with absolutely nothing recorded except the keyboard
characters we select, a text editor is the tool to use.
If using a Microsoft operating system such as DOS or Windows, a couple of text
editors are readily available with the system. In DOS, there is the old Edit text
editing program, which may be invoked by typing edit at the command prompt.
In Windows (3.x/95/98/NT/Me/2k/XP), the Notepad text editor is your stock
choice. Many other text editing programs are available, and some are even free.
| happen to use a free text editor called Vim, and run it under both Windows 95
and Linux operating systems. It matters little which editor you use, so don't
worry if the screenshots in this section don't look like yours; the important
information here is what you type, not which editor you happen to use.
To describe this simple, two-component circuit to SPICE, | will begin by invoking
my text editor program and typing in a "title" line for the circuit:
File Edit Tools Syntax Buffers Window Help
QaBHOBS ve @RRARB SSATFEGPAZAA
My first circuit
"“circuiti.cir" 2L, 18C written
We can describe the battery to the computer by typing in a line of text starting
with the letter "v" (for "Voltage source"), identifying which wire each terminal of
the battery connects to (the node numbers), and the battery's voltage, like this:
File Edit Tools Syntax Buffers Window Help
a2HBS eg GORRBSSATOMPMOZA
“circuiti.cir" 3L, 300 written
This line of text tells SPICE that we have a voltage source connected between
nodes 1 and O, direct current (DC), 10 volts. That's all the computer needs to
know regarding the battery. Now we turn to the resistor: SPICE requires that
resistors be described with a letter "r," the numbers of the two nodes
(connection points), and the resistance in ohms. Since this is a computer
simulation, there is no need to specify a power rating for the resistor. That's one
nice thing about "virtual" components: they can't be harmed by excessive
voltages or currents!
File Edit Tools Syntax Buffers Window Help
abs de 2LRKRB SSATFOEDPAZA
“circuiti.cir" 4L, 38C written
Now, SPICE will know there is a resistor connected between nodes 1 and 0 with
a value of 5 QO. This very brief line of text tells the computer we have a resistor
("r") connected between the same two nodes as the battery (1 and 0), witha
resistance value of 5 Q.
If we add an .end statement to this collection of SPICE commands to indicate
the end of the circuit description, we will have all the information SPICE needs,
collected in one file and ready for processing. This circuit description,
comprised of lines of text in a computer file, is technically known as a neti/ist, or
deck:
File Edit Tools Syntax Buffers Window Help
aS de 2LARRB SSATFOEGPA?IA
"circuiti.cir" 5L, 430 written
Once we have finished typing all the necessary SPICE commands, we need to
"save" them to a file on the computer's hard disk so that SPICE has something
to reference to when invoked. Since this is my first SPICE netlist, I'll save it
under the filename "circuit1.cir" (the actual name being arbitrary). You may
elect to name your first SPICE netlist something completely different, just as
long as you don't violate any filename rules for your operating system, such as
using no more than 8+3 characters (eight characters in the name, and three
characters in the extension: 12345678.123) in DOS.
To invoke SPICE (tell it to process the contents of the circuit1.cir netlist file),
we have to exit from the text editor and access a command prompt (the "DOS
prompt" for Microsoft users) where we can enter text commands for the
computer's operating system to obey. This "primitive" way of invoking a
program may seem archaic to computer users accustomed to a "point-and-click"
graphical environment, but it is a very powerful and flexible way of doing
things. Remember, what you're doing here by using SPICE is a simple form of
computer programming, and the more comfortable you become in giving the
computer text-form commands to follow -- as opposed to simply clicking on icon
images using a mouse -- the more mastery you will have over your computer.
Once at a command prompt, type in this command, followed by an [Enter]
keystroke (this example uses the filename circuitl.cir; if you have chosen a
different filename for your netlist file, substitute it):
Spice < circuitl.cir
Here is how this looks on my computer (running the Linux operating system),
just before | press the [Enter] key:
spice < circuiti.cir
As soon as you press the [Enter] key to issue this command, text from SPICE's
output should scroll by on the computer screen. Here is a screenshot showing
what SPICE outputs on my computer (I've lengthened the "terminal" window to
show you the full text. With a normal-size terminal, the text easily exceeds one
page length):
10 dc 10
ridas
,end
LEXKKHKEKKEKKEKSES {O2 4XXXXKKKKKKAKKKKKKEKEKEK «Spice 5
EXKEKKKKKKAKKKERKEL, SESS SSS SSeS ISS SSS
Omy first circuit
stall signal bias solution ia=1)) @]=1
27.000 deg c
ES SSSSSSSCSSSCSSSCSSSSSSSCSSSCSSSSSSSSSSCSSSSSSSSOSSSSSSSSSSOSS SSS SSS SS OSS SSS SSS
KKKKAAAKEKKKKKKAEEERKE KKK KKAK KARE RRR EEK
node voltage
1)
yoltage source currents
name current
-? ,.000E+00
total power dissipation 2,00E+01 watts
(BG xxxaxeeal3 106 145exxxs
Oe input listing temperature = 27,000 deg c
WESSSSSSSSISSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS SSS SSS
O#error*: end card missing
M [tony@localhost “/liec/DC]$
SPICE begins with a reiteration of the netlist, complete with title line and .end
statement. About halfway through the simulation it displays the voltage at all
nodes with reference to node O. In this example, we only have one node other
than node 0, so it displays the voltage there: 10.0000 volts. Then it displays the
current through each voltage source. Since we only have one voltage source in
the entire circuit, it only displays the current through that one. In this case, the
source current is 2 amps. Due to a quirk in the way SPICE analyzes current, the
value of 2 amps is output as a negative (-) 2 amps.
The last line of text in the computer's analysis report is "total power
dissipation," which in this case is given as "2.,00E+01" watts: 2.00 x 101, or 20
watts. SPICE outputs most figures in scientific notation rather than normal
(fixed-point) notation. While this may seem to be more confusing at first, it is
actually less confusing when very large or very small numbers are involved. The
details of scientific notation will be covered in the next chapter of this book.
One of the benefits of using a "primitive" text-based program such as SPICE is
that the text files dealt with are extremely small compared to other file formats,
especially graphical formats used in other circuit simulation software. Also, the
fact that SPICE's output is plain text means you can direct SPICE's output to
another text file where it may be further manipulated. To do this, we re-issue a
command to the computer's operating system to invoke SPICE, this time
redirecting the output to a file I'll call "output.txt":
[tony@localhost “/liec/DC]$ spice < circuiti.cir > output.txt
SPICE will run "silently" this time, without the stream of text output to the
computer screen as before. A new file, output1.txt, will be created, which you
may open and change using a text editor or word processor. For this illustration,
I'll use the same text editor (Vim) to open this file:
File Edit Tools Syntax Buffers Window Help
ea SS Sue BUG iL Siehite Comes er UI, al
3/15/83 ex
input listing temperature =
¥ 10 dc 10
r1o5s
end
gnal bias solution
@
"output.txt" 54L, 13490
Now, | may freely edit this file, deleting any extraneous text (such as the
"banners" showing date and time), leaving only the text that | feel to be
pertinent to my circuit's analysis:
File Edit Tools Syntax Buffers Window
@®ORREB SSA TODA
total power d ?,00E+01 watts
"output.txt" 17L, > written
Once suitably edited and re-saved under the same filename (output.txt in this
example), the text may be pasted into any kind of document, "plain text" being
a universal file format for almost all computer systems. | can even include it
directly in the text of this book -- rather than as a "screenshot" graphic image --
like this:
my first circuit
v 10 dc 10
r105
.end
node voltage
( 1) 10.0000
voltage source currents
name current
V -2.000E+00
total power dissipation 2.00E+01 watts
Incidentally, this is the preferred format for text output from SPICE simulations
in this book series: as real text, not as graphic screenshot images.
To alter a component value in the simulation, we need to open up the netlist file
(circuitl.cir) and make the required modifications in the text description of the
circuit, then save those changes to the same filename, and re-invoke SPICE at
the command prompt. This process of editing and processing a text file is one
familiar to every computer programmer. One of the reasons | like to teach SPICE
is that it prepares the learner to think and work like a computer programmer,
which is good because computer programming is a significant area of advanced
electronics work.
Earlier we explored the consequences of changing one of the three variables in
an electric circuit (voltage, current, or resistance) using Ohm's Law to
mathematically predict what would happen. Now let's try the same thing using
SPICE to do the math for us.
If we were to triple the voltage in our last example circuit from 10 to 30 volts
and keep the circuit resistance unchanged, we would expect the current to
triple as well. Let's try this, re-naming our netlist file so as to not over-write the
first file. This way, we will have both versions of the circuit simulation stored on
the hard drive of our computer for future use. The following text listing is the
output of SPICE for this modified netlist, formatted as plain text rather than as a
graphic image of my computer screen:
second example circuit
v 10 dc 30
r105
.end
node voltage
( 1) 30.0000
voltage source currents
name current
V -6.000E+00
total power dissipation 1.80E+02 watts
Just as we expected, the current tripled with the voltage increase. Current used
to be 2 amps, but now it has increased to 6 amps (-6.000 x 10°). Note also how
the total power dissipation in the circuit has increased. It was 20 watts before,
but now is 180 watts (1.8 x 102). Recalling that power is related to the square of
the voltage (Joule's Law: P=E2/R), this makes sense. If we triple the circuit
voltage, the power should increase by a factor of nine (32 = 9). Nine times 20 is
indeed 180, so SPICE's output does indeed correlate with what we know about
power in electric circuits.
If we want to see how this simple circuit would respond over a wide range of
battery voltages, we can invoke some of the more advanced options within
SPICE. Here, I'll use the ".dc" analysis option to vary the battery voltage from 0
to 100 volts in 5 volt increments, printing out the circuit voltage and current at
every step. The lines in the SPICE netlist beginning with a star symbol ("*") are
comments. That is, they don't tell the computer to do anything relating to
circuit analysis, but merely serve as notes for any human being reading the
netlist text.
third example circuit
v10
r105
*the ".dc" statement tells spice to sweep the "v" supply
*voltage from 0 to 100 volts in 5 volt steps.
.dc v 0 100 5
.print dc v(1) i(v)
.end
The .print command in this SPICE netlist instructs SPICE to print columns of
numbers corresponding to each step in the analysis:
V i(v)
0.000E+00 0.000E+00
5 .Q000E+00 -1.000E+00
1.000E+01 -2.000E+00
1.500E+01 -3.000E+00
2.Q000E+01 -4.000E+00
2.500E+01 -5.000E+00
3.000E+01 -6.000E+00
3.500E+01 -7.000E+00
4.000E+01 -8.000E+00
4.500E+01 -9.000E+00
5.000E+01 -1.000E+01
5.500E+01 -1.100E+01
6.000E+01 -1.200E+01
6.500E+01 -1.300E+01
7.000E+01 -1.400E+01
7.500E+01 -1.500E+01
8.Q000E+01 -1.600E+01
8.500E+01 -1.700E+01
9.Q00E+01 -1.800E+01
9.500E+01 -1.900E+01
1. 000E+02 -2.000E+01
If | re-edit the netlist file, changing the .print command into a .plot command,
SPICE will output a crude graph made up of text characters:
0.000e+00 0.000e+00 . : +
5.000e+00 -1.000e+00 . : +
1.000e+01 -2.000e+00 . . +
1.500e+01 -3.000e+00 . . +
2.000e+01 -4.000e+00 . . +
2.500e+01 -5.000e+00 . : +
3.000e+01 -6.000e+00 . . +
3.500e+01 -7.000e+00 . F +
4.000e+01 -8.000e+00 . ‘ +
4.500e+01 -9.000e+00 . J+
5.000e+01 -1.000e+01 . +
5.500e+01 -1.100e+01 . +
6.000e+01 -1.200e+01 . +
6.500e+01 -1.300e+01 . +
7.000e+01 -1.400e+01 . +
7.500e+01 -1.500e+01 . +
8.000e+01 -1.600e+01 . +
8.500e+01 -1.700e+01 . +
9.000e+01 -1.800e+01 . +
9.500e+01 -1.900e+01 . +
1.000e+02 -2.000e+01 +
| |
sweep v#branch-2.00e+01 -1.00e+01 0.00e+00
In both output formats, the left-hand column of numbers represents the battery
voltage at each interval, as it increases from 0 volts to 100 volts, 5 volts ata
time. The numbers in the right-hand column indicate the circuit current for
each of those voltages. Look closely at those numbers and you'll see the
proportional relationship between each pair: Ohm's Law (I=E/R) holds true in
each and every case, each current value being 1/5 the respective voltage value,
because the circuit resistance is exactly 5 Q. Again, the negative numbers for
current in this SPICE analysis is more of a quirk than anything else. Just pay
attention to the absolute value of each number unless otherwise specified.
There are even some computer programs able to interpret and convert the non-
graphical data output by SPICE into a graphical plot. One of these programs is
called Nutmeg, and its output looks something like this:
Note how Nutmeg plots the resistor voltage v(1) (voltage between node 1 and
the implied reference point of node 0) as a line with a positive slope (from
lower-left to upper-right).
Whether or not you ever become proficient at using SPICE is not relevant to its
application in this book. All that matters is that you develop an understanding
for what the numbers mean in a SPICE-generated report. In the examples to
come, I'll do my best to annotate the numerical results of SPICE to eliminate
any confusion, and unlock the power of this amazing tool to help you
understand the behavior of electric circuits.
Contributors
Contributors to this chapter are listed in chronological order of their
contributions, from most recent to first. See Appendix 2 (Contributor List) for
dates and contact information.
Larry Cramblett (September 20, 2004): identified serious typographical error
in "Nonlinear conduction" section.
James Boorn (January 18, 2001): identified sentence structure error and
offered correction. Also, identified discrepancy in netlist syntax requirements
between SPICE version 2g6 and version 3f5.
Ben Crowell, Ph.D. (January 13, 2001): suggestions on improving the
technical accuracy of vo/tage and charge definitions.
Jason Starck (June 2000): HTML document formatting, which led to a much
better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt, under
the terms and conditions of the Design Science License.
—| | +4/l—
—/ | 4]
Lessons In Electric Circuits
-- Volume |
Chapter 3
ELECTRICAL SAFETY
The importance of electrical safety
Physiological effects of electricity
Shock current path
Ohm's Law (again!)
Safe practices
Emergency response
Common sources of hazard
Safe circuit design
Safe meter usage
Electric shock data
Contributors
Bibliography
The importance of electrical safety
With this lesson, | hope to avoid a common mistake found in
electronics textbooks of either ignoring or not covering with
sufficient detail the subject of electrical safety. | assume that
whoever reads this book has at least a passing interest in
actually working with electricity, and as such the topic of
safety is of paramount importance. Those authors, editors,
and publishers who fail to incorporate this subject into their
introductory texts are depriving the reader of life-saving
information.
As an instructor of industrial electronics, | soend a full week
with my students reviewing the theoretical and practical
aspects of electrical safety. The same textbooks | found
lacking in technical clarity | also found lacking in coverage
of electrical safety, hence the creation of this chapter. Its
placement after the first two chapters is intentional: in order
for the concepts of electrical safety to make the most sense,
some foundational knowledge of electricity is necessary.
Another benefit of including a detailed lesson on electrical
safety is the practical context it sets for basic concepts of
voltage, current, resistance, and circuit design. The more
relevant a technical topic can be made, the more likely a
student will be to pay attention and comprehend. And what
could be more relevant than application to your own
personal safety? Also, with electrical power being such an
everyday presence in modern life, almost anyone can relate
to the illustrations given in such a lesson. Have you ever
wondered why birds don't get shocked while resting on
power lines? Read on and find out!
Physiological effects of electricity
Most of us have experienced some form of electric "shock,"
where electricity causes our body to experience pain or
trauma. If we are fortunate, the extent of that experience is
limited to tingles or jolts of pain from static electricity
buildup discharging through our bodies. When we are
working around electric circuits capable of delivering high
power to loads, electric shock becomes a much more serious
issue, and pain is the least significant result of shock.
As electric current is conducted through a material, any
opposition to that flow of electrons (resistance) results in a
dissipation of energy, usually in the form of heat. This is the
most basic and easy-to-understand effect of electricity on
living tissue: current makes it heat up. If the amount of heat
generated is sufficient, the tissue may be burnt. The effect is
physiologically the same as damage caused by an open
flame or other high-temperature source of heat, except that
electricity has the ability to burn tissue well beneath the
Skin of a victim, even burning internal organs.
Another effect of electric current on the body, perhaps the
most significant in terms of hazard, regards the nervous
system. By "nervous system" | mean the network of special
cells in the body called "nerve cells" or "neurons" which
process and conduct the multitude of signals responsible for
regulation of many body functions. The brain, spinal cord,
and sensory/motor organs in the body function together to
allow it to sense, move, respond, think, and remember.
Nerve cells communicate to each other by acting as
“transducers:" creating electrical signals (very small
voltages and currents) in response to the input of certain
chemical compounds called neurotransmitters, and
releasing neurotransmitters when stimulated by electrical
signals. If electric current of sufficient magnitude is
conducted through a living creature (human or otherwise),
its effect will be to override the tiny electrical impulses
normally generated by the neurons, overloading the nervous
system and preventing both reflex and volitional signals
from being able to actuate muscles. Muscles triggered by an
external (shock) current will involuntarily contract, and
there's nothing the victim can do about it.
This problem is especially dangerous if the victim contacts
an energized conductor with his or her hands. The forearm
muscles responsible for bending fingers tend to be better
developed than those muscles responsible for extending
fingers, and so if both sets of muscles try to contract
because of an electric current conducted through the
person's arm, the "bending" muscles will win, clenching the
fingers into a fist. If the conductor delivering current to the
victim faces the palm of his or her hand, this clenching
action will force the hand to grasp the wire firmly, thus
worsening the situation by securing excellent contact with
the wire. The victim will be completely unable to let go of
the wire.
Medically, this condition of involuntary muscle contraction is
called tetanus. Electricians familiar with this effect of
electric shock often refer to an immobilized victim of electric
shock as being "froze on the circuit." Shock-induced tetanus
can only be interrupted by stopping the current through the
victim.
Even when the current is stopped, the victim may not regain
voluntary control over their muscles for a while, as the
neurotransmitter chemistry has been thrown into disarray.
This principle has been applied in "stun gun" devices such
as Tasers, which on the principle of momentarily shocking a
victim with a high-voltage pulse delivered between two
electrodes. A well-placed shock has the effect of temporarily
(a few minutes) immobilizing the victim.
Electric current is able to affect more than just skeletal
muscles in a shock victim, however. The diaphragm muscle
controlling the lungs, and the heart -- which is a muscle in
itself -- can also be "frozen" in a state of tetanus by electric
current. Even currents too low to induce tetanus are often
able to scramble nerve cell signals enough that the heart
cannot beat properly, sending the heart into a condition
known as fibrillation. A fibrillating heart flutters rather than
beats, and is ineffective at pumping blood to vital organs in
the body. In any case, death from asphyxiation and/or
Cardiac arrest will surely result from a strong enough electric
current through the body. Ironically, medical personnel use a
strong jolt of electric current applied across the chest of a
victim to "jump start" a fibrillating heart into a normal
beating pattern.
That last detail leads us into another hazard of electric
shock, this one peculiar to public power systems. Though
our initial study of electric circuits will focus almost
exclusively on DC (Direct Current, or electricity that moves
in a continuous direction in a circuit), modern power
systems utilize alternating current, or AC. The technical
reasons for this preference of AC over DC in power systems
are irrelevant to this discussion, but the special hazards of
each kind of electrical power are very important to the topic
of safety.
How AC affects the body depends largely on frequency.
Low-frequency (50- to 60-Hz) AC is used in US (60 Hz)
and European (50 Hz) households; it can be more
dangerous than high-frequency AC and is 3 to 5 times
more dangerous than DC of the same voltage and
amperage. Low-frequency AC produces extended muscle
contraction (tetany), which may freeze the hand to the
current's source, prolonging exposure. DC is most likely
to cause a single convulsive contraction, which often
forces the victim away from the current's source.
[MMOM]
AC's alternating nature has a greater tendency to throw the
heart's pacemaker neurons into a condition of fibrillation,
whereas DC tends to just make the heart stand still. Once
the shock current is halted, a "frozen" heart has a better
chance of regaining a normal beat pattern than a fibrillating
heart. This is why "defibrillating" equipment used by
emergency medics works: the jolt of current supplied by the
defibrillator unit is DC, which halts fibrillation and gives the
heart a chance to recover.
In either case, electric currents high enough to cause
involuntary muscle action are dangerous and are to be
avoided at all costs. In the next section, we'll take a look at
how such currents typically enter and exit the body, and
examine precautions against such occurrences.
REVIEW:
Electric current is capable of producing deep and severe
burns in the body due to power dissipation across the
body's electrical resistance.
Tetanus is the condition where muscles involuntarily
contract due to the passage of external electric current
through the body. When involuntary contraction of
muscles controlling the fingers causes a victim to be
unable to let go of an energized conductor, the victim is
said to be "froze on the circuit."
Diaphragm (lung) and heart muscles are similarly
affected by electric current. Even currents too small to
induce tetanus can be strong enough to interfere with
the heart's pacemaker neurons, causing the heart to
flutter instead of strongly beat.
Direct current (DC) is more likely to cause muscle
tetanus than alternating current (AC), making DC more
likely to "freeze" a victim in a shock scenario. However,
AC is more likely to cause a victim's heart to fibrillate,
which is a more dangerous condition for the victim after
the shocking current has been halted.
Shock current path
As we've already learned, electricity requires a complete
path (circuit) to continuously flow. This is why the shock
received from static electricity is only a momentary jolt: the
flow of electrons is necessarily brief when static charges are
equalized between two objects. Shocks of self-limited
duration like this are rarely hazardous.
Without two contact points on the body for current to enter
and exit, respectively, there is no hazard of shock. This is
why birds can safely rest on high-voltage power lines
without getting shocked: they make contact with the circuit
at only one point.
bird (not shocked)
High voltage
across source
and load
In order for electrons to flow through a conductor, there
must be a voltage present to motivate them. Voltage, as you
should recall, is a/ways relative between two points. There is
no such thing as voltage "on" or "at" a single point in the
circuit, and so the bird contacting a single point in the
above circuit has no voltage applied across its body to
establish a current through it. Yes, even though they rest on
two feet, both feet are touching the same wire, making them
electrically common. Electrically speaking, both of the bird's
feet touch the same point, hence there is no voltage
between them to motivate current through the bird's body.
This might lend one to believe that its impossible to be
shocked by electricity by only touching a single wire. Like
the birds, if we're sure to touch only one wire at a time, we'll
be safe, right? Unfortunately, this is not correct. Unlike birds,
people are usually standing on the ground when they
contact a "live" wire. Many times, one side of a power system
will be intentionally connected to earth ground, and so the
person touching a single wire is actually making contact
between two points in the circuit (the wire and earth
ground):
bird (not shocked)
person (SHOCKED!)
High voltage
across source
and load
path for current through the dirt
The ground symbol is that set of three horizontal bars of
decreasing width located at the lower-left of the circuit
shown, and also at the foot of the person being shocked. In
real life the power system ground consists of some kind of
metallic conductor buried deep in the ground for making
maximum contact with the earth. That conductor is
electrically connected to an appropriate connection point on
the circuit with thick wire. The victim's ground connection is
through their feet, which are touching the earth.
A few questions usually arise at this point in the mind of the
student:
e If the presence of a ground point in the circuit provides
an easy point of contact for someone to get shocked,
why have it in the circuit at all? Wouldn't a ground-less
circuit be safer?
e The person getting shocked probably isn't bare-footed. If
rubber and fabric are insulating materials, then why
aren't their shoes protecting them by preventing a
circuit from forming?
e How good of a conductor can dirt be? If you can get
shocked by current through the earth, why not use the
earth as a conductor in our power circuits?
In answer to the first question, the presence of an
intentional "grounding" point in an electric circuit is
intended to ensure that one side of it /s safe to come in
contact with. Note that if our victim in the above diagram
were to touch the bottom side of the resistor, nothing would
happen even though their feet would still be contacting
ground:
bird (not shocked)
ye
High voltage
across source
and load
person (not shocked)
s wm
~
~=—.
~_.
_—
=<.
-
Because the bottom side of the circuit is firmly connected to
ground through the grounding point on the lower-left of the
circuit, the lower conductor of the circuit is made e/ectrically
common with earth ground. Since there can be no voltage
between electrically common points, there will be no voltage
applied across the person contacting the lower wire, and
they will not receive a shock. For the same reason, the wire
connecting the circuit to the grounding rod/plates is usually
left bare (no insulation), so that any metal object it brushes
up against will similarly be electrically common with the
earth.
Circuit grounding ensures that at least one point in the
circuit will be safe to touch. But what about leaving a circuit
completely ungrounded? Wouldn't that make any person
touching just a single wire as safe as the bird sitting on just
one? Ideally, yes. Practically, no. Observe what happens with
no ground at all:
bird (not shocked)
Pe person (not shocked)
——| High voltage
——__ across source
—=— and load
Despite the fact that the person's feet are still contacting
ground, any single point in the circuit should be safe to
touch. Since there is no complete path (circuit) formed
through the person's body from the bottom side of the
voltage source to the top, there is no way for a current to be
established through the person. However, this could all
change with an accidental ground, such as a tree branch
touching a power line and providing connection to earth
ground:
bird (not shocked)
a “person (SHOCKED!)
—| High voltage
_— across source
— and load
|
T
!
'
'
'
!
'
'
'
'
'
'
'
!
'
accidental ground path through tree
(touching wire) completes the circuit
for shock current through the victim.
Such an accidental connection between a power system
conductor and the earth (ground) is called a ground fault.
Ground faults may be caused by many things, including dirt
buildup on power line insulators (creating a dirty-water path
for current from the conductor to the pole, and to the
ground, when it rains), ground water infiltration in buried
power line conductors, and birds landing on power lines,
bridging the line to the pole with their wings. Given the
many causes of ground faults, they tend to be
unpredicatable. In the case of trees, no one can guarantee
which wire their branches might touch. If a tree were to
brush up against the top wire in the circuit, it would make
the top wire safe to touch and the bottom one dangerous --
just the opposite of the previous scenario where the tree
contacts the bottom wire:
bird (not shocked)
Pa person (not shocked)
— High voltage
_— _ across source
— and load
— person (GHOCKED!)
accidental ground path through tree
fous wire) completes the circuit
for shock current through the victim.
With a tree branch contacting the top wire, that wire
becomes the grounded conductor in the circuit, electrically
common with earth ground. Therefore, there is no voltage
between that wire and ground, but full (high) voltage
between the bottom wire and ground. As mentioned
previously, tree branches are only one potential source of
ground faults in a power system. Consider an ungrounded
power system with no trees in contact, but this time with
two people touching single wires:
bird (not shocked)
— High voltage
=. - @OTOSs BOUIe: “Se = ele eee
—- and load i
With each person standing on the ground, contacting
different points in the circuit, a path for shock current is
made through one person, through the earth, and through
the other person. Even though each person thinks they're
safe in only touching a single point in the circuit, their
combined actions create a deadly scenario. In effect, one
person acts as the ground fault which makes it unsafe for
the other person. This is exactly why ungrounded power
systems are dangerous: the voltage between any point in
the circuit and ground (earth) is unpredictable, because a
ground fault could appear at any point in the circuit at any
time. The only character guaranteed to be safe in these
scenarios is the bird, who has no connection to earth ground
at all! By firmly connecting a designated point in the circuit
to earth ground ("grounding" the circuit), at least safety can
be assured at that one point. This is more assurance of
safety than having no ground connection at all.
In answer to the second question, rubber-soled shoes do
indeed provide some electrical insulation to help protect
someone from conducting shock current through their feet.
However, most common shoe designs are not intended to be
electrically "safe," their soles being too thin and not of the
right substance. Also, any moisture, dirt, or conductive salts
from body sweat on the surface of or permeated through the
soles of shoes will compromise what little insulating value
the shoe had to begin with. There are shoes specifically
made for dangerous electrical work, as well as thick rubber
mats made to stand on while working on live circuits, but
these special pieces of gear must be in absolutely clean, dry
condition in order to be effective. Suffice it to say, normal
footwear is not enough to guarantee protection against
electric shock from a power system.
Research conducted on contact resistance between parts of
the human body and points of contact (Such as the ground)
shows a wide range of figures (See end of chapter for
information on the source of this data):
e Hand or foot contact, insulated with rubber: 20 MQ
typical.
e Foot contact through leather shoe sole (dry): 100 kQ to
500 kQ
e Foot contact through leather shoe sole (wet): 5 kQ to 20
kO
As you Can see, not only is rubber a far better insulating
material than leather, but the presence of water in a porous
substance such as leather greatly reduces electrical
resistance.
In answer to the third question, dirt is not a very good
conductor (at least not when its dry!). It is too poor of a
conductor to support continuous current for powering a load.
However, as we will see in the next section, it takes very
little current to injure or kill a human being, so even the
poor conductivity of dirt is enough to provide a path for
deadly current when there is sufficient voltage available, as
there usually is in power systems.
Some ground surfaces are better insulators than others.
Asphalt, for instance, being oil-based, has a much greater
resistance than most forms of dirt or rock. Concrete, on the
other hand, tends to have fairly low resistance due to its
intrinsic water and electrolyte (conductive chemical)
content.
e REVIEW:
e Electric shock can only occur when contact is made
between two points of a circuit; when voltage is applied
across a victim's body.
e Power circuits usually have a designated point that is
“grounded:" firmly connected to metal rods or plates
buried in the dirt to ensure that one side of the circuit is
always at ground potential (zero voltage between that
point and earth ground).
e A ground fault is an accidental connection between a
circuit conductor and the earth (ground).
e Special, insulated shoes and mats are made to protect
persons from shock via ground conduction, but even
these pieces of gear must be in clean, dry condition to
be effective. Normal footwear is not good enough to
provide protection from shock by insulating its wearer
from the earth.
e Though dirt is a poor conductor, it can conduct enough
current to injure or kill a human being.
Ohm's Law (again! )
A common phrase heard in reference to electrical safety
goes something like this: "/t's not voltage that kills, its
current!" While there is an element of truth to this, there's
more to understand about shock hazard than this simple
adage. If voltage presented no danger, no one would ever
print and display signs saying: DANGER -- HIGH VOLTAGE!
The principle that "current kills" is essentially correct. It is
electric current that burns tissue, freezes muscles, and
fibrillates hearts. However, electric current doesn't just occur
on its own: there must be voltage available to motivate
electrons to flow through a victim. A person's body also
presents resistance to current, which must be taken into
account.
Taking Ohm's Law for voltage, current, and resistance, and
expressing it in terms of current for a given voltage and
resistance, we have this equation:
Ohm's Law
1 Voltage
Current = ————_+——_
Resistance
l=
The amount of current through a body is equal to the
amount of voltage applied between two points on that body,
divided by the electrical resistance offered by the body
between those two points. Obviously, the more voltage
available to cause electrons to flow, the easier they will flow
through any given amount of resistance. Hence, the danger
of high voltage: high voltage means potential for large
amounts of current through your body, which will injure or
kill you. Conversely, the more resistance a body offers to
current, the slower electrons will flow for any given amount
of voltage. Just how much voltage is dangerous depends on
how much total resistance is in the circuit to oppose the flow
of electrons.
Body resistance is not a fixed quantity. It varies from person
to person and from time to time. There's even a body fat
measurement technique based on a measurement of
electrical resistance between a person's toes and fingers.
Differing percentages of body fat give provide different
resistances: just one variable affecting electrical resistance
in the human body. In order for the technique to work
accurately, the person must regulate their fluid intake for
several hours prior to the test, indicating that body
hydration is another factor impacting the body's electrical
resistance.
Body resistance also varies depending on how contact is
made with the skin: is it from hand-to-hand, hand-to-foot,
foot-to-foot, hand-to-elbow, etc.? Sweat, being rich in salts
and minerals, is an excellent conductor of electricity for
being a liquid. So is blood, with its similarly high content of
conductive chemicals. Thus, contact with a wire made by a
Sweaty hand or open wound will offer much less resistance
to current than contact made by clean, dry skin.
Measuring electrical resistance with a sensitive meter, |
measure approximately 1 million ohms of resistance (1 MQ)
between my two hands, holding on to the meter's metal
probes between my fingers. The meter indicates less
resistance when | squeeze the probes tightly and more
resistance when | hold them loosely. Sitting here at my
computer, typing these words, my hands are clean and dry.
If | were working in some hot, dirty, industrial environment,
the resistance between my hands would likely be much less,
presenting less opposition to deadly current, and a greater
threat of electrical shock.
But how much current is harmful? The answer to that
question also depends on several factors. Individual body
chemistry has a significant impact on how electric current
affects an individual. Some people are highly sensitive to
current, experiencing involuntary muscle contraction with
shocks from static electricity. Others can draw large sparks
from discharging static electricity and hardly feel it, much
less experience a muscle spasm. Despite these differences,
approximate guidelines have been developed through tests
which indicate very little current being necessary to
manifest harmful effects (again, see end of chapter for
information on the source of this data). All current figures
given in milliamps (a milliamp is equal to 1/1000 of an
amp):
BODILY EFFECT DIRECT CURRENT (DC) 60 Hz AC 10 kHz
AC
Slight sensation Men = 1.0 mA 0.4 mA 7 mA
felt at hand(s) Women = 0.6 mA 0.3 mA 5 mA
Threshold of Men = 5.2 mA 1.1 mA 12 mA
perception Women = 3.5 mA 0.7 mA 8 mA
Painful, but Men = 62 mA 9 mA 55 mA
voluntary muscle Women = 41 mA 6 mA 37 mA
control maintained
Painful, unable Men = 76 mA 16 mA 75 mA
to let go of wires Women = 51 mA 10.5 mA 50 mA
Severe pain, Men = 90 mA 23 mA 94 mA
difficulty Women = 60 mA 15 mA 63 mA
breathing
Possible heart Men = 500 mA 100 mA
fibrillation Women = 500 mA 100 mA
after 3 seconds
"Hz" stands for the unit of Hertz, the measure of how rapidly
alternating current alternates, a measure otherwise known
as frequency. So, the column of figures labeled "60 Hz AC"
refers to current that alternates at a frequency of 60 cycles
(1 cycle = period of time where electrons flow one direction,
then the other direction) per second. The last column,
labeled "10 kHz AC," refers to alternating current that
completes ten thousand (10,000) back-and-forth cycles
each and every second.
Keep in mind that these figures are only approximate, as
individuals with different body chemistry may react
differently. It has been suggested that an across-the-chest
current of only 17 milliamps AC is enough to induce
fibrillation in a human subject under certain conditions.
Most of our data regarding induced fibrillation comes from
animal testing. Obviously, it is not practical to perform tests
of induced ventricular fibrillation on human subjects, so the
available data is sketchy. Oh, and in case you're wondering, |
have no idea why women tend to be more susceptible to
electric currents than men!
Suppose | were to place my two hands across the terminals
of an AC voltage source at 60 Hz (60 cycles, or alternations
back-and-forth, per second). How much voltage would be
necessary in this clean, dry state of skin condition to
produce a current of 20 milliamps (enough to cause me to
become unable to let go of the voltage source)? We can use
Ohm's Law (E=IR) to determine this:
E = (20 mA)(1 MQ)
E = 20,000 volts, or 20 kV
Bear in mind that this is a "best case" scenario (clean, dry
Skin) from the standpoint of electrical safety, and that this
figure for voltage represents the amount necessary to
induce tetanus. Far less would be required to cause a painful
shock! Also keep in mind that the physiological effects of
any particular amount of current can vary significantly from
person to person, and that these calculations are rough
estimates only.
With water sprinkled on my fingers to simulate sweat, | was
able to measure a hand-to-hand resistance of only 17,000
ohms (17 kQ). Bear in mind this is only with one finger of
each hand contacting a thin metal wire. Recalculating the
voltage required to cause a current of 20 milliamps, we
obtain this figure:
E = (20 mA)(17 kQ)
E = 340 volts
In this realistic condition, it would only take 340 volts of
potential from one of my hands to the other to cause 20
milliamps of current. However, it is still possible to receive a
deadly shock from less voltage than this. Provided a much
lower body resistance figure augmented by contact with a
ring (a band of gold wrapped around the circumference of
one's finger makes an exce//ent contact point for electrical
shock) or full contact with a large metal object such asa
pipe or metal handle of a tool, the body resistance figure
could drop as low as 1,000 ohms (1 kQ), allowing an even
lower voltage to present a potential hazard:
E = (20 mA)(1 kQ)
E = 20 volts
Notice that in this condition, 20 volts is enough to produce a
current of 20 milliamps through a person: enough to induce
tetanus. Remember, it has been suggested a current of only
17 milliamps may induce ventricular (heart) fibrillation. With
a hand-to-hand resistance of 1000 Q, it would only take 17
volts to create this dangerous condition:
E = (17 mA)(1 kQ)
E = 17 volts
Seventeen volts is not very much as far as electrical systems
are concerned. Granted, this is a "worst-case" scenario with
60 Hz AC voltage and excellent bodily conductivity, but it
does stand to show how little voltage may present a serious
threat under certain conditions.
The conditions necessary to produce 1,000 O of body
resistance don't have to be as extreme as what was
presented, either (sweaty skin with contact made on a gold
ring). Body resistance may decrease with the application of
voltage (especially if tetanus causes the victim to maintain a
tighter grip on a conductor) so that with constant voltage a
shock may increase in severity after initial contact. What
begins as a mild shock -- just enough to "freeze" a victim so
they can't let go -- may escalate into something severe
enough to kill them as their body resistance decreases and
Current correspondingly increases.
Research has provided an approximate set of figures for
electrical resistance of human contact points under different
conditions (see end of chapter for information on the source
of this data):
e Wire touched by finger: 40,000 © to 1,000,000 OQ dry,
4,000 © to 15,000 Q wet.
Wire held by hand: 15,000 Q to 50,000 Q dry, 3,000 Q to
5,000 © wet.
e Metal pliers held by hand: 5,000 Oto 10,000 Q dry,
1,000 Q to 3,000 Q wet.
e Contact with palm of hand: 3,000 Q to 8,000 QO dry,
1,000 Q to 2,000 © wet.
e 1.5 inch metal pipe grasped by one hand: 1,000 Q to
3,000 QO dry, 500 1 to 1,500 Q wet.
e 1.5 inch metal pipe grasped by two hands: 500 Q to
1,500 kQ dry, 250 0 to 750 Q wet.
e Hand immersed in conductive liquid: 200 Q to 500 Q.
e Foot immersed in conductive liquid: 100 QO to 300 Q.
Note the resistance values of the two conditions involving a
1.5 inch metal pipe. The resistance measured with two
hands grasping the pipe is exactly one-half the resistance of
one hand grasping the pipe.
1.5" metal pipe
With two hands, the bodily contact area is twice as great as
with one hand. This is an important lesson to learn: electrical
resistance between any contacting objects diminishes with
increased contact area, all other factors being equal. With
two hands holding the pipe, electrons have two, para/le/
routes through which to flow from the pipe to the body (or
vice-versa).
1.5” metal pipe
Two 2kQ contact points in "parallel"
with each other gives 1 kQ total
pipe-to-body resistance.
As we will see in a later chapter, para//e/ circuit pathways
always result in less overall resistance than any single
pathway considered alone.
In industry, 30 volts is generally considered to be a
conservative threshold value for dangerous voltage. The
cautious person should regard any voltage above 30 volts as
threatening, not relying on normal body resistance for
protection against shock. That being said, it is still an
excellent idea to keep one's hands clean and dry, and
remove all metal jewelry when working around electricity.
Even around lower voltages, metal jewelry can present a
hazard by conducting enough current to burn the skin if
brought into contact between two points in a circuit. Metal
rings, especially, have been the cause of more than a few
burnt fingers by bridging between points in a low-voltage,
high-current circuit.
Also, voltages lower than 30 can be dangerous if they are
enough to induce an unpleasant sensation, which may
cause you to jerk and accidently come into contact across a
higher voltage or some other hazard. | recall once working
on a automobile on a hot summer day. | was wearing shorts,
my bare leg contacting the chrome bumper of the vehicle as
| tightened battery connections. When | touched my metal
wrench to the positive (ungrounded) side of the 12 volt
battery, | could feel a tingling sensation at the point where
my leg was touching the bumper. The combination of firm
contact with metal and my sweaty skin made it possible to
feel a shock with only 12 volts of electrical potential.
Thankfully, nothing bad happened, but had the engine been
running and the shock felt at my hand instead of my leg, |
might have reflexively jerked my arm into the path of the
rotating fan, or dropped the metal wrench across the battery
terminals (producing /arge amounts of current through the
wrench with lots of accompanying sparks). This illustrates
another important lesson regarding electrical safety; that
electric current itself may be an indirect cause of injury by
causing you to jump or spasm parts of your body into harm's
way.
The path current takes through the human body makes a
difference as to how harmful it is. Current will affect
whatever muscles are in its path, and since the heart and
lung (diaphragm) muscles are probably the most critical to
one's survival, shock paths traversing the chest are the most
dangerous. This makes the hand-to-hand shock current path
a very likely mode of injury and fatality.
To guard against such an occurrence, it is advisable to only
use one hand to work on live circuits of hazardous voltage,
keeping the other hand tucked into a pocket so as to not
accidently touch anything. Of course, it is a/ways safer to
work on a circuit when it is unpowered, but this is not always
practical or possible. For one-handed work, the right hand is
generally preferred over the left for two reasons: most
people are right-handed (thus granting additional
coordination when working), and the heart is usually
situated to the left of center in the chest cavity.
For those who are left-handed, this advice may not be the
best. If such a person is sufficiently uncoordinated with their
right hand, they may be placing themselves in greater
danger by using the hand they're least comfortable with,
even if shock current through that hand might present more
of a hazard to their heart. The relative hazard between
shock through one hand or the other is probably less than
the hazard of working with less than optimal coordination,
so the choice of which hand to work with is best left to the
individual.
The best protection against shock from a live circuit is
resistance, and resistance can be added to the body through
the use of insulated tools, gloves, boots, and other gear.
Current in a circuit is a function of available voltage divided
by the tota/ resistance in the path of the flow. As we will
investigate in greater detail later in this book, resistances
have an additive effect when they're stacked up so that
there's only one path for electrons to flow:
—_—— |
Body resistance
—
Person in direct contact with voltage source:
current limited only by body resistance.
Becc
Now we'll see an equivalent circuit for a person wearing
insulated gloves and boots:
|
Glove resistance
Body resistance
Boot resistance
|-—_—>
Person wearing insulating gloves and boots:
current now limited by total circuit resistance.
E
- Ricay +R
l=
R
glove boot
Because electric current must pass through the boot and the
body and the glove to complete its circuit back to the
battery, the combined total (sum) of these resistances
opposes the flow of electrons to a greater degree than any of
the resistances considered individually.
Safety is one of the reasons electrical wires are usually
covered with plastic or rubber insulation: to vastly increase
the amount of resistance between the conductor and
whoever or whatever might contact it. Unfortunately, it
would be prohibitively expensive to enclose power line
conductors in sufficient insulation to provide safety in case
of accidental contact, so safety is maintained by keeping
those lines far enough out of reach so that no one can
accidently touch them.
e REVIEW:
e Harm to the body is a function of the amount of shock
current. Higher voltage allows for the production of
higher, more dangerous currents. Resistance opposes
current, making high resistance a good protective
measure against shock.
e Any voltage above 30 is generally considered to be
capable of delivering dangerous shock currents.
e Metal jewelry is definitely bad to wear when working
around electric circuits. Rings, watchbands, necklaces,
bracelets, and other such adornments provide excellent
electrical contact with your body, and can conduct
current themselves enough to produce skin burns, even
with low voltages.
Low voltages can still be dangerous even if they're too
low to directly cause shock injury. They may be enough
to startle the victim, causing them to jerk back and
contact something more dangerous in the near vicinity.
When necessary to work on a "live" circuit, it is best to
perform the work with one hand so as to prevent a
deadly hand-to-hand (through the chest) shock current
path.
Safe practices
If at all possible, shut off the power to a circuit before
performing any work on it. You must secure all sources of
harmful energy before a system may be considered safe to
work on. In industry, securing a circuit, device, or system in
this condition is commonly known as placing it in a Zero
Energy State. The focus of this lesson is, of course, electrical
safety. However, many of these principles apply to non-
electrical systems as well.
Securing something in a Zero Energy State means ridding it
of any sort of potential or stored energy, including but not
limited to:
Dangerous voltage
Spring pressure
Hydraulic (liquid) pressure
Pneumatic (air) pressure
Suspended weight
Chemical energy (flammable or otherwise reactive
substances)
e Nuclear energy (radioactive or fissile substances)
Voltage by its very nature is a manifestation of potential
energy. In the first chapter | even used elevated liquid as an
analogy for the potential energy of voltage, having the
Capacity (potential) to produce current (flow), but not
necessarily realizing that potential until a suitable path for
flow has been established, and resistance to flow is
overcome. A pair of wires with high voltage between them
do not look or sound dangerous even though they harbor
enough potential energy between them to push deadly
amounts of current through your body. Even though that
voltage isn't presently doing anything, it has the potential
to, and that potential must be neutralized before it is safe to
physically contact those wires.
All properly designed circuits have "disconnect" switch
mechanisms for securing voltage from a circuit. Sometimes
these "disconnects" serve a dual purpose of automatically
opening under excessive current conditions, in which case
we call them "circuit breakers." Other times, the
disconnecting switches are strictly manually-operated
devices with no automatic function. In either case, they are
there for your protection and must be used properly. Please
note that the disconnect device should be separate from the
regular switch used to turn the device on and off. Itisa
safety switch, to be used only for securing the system ina
Zero Energy State:
Disconnect On/Off
switch switch
rf
Power i a
source —— Load
With the disconnect switch in the "open" position as shown
(no continuity), the circuit is broken and no current will
exist. There will be zero voltage across the load, and the full
voltage of the source will be dropped across the open
contacts of the disconnect switch. Note how there is no need
for a disconnect switch in the lower conductor of the circuit.
Because that side of the circuit is firmly connected to the
earth (ground), it is electrically common with the earth and
is best left that way. For maximum safety of personnel
working on the load of this circuit, a temporary ground
connection could be established on the top side of the load,
to ensure that no voltage could ever be dropped across the
load:
Disconnect On/Off
switch switch
Power =
— temporar Load
source — ground”
With the temporary ground connection in place, both sides
of the load wiring are connected to ground, securing a Zero
Energy State at the load.
Since a ground connection made on both sides of the load is
electrically equivalent to short-circuiting across the load
with a wire, that is another way of accomplishing the same
goal of maximum safety:
Disconnect On/Off
switch
switch
Power i saci
source ——
*
zero voltage
ensured here
ms
Load
temporary
shorting wire
Either way, both sides of the load will be electrically
common to the earth, allowing for no voltage (potential
energy) between either side of the load and the ground
people stand on. This technique of temporarily grounding
conductors in a de-energized power system is very common
in maintenance work performed on high voltage power
distribution systems.
A further benefit of this precaution is protection against the
possibility of the disconnect switch being closed (turned
"on" so that circuit continuity is established) while people
are still contacting the load. The temporary wire connected
across the load would create a short-circuit when the
disconnect switch was closed, immediately tripping any
overcurrent protection devices (circuit breakers or fuses) in
the circuit, which would shut the power off again. Damage
may very well be sustained by the disconnect switch if this
were to happen, but the workers at the load are kept safe.
It would be good to mention at this point that overcurrent
devices are not intended to provide protection against
electric shock. Rather, they exist solely to protect
conductors from overheating due to excessive currents. The
temporary shorting wires just described would indeed cause
any overcurrent devices in the circuit to "trip" if the
disconnect switch were to be closed, but realize that electric
shock protection is not the intended function of those
devices. Their primary function would merely be leveraged
for the purpose of worker protection with the shorting wire in
place.
Since it is obviously important to be able to secure any
disconnecting devices in the open (off) position and make
sure they stay that way while work is being done on the
circuit, there is need for a structured safety system to be put
into place. Such a system is commonly used in industry and
it is called Lock-out/Tag-out.
A lock-out/tag-out procedure works like this: all individuals
working on a secured circuit have their own personal
padlock or combination lock which they set on the control
lever of a disconnect device prior to working on the system.
Additionally, they must fill out and sign a tag which they
hang from their lock describing the nature and duration of
the work they intend to perform on the system. If there are
multiple sources of energy to be "locked out" (multiple
disconnects, both electrical and mechanical energy sources
to be secured, etc.), the worker must use as many of his or
her locks as necessary to secure power from the system
before work begins. This way, the system is maintained ina
Zero Energy State until every last lock is removed from all
the disconnect and shutoff devices, and that means every
last worker gives consent by removing their own personal
locks. If the decision is made to re-energize the system and
one person's lock(s) still remain in place after everyone
present removes theirs, the tag(s) will show who that person
is and what it is they're doing.
Even with a good lock-out/tag-out safety program in place,
there is still need for diligence and common-sense
precaution. This is especially true in industrial settings
where a multitude of people may be working on a device or
system at once. Some of those people might not know about
proper lock-out/tag-out procedure, or might know about it
but are too complacent to follow it. Don't assume that
everyone has followed the safety rules!
After an electrical system has been locked out and tagged
with your own personal lock, you must then double-check to
see if the voltage really has been secured in a zero state.
One way to check is to see if the machine (or whatever it is
that's being worked on) will start up if the Start switch or
button is actuated. If it starts, then you know you haven't
successfully secured the electrical power from it.
Additionally, you should a/ways check for the presence of
dangerous voltage with a measuring device before actually
touching any conductors in the circuit. To be safest, you
should follow this procedure of checking, using, and then
checking your meter:
e Check to see that your meter indicates properly ona
known source of voltage.
e Use your meter to test the locked-out circuit for any
dangerous voltage.
e Check your meter once more on a known source of
voltage to see that it still indicates as it should.
While this may seem excessive or even paranoid, it isa
proven technique for preventing electrical shock. | once had
a meter fail to indicate voltage when it should have while
checking a circuit to see if it was "dead." Had | not used
other means to check for the presence of voltage, | might
not be alive today to write this. There's always the chance
that your voltage meter will be defective just when you need
it to check for a dangerous condition. Following these steps
will help ensure that you're never misled into a deadly
situation by a broken meter.
Finally, the electrical worker will arrive at a point in the
safety check procedure where it is deemed safe to actually
touch the conductor(s). Bear in mind that after all of the
precautionary steps have taken, it is still possible (although
very unlikely) that a dangerous voltage may be present. One
final precautionary measure to take at this point is to make
momentary contact with the conductor(s) with the back of
the hand before grasping it or a metal tool in contact with it.
Why? If, for some reason there is still voltage present
between that conductor and earth ground, finger motion
from the shock reaction (clenching into a fist) will break
contact with the conductor. Please note that this is
absolutely the /ast step that any electrical worker should
ever take before beginning work on a power system, and
should never be used as an alternative method of checking
for dangerous voltage. If you ever have reason to doubt the
trustworthiness of your meter, use another meter to obtain a
“second opinion."
REVIEW:
Zero Energy State: When a circuit, device, or system has
been secured so that no potential energy exists to harm
someone working on it.
Disconnect switch devices must be present in a properly
designed electrical system to allow for convenient
readiness of a Zero Energy State.
Temporary grounding or shorting wires may be
connected to a load being serviced for extra protection
to personnel working on that load.
Lock-out/Tag-out works like this: when working ona
system in a Zero Energy State, the worker places a
personal padlock or combination lock on every energy
disconnect device relevant to his or her task on that
system. Also, a tag is hung on every one of those locks
describing the nature and duration of the work to be
done, and who is doing it.
Always verify that a circuit has been secured in a Zero
Energy State with test equipment after "locking it out."
Be sure to test your meter before and after checking the
circuit to verify that it is working properly.
When the time comes to actually make contact with the
conductor(s) of a Supposedly dead power system, do so
first with the back of one hand, so that if a shock should
occur, the muscle reaction will pull the fingers away
from the conductor.
Emergency response
Despite lock-out/tag-out procedures and multiple repetitions
of electrical safety rules in industry, accidents still do occur.
The vast majority of the time, these accidents are the result
of not following proper safety procedures. But however they
may occur, they still do happen, and anyone working around
electrical systems should be aware of what needs to be done
for a victim of electrical shock.
If you see someone lying unconscious or "froze on the
circuit," the very first thing to do is shut off the power by
opening the appropriate disconnect switch or circuit breaker.
If someone touches another person being shocked, there
may be enough voltage dropped across the body of the
victim to shock the would-be rescuer, thereby "freezing" two
people instead of one. Don't be a hero. Electrons don't
respect heroism. Make sure the situation is safe for you to
step into, or elSe you wi// be the next victim, and nobody will
benefit from your efforts.
One problem with this rule is that the source of power may
not be known, or easily found in time to save the victim of
shock. If a shock victim's breathing and heartbeat are
paralyzed by electric current, their survival time is very
limited. If the shock current is of sufficient magnitude, their
flesh and internal organs may be quickly roasted by the
power the current dissipates as it runs through their body.
If the power disconnect switch cannot be located quickly
enough, it may be possible to dislodge the victim from the
circuit they're frozen on to by prying them or hitting them
away with a dry wooden board or piece of nonmetallic
conduit, common items to be found in industrial
construction scenes. Another item that could be used to
safely drag a "frozen" victim away from contact with power
is an extension cord. By looping a cord around their torso
and using it as a rope to pull them away from the circuit,
their grip on the conductor(s) may be broken. Bear in mind
that the victim will be holding on to the conductor with all
their strength, so pulling them away probably won't be easy!
Once the victim has been safely disconnected from the
source of electric power, the immediate medical concerns for
the victim should be respiration and circulation (breathing
and pulse). If the rescuer is trained in CPR, they should
follow the appropriate steps of checking for breathing and
pulse, then applying CPR as necessary to keep the victim's
body from deoxygenating. The cardinal rule of CPR is to
keep going until you have been relieved by qualified
personnel.
If the victim is conscious, it is best to have them lie still until
qualified emergency response personnel arrive on the scene.
There is the possibility of the victim going into a state of
physiological shock -- a condition of insufficient blood
circulation different from electrical shock -- and so they
should be kept as warm and comfortable as possible. An
electrical shock insufficient to cause immediate interruption
of the heartbeat may be strong enough to cause heart
irregularities or a heart attack up to several hours later, so
the victim should pay close attention to their own condition
after the incident, ideally under supervision.
e REVIEW:
e A person being shocked needs to be disconnected from
the source of electrical power. Locate the disconnecting
switch/breaker and turn it off. Alternatively, if the
disconnecting device cannot be located, the victim can
be pried or pulled from the circuit by an insulated object
such as a dry wood board, piece of nonmetallic conduit,
or rubber electrical cord.
e Victims need immediate medical response: check for
breathing and pulse, then apply CPR as necessary to
maintain oxygenation.
e If a victim is still conscious after having been shocked,
they need to be closely monitored and cared for until
trained emergency response personnel arrive. There is
danger of physiological shock, so keep the victim warm
and comfortable.
e Shock victims may suffer heart trouble up to several
hours after being shocked. The danger of electric shock
does not end after the immediate medical attention.
Common sources of hazard
Of course there is danger of electrical shock when directly
performing manual work on an electrical power system.
However, electric shock hazards exist in many other places,
thanks to the widespread use of electric power in our lives.
As we Saw earlier, skin and body resistance has a lot to do
with the relative hazard of electric circuits. The higher the
body's resistance, the less likely harmful current will result
from any given amount of voltage. Conversely, the lower the
body's resistance, the more likely for injury to occur from the
application of a voltage.
The easiest way to decrease skin resistance is to get it wet.
Therefore, touching electrical devices with wet hands, wet
feet, or especially in a sweaty condition (salt water is a much
better conductor of electricity than fresh water) is
dangerous. In the household, the bathroom is one of the
more likely places where wet people may contact electrical
appliances, and so shock hazard is a definite threat there.
Good bathroom design will locate power receptacles away
from bathtubs, showers, and sinks to discourage the use of
appliances nearby. Telephones that plug into a wall socket
are also sources of hazardous voltage (the open circuit
voltage is 48 volts DC, and the ringing signal is 150 volts AC
-- remember that any voltage over 30 is considered
potentially dangerous!). Appliances such as telephones and
radios should never, ever be used while sitting in a bathtub.
Even battery-powered devices should be avoided. Some
battery-operated devices employ voltage-increasing
circuitry capable of generating lethal potentials.
Swimming pools are another source of trouble, since people
often operate radios and other powered appliances nearby.
The National Electrical Code requires that special shock-
detecting receptacles called Ground-Fault Current
Interrupting (GFI or GFCI) be installed in wet and outdoor
areas to help prevent shock incidents. More on these devices
in a later section of this chapter. These special devices have
no doubt saved many lives, but they can be no substitute for
common sense and diligent precaution. As with firearms, the
best "safety" is an informed and conscientious operator.
Extension cords, so commonly used at home and in industry,
are also sources of potential hazard. All cords should be
regularly inspected for abrasion or cracking of insulation,
and repaired immediately. One sure method of removing a
damaged cord from service is to unplug it from the
receptacle, then cut off that plug (the "male" plug) with a
pair of side-cutting pliers to ensure that no one can use it
until it is fixed. This is important on jobsites, where many
people share the same equipment, and not all people there
may be aware of the hazards.
Any power tool showing evidence of electrical problems
should be immediately serviced as well. I've heard several
horror stories of people who continue to work with hand
tools that periodically shock them. Remember, e/ectricity
can kill, and the death it brings can be gruesome. Like
extension cords, a bad power tool can be removed from
service by unplugging it and cutting off the plug at the end
of the cord.
Downed power lines are an obvious source of electric shock
hazard and should be avoided at all costs. The voltages
present between power lines or between a power line and
earth ground are typically very high (2400 volts being one
of the lowest voltages used in residential distribution
systems). If a power line is broken and the metal conductor
falls to the ground, the immediate result will usually be a
tremendous amount of arcing (Sparks produced), often
enough to dislodge chunks of concrete or asphalt from the
road surface, and reports rivaling that of a rifle or shotgun.
To come into direct contact with a downed power line is
almost sure to cause death, but other hazards exist which
are not so obvious.
When a line touches the ground, current travels between
that downed conductor and the nearest grounding point in
the system, thus establishing a circuit:
TT TTT
downed power al
a
current through the earth
The earth, being a conductor (if only a poor one), will
conduct current between the downed line and the nearest
system ground point, which will be some kind of conductor
buried in the ground for good contact. Being that the earth
iS a much poorer conductor of electricity than the metal
cables strung along the power poles, there will be
substantial voltage dropped between the point of cable
contact with the ground and the grounding conductor, and
little voltage dropped along the length of the cabling (the
following figures are very approximate):
= 2390 ok
' volts '
- - >
current through the earth
If the distance between the two ground contact points (the
downed cable and the system ground) is small, there will be
substantial voltage dropped along short distances between
the two points. Therefore, a person standing on the ground
between those two points will be in danger of receiving an
electric shock by intercepting a voltage between their two
feet!
i person downed power li
(SHOCKED!)
- ~ - -
current through the earth —-— ~+—250 wlts
2390
volts
Again, these voltage figures are very approximate, but they
serve to illustrate a potential hazard: that a person can
become a victim of electric shock from a downed power line
without even coming into contact with that line!
One practical precaution a person could take if they see a
power line falling towards the ground is to only contact the
ground at one point, either by running away (when you run,
only one foot contacts the ground at any given time), or if
there's nowhere to run, by standing on one foot. Obviously,
if there's somewhere safer to run, running is the best option.
By eliminating two points of contact with the ground, there
will be no chance of applying deadly voltage across the
body through both legs.
e REVIEW:
e Wet conditions increase risk of electric shock by
lowering skin resistance.
e Immediately replace worn or damaged extension cords
and power tools. You can prevent innocent use of a bad
cord or tool by cutting the male plug off the cord (while
its unplugged from the receptacle, of course).
e Power lines are very dangerous and should be avoided
at all costs. If you see a line about to hit the ground,
stand on one foot or run (only one foot contacting the
ground) to prevent shock from voltage dropped across
the ground between the line and the system ground
point.
Safe circuit design
As we Saw earlier, a power system with no secure connection
to earth ground is unpredictable from a safety perspective:
there's no way to guarantee how much or how little voltage
will exist between any point in the circuit and earth ground.
By grounding one side of the power system's voltage source,
at least one point in the circuit can be assured to be
electrically common with the earth and therefore present no
shock hazard. In a simple two-wire electrical power system,
the conductor connected to ground is called the neutra/, and
the other conductor is called the hot, also known as the /ive
or the active:
"Hot" conductor
Source — Load
— "Neutral" conductor
Ground point
As far as the voltage source and load are concerned,
grounding makes no difference at all. It exists purely for the
sake of personnel safety, by guaranteeing that at least one
point in the circuit will be safe to touch (zero voltage to
ground). The "Hot" side of the circuit, named for its potential
for shock hazard, will be dangerous to touch unless voltage
is secured by proper disconnection from the source (ideally,
using a systematic lock-out/tag-out procedure).
This imbalance of hazard between the two conductors in a
simple power circuit is important to understand. The
following series of illustrations are based on common
household wiring systems (using DC voltage sources rather
than AC for simplicity).
If we take a look at a simple, household electrical appliance
such as a toaster with a conductive metal case, we can see
that there should be no shock hazard when it is operating
properly. The wires conducting power to the toaster's
heating element are insulated from touching the metal case
(and each other) by rubber or plastic.
Electrical
"Hot" appliance
to | pt
Source —
120 V i
"Neutral" metal ae }
Ground point
no voltage
between case
and ground
a
However, if one of the wires inside the toaster were to
accidently come in contact with the metal case, the case will
be made electrically common to the wire, and touching the
case will be just as hazardous as touching the wire bare.
Whether or not this presents a shock hazard depends on
which wire accidentally touches:
accidental
contact
"Hot" re
| | plug
Source —
120 V
= “Neutral” voltage between
Ground point case and ground!
If the "hot" wire contacts the case, it places the user of the
toaster in danger. On the other hand, if the neutral wire
contacts the case, there is no danger of shock:
"Hot”
Source —_
120 V Tt
— "Neutral"
Ground point no voltage between
case and ground!
accidental
contact
To help ensure that the former failure is less likely than the
latter, engineers try to design appliances in such a way as to
minimize hot conductor contact with the case. Ideally, of
course, you don't want either wire accidently coming in
contact with the conductive case of the appliance, but there
are uSually ways to design the layout of the parts to make
accidental contact less likely for one wire than for the other.
However, this preventative measure is effective only if
power plug polarity can be guaranteed. If the plug can be
reversed, then the conductor more likely to contact the case
might very well be the "hot" one:
[Le
Lo | pus
Source — accidental
es
120 V tL _J-
"Neutral"
voltage between
Ground point case and ground!
Appliances designed this way usually come with "polarized"
plugs, one prong of the plug being slightly narrower than
the other. Power receptacles are also designed like this, one
slot being narrower than the other. Consequently, the plug
cannot be inserted "backwards," and conductor identity
inside the appliance can be guaranteed. Remember that this
has no effect whatsoever on the basic function of the
appliance: its strictly for the sake of user safety.
Some engineers address the safety issue simply by making
the outside case of the appliance nonconductive. Such
appliances are called double-insulated, since the insulating
case serves as a second layer of insulation above and
beyond that of the conductors themselves. If a wire inside
the appliance accidently comes in contact with the case,
there is no danger presented to the user of the appliance.
Other engineers tackle the problem of safety by maintaining
a conductive case, but using a third conductor to firmly
connect that case to ground:
"Hot"
3-prong
| | plug
Source —
120 V a
"Neutral" |
Grounded case
ensures zero
voltage between
a case and ground
Ground point |
"Ground"
=e
The third prong on the power cord provides a direct
electrical connection from the appliance case to earth
ground, making the two points electrically common with
each other. If they're electrically common, then there cannot
be any voltage dropped between them. At least, that's how
it is supposed to work. If the hot conductor accidently
touches the metal appliance case, it will create a direct
short-circuit back to the voltage source through the ground
wire, tripping any overcurrent protection devices. The user
of the appliance will remain safe.
This is why its so important never to cut the third prong offa
power plug when trying to fit it into a two-prong receptacle.
If this is done, there will be no grounding of the appliance
case to keep the user(s) safe. The appliance will still function
properly, but if there is an internal fault bringing the hot
wire in contact with the case, the results can be deadly. If a
two-prong receptacle must be used, a two- to three-prong
receptacle adapter can be installed with a grounding wire
attached to the receptacle's grounded cover screw. This will
maintain the safety of the grounded appliance while
plugged in to this type of receptacle.
Electrically safe engineering doesn't necessarily end at the
load, however. A final safeguard against electrical shock can
be arranged on the power supply side of the circuit rather
than the appliance itself. This safeguard is called grouna-
fault detection, and it works like this:
Source — _—
120 V fi as aoe
—= “Neutral” no voltage
Ground point between case
and ground
In a properly functioning appliance (Shown above), the
current measured through the hot conductor should be
exactly equal to the current through the neutral conductor,
because there's only one path for electrons to flow in the
circuit. With no fault inside the appliance, there is no
connection between circuit conductors and the person
touching the case, and therefore no shock.
If, however, the hot wire accidently contacts the metal case,
there will be current through the person touching the case.
The presence of a shock current will be manifested as a
difference of current between the two power conductors at
the receptacle:
accidental
a
| | (mo ie!
Source —
120 V
ae
=_ "Neutral"
T Shock current}
! | Shock current
Shock current —-
This difference in current between the "hot" and "neutral"
conductors will only exist if there is current through the
ground connection, meaning that there is a fault in the
system. Therefore, such a current difference can be used as
a way to detect a fault condition. If a device is set up to
measure this difference of current between the two power
conductors, a detection of current imbalance can be used to
trigger the opening of a disconnect switch, thus cutting
power off and preventing serious shock:
"Hot"
ae ee
Source —
120 V arpa
— "Neutral"
: switches open automatically
if the difference between the
two currents becomes too =
great.
Such devices are called Ground Fault Current Interruptors, or
GFCls for short. Outside North America, the GFCI is variously
known as a Safety switch, a residual current device (RCD), an
RCBO or RCD/MCB if combined with a miniature circuit
breaker, or earth leakage circuit breaker (ELCB). They are
compact enough to be built into a power receptacle. These
receptacles are easily identified by their distinctive "Test"
and "Reset" buttons. The big advantage with using this
approach to ensure safety is that it works regardless of the
appliance's design. Of course, using a double-insulated or
grounded appliance in addition to a GFCI receptacle would
be better yet, but its comforting to know that something can
be done to improve safety above and beyond the design and
condition of the appliance.
The arc fault circuit interrupter (AFCI), a circuit breaker
designed to prevent fires, is designed to open on
intermittent resistive short circuits. For example, a normal
15 A breaker is designed to open circuit quickly if loaded
well beyond the 15 A rating, more slowly a little beyond the
rating. While this protects against direct shorts and several
seconds of overload, respectively, it does not protect against
arcs- similar to arc-welding. An arc is a highly variable load,
repetitively peaking at over 70 A, open circuiting with
alternating current zero-crossings. Though, the average
current is not enough to trip a standard breaker, it is enough
to start a fire. This arc could be created by a metalic short
circuit which burns the metal open, leaving a resistive
sputtering plasma of ionized gases.
The AFCI contains electronic circuitry to sense this
intermittent resistive short circuit. It protects against both
hot to neutral and hot to ground arcs. The AFCI does not
protect against personal shock hazards like a GFCI does.
Thus, GFCls still need to be installed in kitchen, bath, and
outdoors circuits. Since the AFCI often trips upon starting
large motors, and more generally on brushed motors, its
installation is limited to bedroom circuits by the U.S.
National Electrical code. Use of the AFCI should reduce the
number of electrical fires. However, nuisance-trips when
running appliances with motors on AFCI circuits is a
problem.
e REVIEW:
e Power systems often have one side of the voltage supply
connected to earth ground to ensure safety at that
point.
e The "grounded" conductor in a power system is called
the neutra/ conductor, while the ungrounded conductor
is called the hot.
e Grounding in power systems exists for the sake of
personnel safety, not the operation of the load(s).
Electrical safety of an appliance or other load can be
improved by good engineering: polarized plugs, double
insulation, and three-prong "grounding" plugs are all
ways that safety can be maximized on the load side.
e Ground Fault Current Interruptors (GFCls) work by
sensing a difference in current between the two
conductors supplying power to the load. There should be
no difference in current at all. Any difference means that
current must be entering or exiting the load by some
means other than the two main conductors, which is not
good. A significant current difference will automatically
open a disconnecting switch mechanism, cutting power
off completely.
Safe meter usage
Using an electrical meter safely and efficiently is perhaps
the most valuable skill an electronics technician can master,
both for the sake of their own personal safety and for
proficiency at their trade. It can be daunting at first to use a
meter, knowing that you are connecting it to live circuits
which may harbor life-threatening levels of voltage and
current. This concern is not unfounded, and it is always best
to proceed cautiously when using meters. Carelessness more
than any other factor is what causes experienced
technicians to have electrical accidents.
The most common piece of electrical test equipment is a
meter called the mu/timeter. Multimeters are so named
because they have the ability to measure a multiple of
variables: voltage, current, resistance, and often many
others, some of which cannot be explained here due to their
complexity. In the hands of a trained technician, the
multimeter is both an efficient work tool and a safety device.
In the hands of someone ignorant and/or careless, however,
the multimeter may become a source of danger when
connected to a "live" circuit.
There are many different brands of multimeters, with
multiple models made by each manufacturer sporting
different sets of features. The multimeter shown here in the
following illustrations is a "generic" design, not specific to
any manufacturer, but general enough to teach the basic
principles of use:
Multimeter
HHE.H
You will notice that the display of this meter is of the
"digital" type: showing numerical values using four digits in
a manner similar to a digital clock. The rotary selector switch
(now set in the Off position) has five different measurement
positions it can be set in: two "V" settings, two "A" settings,
and one setting in the middle with a funny-looking
"horseshoe" symbol on it representing "resistance." The
"horseshoe" symbol is the Greek letter "Omega" (Q), which is
the common symbol for the electrical unit of ohms.
Of the two "V" settings and two "A" settings, you will notice
that each pair is divided into unique markers with either a
pair of horizontal lines (one solid, one dashed), or a dashed
line with a squiggly curve over it. The parallel lines
represent "DC" while the squiggly curve represents "AC."
The "V" of course stands for "voltage" while the "A" stands
for "amperage" (current). The meter uses different
techniques, internally, to measure DC than it uses to
measure AC, and so it requires the user to select which type
of voltage (V) or current (A) is to be measured. Although we
haven't discussed alternating current (AC) in any technical
detail, this distinction in meter settings is an important one
to bear in mind.
There are three different sockets on the multimeter face into
which we can plug our test /eads. Test leads are nothing
more than specially-prepared wires used to connect the
meter to the circuit under test. The wires are coated ina
color-coded (either black or red) flexible insulation to
prevent the user's hands from contacting the bare
conductors, and the tips of the probes are sharp, stiff pieces
of wire:
tip
HBB. robe
lead
plug
vaQ
lead
plug
probe
tip
The black test lead a/ways plugs into the black socket on the
multimeter: the one marked "COM" for "common." The red
test lead plugs into either the red socket marked for voltage
and resistance, or the red socket marked for current,
depending on which quantity you intend to measure with
the multimeter.
To see how this works, let's look at a couple of examples
showing the meter in use. First, we'll set up the meter to
measure DC voltage from a battery:
Note that the two test leads are plugged into the
appropriate sockets on the meter for voltage, and the
selector switch has been set for DC "V". Now, we'll take a
look at an example of using the multimeter to measure AC
voltage from a household electrical power receptacle (wall
socket):
| 14.6
The only difference in the setup of the meter is the
placement of the selector switch: it is now turned to AC "V".
Since we're still measuring voltage, the test leads will
remain plugged in the same sockets. In both of these
examples, it is /mperative that you not let the probe tips
come in contact with one another while they are both in
contact with their respective points on the circuit. If this
happens, a short-circuit will be formed, creating a spark and
perhaps even a ball of flame if the voltage source is capable
of supplying enough current! The following image illustrates
the potential for hazard:
| 19.6
large spark
from short
circuit!
This is just one of the ways that a meter can become a
source of hazard if used improperly.
Voltage measurement is perhaps the most common function
a multimeter is used for. It is certainly the primary
measurement taken for safety purposes (part of the lock-
out/tag-out procedure), and it should be well understood by
the operator of the meter. Being that voltage is always
relative between two points, the meter must be firmly
connected to two points in a circuit before it will provide a
reliable measurement. That usually means both probes must
be grasped by the user's hands and held against the proper
contact points of a voltage source or circuit while measuring.
Because a hand-to-hand shock current path is the most
dangerous, holding the meter probes on two points ina
high-voltage circuit in this manner is always a potential
hazard. If the protective insulation on the probes is worn or
cracked, it is possible for the user's fingers to come into
contact with the probe conductors during the time of test,
causing a bad shock to occur. If it is possible to use only one
hand to grasp the probes, that is a safer option. Sometimes
it is possible to "latch" one probe tip onto the circuit test
point so that it can be let go of and the other probe set in
place, using only one hand. Special probe tip accessories
such as spring clips can be attached to help facilitate this.
Remember that meter test leads are part of the whole
equipment package, and that they should be treated with
the same care and respect that the meter itself is. If you
need a special accessory for your test leads, such as a spring
clip or other special probe tip, consult the product catalog of
the meter manufacturer or other test equipment
manufacturer. Do not try to be creative and make your own
test probes, as you may end up placing yourself in danger
the next time you use them on a live circuit.
Also, it must be remembered that digital multimeters usually
do a good job of discriminating between AC and DC
measurements, as they are set for one or the other when
checking for voltage or current. As we have seen earlier,
both AC and DC voltages and currents can be deadly, so
when using a multimeter as a safety check device you
should always check for the presence of both AC and DC,
even if you're not expecting to find both! Also, when
checking for the presence of hazardous voltage, you should
be sure to check a// pairs of points in question.
For example, suppose that you opened up an electrical
wiring cabinet to find three large conductors supplying AC
power to a load. The circuit breaker feeding these wires
(Supposedly) has been shut off, locked, and tagged. You
double-checked the absence of power by pressing the Start
button for the load. Nothing happened, so now you move on
to the third phase of your safety check: the meter test for
voltage.
First, you check your meter on a Known source of voltage to
see that its working properly. Any nearby power receptacle
should provide a convenient source of AC voltage for a test.
You do so and find that the meter indicates as it should.
Next, you need to check for voltage among these three wires
in the cabinet. But voltage is measured between two points,
so where do you check?
The answer is to check between all combinations of those
three points. As you can see, the points are labeled "A", "B",
and "C" in the illustration, so you would need to take your
multimeter (set in the voltmeter mode) and check between
pointsA&B,B&C, andA & C. If you find voltage between
any of those pairs, the circuit is not in a Zero Energy State.
But wait! Remember that a multimeter will not register DC
voltage when its in the AC voltage mode and vice versa, so
you need to check those three pairs of points in each mode
for a total of six voltage checks in order to be complete!
However, even with all that checking, we still haven't
covered all possibilities yet. Remember that hazardous
voltage can appear between a single wire and ground (in
this case, the metal frame of the cabinet would be a good
ground reference point) in a power system. So, to be
perfectly safe, we not only have to check between A & B, B
& C, and A & C (in both AC and DC modes), but we also have
to check between A & ground, B & ground, and C & ground
(in both AC and DC modes)! This makes for a grand total of
twelve voltage checks for this seemingly simple scenario of
only three wires. Then, of course, after we've completed all
these checks, we need to take our multimeter and re-test it
against a known source of voltage such as a power
receptacle to ensure that its still in good working order.
Using a multimeter to check for resistance is a much simpler
task. The test leads will be kept plugged in the same sockets
as for the voltage checks, but the selector switch will need
to be turned until it points to the "horseshoe" resistance
symbol. Touching the probes across the device whose
resistance is to be measured, the meter should properly
display the resistance in ohms:
B |.cA
eareer Som pasian
resisto
(0 * | [cme
One very important thing to remember about measuring
resistance is that it must only be done on de-energized
components! When the meter is in "resistance" mode, it uses
a small internal battery to generate a tiny current through
the component to be measured. By sensing how difficult it is
to move this current through the component, the resistance
of that component can be determined and displayed. If there
is any additional source of voltage in the meter-lead-
component-lead-meter loop to either aid or oppose the
resistance-measuring current produced by the meter, faulty
readings will result. In a worse-case situation, the meter may
even be damaged by the external voltage.
The "resistance" mode of a multimeter is very useful in
determining wire continuity as well as making precise
measurements of resistance. When there is a good, solid
connection between the probe tips (simulated by touching
them together), the meter shows almost zero Q. If the test
leads had no resistance in them, it would read exactly zero:
If the leads are not in contact with each other, or touching
opposite ends of a broken wire, the meter will indicate
infinite resistance (usually by displaying dashed lines or the
abbreviation "O.L." which stands for "open loop"):
By far the most hazardous and complex application of the
multimeter is in the measurement of current. The reason for
this is quite simple: in order for the meter to measure
current, the current to be measured must be forced to go
through the meter. This means that the meter must be made
part of the current path of the circuit rather than just be
connected off to the side somewhere as is the case when
measuring voltage. In order to make the meter part of the
current path of the circuit, the original circuit must be
"broken" and the meter connected across the two points of
the open break. To set the meter up for this, the selector
switch must point to either AC or DC "A" and the red test
lead must be plugged in the red socket marked "A". The
following illustration shows a meter all ready to measure
Current and a circuit to be tested:
al O00 simple battery-lamp circuit
*
Ale
Now, the circuit is broken in preparation for the meter to be
connected:
0.000
lamp goes out
Ale
The next step is to insert the meter in-line with the circuit by
connecting the two probe tips to the broken ends of the
circuit, the black probe to the negative (-) terminal of the 9-
volt battery and the red probe to the loose wire end leading
to the lamp:
circuit current now has to
go through the meter
This example shows a very safe circuit to work with. 9 volts
hardly constitutes a shock hazard, and so there is little to
fear in breaking this circuit open (bare handed, no less!) and
connecting the meter in-line with the flow of electrons.
However, with higher power circuits, this could bea
hazardous endeavor indeed. Even if the circuit voltage was
low, the normal current could be high enough that an
injurious spark would result the moment the last meter
probe connection was established.
Another potential hazard of using a multimeter in its
current-measuring ("ammeter") mode is failure to properly
put it back into a voltage-measuring configuration before
measuring voltage with it. The reasons for this are specific to
ammeter design and operation. When measuring circuit
current by placing the meter directly in the path of current,
it is best to have the meter offer little or no resistance
against the flow of electrons. Otherwise, any additional
resistance offered by the meter would impede the electron
flow and alter the circuits operation. Thus, the multimeter is
designed to have practically zero ohms of resistance
between the test probe tips when the red probe has been
plugged into the red "A" (current-measuring) socket. In the
voltage-measuring mode (red lead plugged into the red "V"
socket), there are many mega-ohms of resistance between
the test probe tips, because voltmeters are designed to have
close to infinite resistance (so that they don't draw any
appreciable current from the circuit under test).
When switching a multimeter from current- to voltage-
measuring mode, its easy to spin the selector switch from
the "A" to the "V" position and forget to correspondingly
switch the position of the red test lead plug from "A" to "V",
The result -- if the meter is then connected across a source of
substantial voltage -- will be a short-circuit through the
meter!
OL
SHORT- CIRCUIT!
To help prevent this, most multimeters have a warning
feature by which they beep if ever there's a lead plugged in
the "A" socket and the selector switch is set to "V". As
convenient as features like these are, though, they are still
no substitute for clear thinking and caution when using a
multimeter.
All good-quality multimeters contain fuses inside that are
engineered to "blow" in the event of excessive current
through them, such as in the case illustrated in the last
image. Like all overcurrent protection devices, these fuses
are primarily designed to protect the equipment (in this
case, the meter itself) from excessive damage, and only
secondarily to protect the user from harm. A multimeter can
be used to check its own current fuse by setting the selector
switch to the resistance position and creating a connection
between the two red sockets like this:
Indication with a good fuse Indication with a "blown" fuse
O.506
A good fuse will indicate very little resistance while a blown
fuse will always show "O.L." (or whatever indication that
model of multimeter uses to indicate no continuity). The
actual number of ohms displayed for a good fuse is of little
consequence, so long as its an arbitrarily low figure.
So now that we've seen how to use a multimeter to measure
voltage, resistance, and current, what more is there to
know? Plenty! The value and capabilities of this versatile
test instrument will become more evident as you gain skill
and familiarity using it. There is no substitute for regular
practice with complex instruments such as these, so feel free
to experiment on safe, battery-powered circuits.
REVIEW:
A meter capable of checking for voltage, current, and
resistance is called a multimeter.
As voltage is always relative between two points, a
voltage-measuring meter ("voltmeter") must be
connected to two points in a circuit in order to obtain a
good reading. Be careful not to touch the bare probe tips
together while measuring voltage, as this will create a
short-circuit!
Remember to always check for both AC and DC voltage
when using a multimeter to check for the presence of
hazardous voltage on a circuit. Make sure you check for
voltage between all pair-combinations of conductors,
including between the individual conductors and
ground!
When in the voltage-measuring ("voltmeter") mode,
multimeters have very high resistance between their
leads.
Never try to read resistance or continuity with a
multimeter on a circuit that is energized. At best, the
resistance readings you obtain from the meter will be
inaccurate, and at worst the meter may be damaged and
you may be injured.
Current measuring meters ("ammeters") are always
connected in a circuit so the electrons have to flow
through the meter.
When in the current-measuring ("ammeter") mode,
multimeters have practically no resistance between their
leads. This is intended to allow electrons to flow through
the meter with the least possible difficulty. If this were
not the case, the meter would add extra resistance in
the circuit, thereby affecting the current.
Electric shock data
The table of electric currents and their various bodily effects
was obtained from online (Internet) sources: the safety page
of Massachusetts Institute of Technology (website: [*]), anda
safety handbook published by Cooper Bussmann, Inc
(website: [*]). In the Bussmann handbook, the table is
appropriately entitled De/eterious Effects of Electric Shock,
and credited to a Mr. Charles F. Dalziel. Further research
revealed Dalziel to be both a scientific pioneer and an
authority on the effects of electricity on the human body.
The table found in the Bussmann handbook differs slightly
from the one available from MIT: for the DC threshold of
perception (men), the MIT table gives 5.2 mA while the
Bussmann table gives a slightly greater figure of 6.2 mA.
Also, for the "unable to let go" 60 Hz AC threshold (men),
the MIT table gives 20 mA while the Bussmann table gives a
lesser figure of 16 mA. As | have yet to obtain a primary
copy of Dalziel's research, the figures cited here are
conservative: | have listed the lowest values in my table
where any data sources differ.
These differences, of course, are academic. The point here is
that relatively small magnitudes of electric current through
the body can be harmful if not lethal.
Data regarding the electrical resistance of body contact
points was taken from a safety page (document 16.1) from
the Lawrence Livermore National Laboratory (website [*]),
citing Ralph H. Lee as the data source. Lee's work was listed
here in a document entitled "Human Electrical Sheet,"
composed while he was an IEEE Fellow at E.I. duPont de
Nemours & Co., and also in an article entitled "Electrical
Safety in Industrial Plants" found in the June 1971 issue of
IEEE Spectrum magazine.
For the morbidly curious, Charles Dalziel's experimentation
conducted at the University of California (Berkeley) began
with a state grant to investigate the bodily effects of sub-
lethal electric current. His testing method was as follows:
healthy male and female volunteer subjects were asked to
hold a copper wire in one hand and place their other hand
on around, brass plate. A voltage was then applied between
the wire and the plate, causing electrons to flow through the
subject's arms and chest. The current was stopped, then
resumed at a higher level. The goal here was to see how
much current the subject could tolerate and still keep their
hand pressed against the brass plate. When this threshold
was reached, laboratory assistants forcefully held the
subject's hand in contact with the plate and the current was
again increased. The subject was asked to release the wire
they were holding, to see at what current level involuntary
muscle contraction (tetanus) prevented them from doing so.
For each subject the experiment was conducted using DC
and also AC at various frequencies. Over two dozen human
volunteers were tested, and later studies on heart fibrillation
were conducted using animal subjects.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Bibliography
1. [MMOM]Robert S. Porter, MD, editor, “The Merck Manuals
Online Medical Library”, “Electrical Injuries,” at
http://www.merck.com/mmpe/sec21/ch316/ch316b.html
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4]
—||4/]l—
Lessons In Electric Circuits
-- Volume |
Chapter 4
SCIENTIFIC NOTATION AND
METRIC PREFIXES
Scientific notation
Arithmetic with scientific notation
Metric notation
Metric prefix conversions
Hand calculator use
Scientific notation in SPICE
Contributors
Scientific notation
In many disciplines of science and engineering, very large and
very small numerical quantities must be managed. Some of
these quantities are mind-boggling in their size, either
extremely small or extremely large. Take for example the mass
of a proton, one of the constituent particles of an atom's
nucleus:
Proton mass = 0.00000000000000000000000167 grams
Or, consider the number of electrons passing by a point ina
circuit every second with a steady electric current of 1 amp:
1 amp = 6,250,000,000,000,000,000 electrons per second
A lot of zeros, isn't it? Obviously, it can get quite confusing to
have to handle so many zero digits in numbers such as this,
even with the help of calculators and computers.
Take note of those two numbers and of the relative sparsity of
non-zero digits in them. For the mass of the proton, all we have
isa "167" preceded by 23 zeros before the decimal point. For
the number of electrons per second in 1 amp, we have "625"
followed by 16 zeros. We call the span of non-zero digits (from
first to last), plus any zero digits not merely used for
placeholding, the "significant digits" of any number.
The significant digits in a real-world measurement are typically
reflective of the accuracy of that measurement. For example, if
we were to say that a car weighs 3,000 pounds, we probably
don't mean that the car in question weighs exactly 3,000
pounds, but that we've rounded its weight to a value more
convenient to say and remember. That rounded figure of 3,000
has only one significant digit: the "3" in front -- the zeros
merely serve as placeholders. However, if we were to say that
the car weighed 3,005 pounds, the fact that the weight is not
rounded to the nearest thousand pounds tells us that the two
zeros in the middle aren't just placeholders, but that all four
digits of the number "3,005" are significant to its
representative accuracy. Thus, the number "3,005" is said to
have four significant figures.
In like manner, numbers with many zero digits are not
necessarily representative of a real-world quantity all the way
to the decimal point. When this is known to be the case, such a
number can be written in a kind of mathematical "shorthand"
to make it easier to deal with. This "shorthand" is called
scientific notation.
With scientific notation, a number is written by representing its
significant digits as a quantity between 1 and 10 (or -1 and
-10, for negative numbers), and the "placeholder" zeros are
accounted for by a power-of-ten multiplier. For example:
1 amp = 6,250,000,000,000,000,000 electrons per second
... can be expressed as...
1 amp = 6.25 x 1018 electrons per second
10 to the 18th power (1018) means 10 multiplied by itself 18
times, ora "1" followed by 18 zeros. Multiplied by 6.25, it looks
like "625" followed by 16 zeros (take 6.25 and skip the decimal
point 18 places to the right). The advantages of scientific
notation are obvious: the number isn't as unwieldy when
written on paper, and the significant digits are plain to identify.
But what about very small numbers, like the mass of the
proton in grams? We can still use scientific notation, except
with a negative power-of-ten instead of a positive one, to shift
the decimal point to the left instead of to the right:
Proton mass = 0.00000000000000000000000167 grams
...Can be expressed as...
Proton mass = 1.67 x 10°24 grams
10 to the -24th power (10°24) means the inverse (1/x) of 10
multiplied by itself 24 times, ora "1" preceded by a decimal
point and 23 zeros. Multiplied by 1.67, it looks like "167"
preceded by a decimal point and 23 zeros. Just as in the case
with the very large number, it is a lot easier for a human being
to deal with this "shorthand" notation. As with the prior case,
the significant digits in this quantity are clearly expressed.
Because the significant digits are represented "on their own,"
away from the power-of-ten multiplier, it is easy to show a level
of precision even when the number looks round. Taking our
3,000 pound car example, we could express the rounded
number of 3,000 in scientific notation as such:
car weight = 3 x 103 pounds
If the car actually weighed 3,005 pounds (accurate to the
nearest pound) and we wanted to be able to express that full
accuracy of measurement, the scientific notation figure could
be written like this:
car weight = 3.005 x 10? pounds
However, what if the car actually did weigh 3,000 pounds,
exactly (to the nearest pound)? If we were to write its weight in
"normal" form (3,000 Ibs), it wouldn't necessarily be clear that
this number was indeed accurate to the nearest pound and not
just rounded to the nearest thousand pounds, or to the nearest
hundred pounds, or to the nearest ten pounds. Scientific
notation, on the other hand, allows us to show that all four
digits are significant with no misunderstanding:
car weight = 3.000 x 103 pounds
Since there would be no point in adding extra zeros to the
right of the decimal point (placeholding zeros being
unnecessary with scientific notation), we know those zeros
must be significant to the precision of the figure.
Arithmetic with scientific notation
The benefits of scientific notation do not end with ease of
writing and expression of accuracy. Such notation also lends
itself well to mathematical problems of multiplication and
division. Let's say we wanted to know how many electrons
would flow past a point in a circuit carrying 1 amp of electric
current in 25 seconds. If we know the number of electrons per
second in the circuit (which we do), then all we need to do is
multiply that quantity by the number of seconds (25) to arrive
at an answer of total electrons:
(6,250,000,000,000,000,000 electrons per second) x (25
seconds) =
156,250,000,000,000,000,000 electrons passing by in 25
seconds
Using scientific notation, we can write the problem like this:
(6.25 x 1018 electrons per second) x (25 seconds)
If we take the "6.25" and multiply it by 25, we get 156.25. So,
the answer could be written as:
156.25 x 1018 electrons
However, if we want to hold to standard convention for
scientific notation, we must represent the significant digits as
a number between 1 and 10. In this case, we'd say "1.5625"
multiplied by some power-of-ten. To obtain 1.5625 from
156.25, we have to skip the decimal point two places to the
left. To compensate for this without changing the value of the
number, we have to raise our power by two notches (10 to the
20th power instead of 10 to the 18th):
1.5625 x 102° electrons
What if we wanted to see how many electrons would pass by in
3,600 seconds (1 hour)? To make our job easier, we could put
the time in scientific notation as well:
(6.25 x 1028 electrons per second) x (3.6 x 103 seconds)
To multiply, we must take the two significant sets of digits
(6.25 and 3.6) and multiply them together; and we need to
take the two powers-of-ten and multiply them together. Taking
6.25 times 3.6, we get 22.5. Taking 1018 times 103, we get
102! (exponents with common base numbers add). So, the
answer Is:
22.5 x 102! electrons
...Or more properly...
2.25 x 1022 electrons
To illustrate how division works with scientific notation, we
could figure that last problem "backwards" to find out how
long it would take for that many electrons to pass by ata
current of 1 amp:
(2.25 x 1022 electrons) / (6.25 x 1018 electrons per second)
Just as in multiplication, we can handle the significant digits
and powers-of-ten in separate steps (remember that you
subtract the exponents of divided powers-of-ten):
(2.25 / 6.25) x (1022 / 1018)
And the answer is: 0.36 x 104, or 3.6 x 103, seconds. You can
see that we arrived at the same quantity of time (3600
seconds). Now, you may be wondering what the point of all
this is when we have electronic calculators that can handle the
math automatically. Well, back in the days of scientists and
engineers using "slide rule" analog computers, these
techniques were indispensable. The "hard" arithmetic (dealing
with the significant digit figures) would be performed with the
slide rule while the powers-of-ten could be figured without any
help at all, being nothing more than simple addition and
subtraction.
REVIEW:
Significant digits are representative of the real-world
accuracy of a number.
Scientific notation is a "shorthand" method to represent
very large and very small numbers in easily-handled form.
When multiplying two numbers in scientific notation, you
can multiply the two significant digit figures and arrive at
a power-of-ten by adding exponents.
When dividing two numbers in scientific notation, you can
divide the two significant digit figures and arrive ata
power-of-ten by subtracting exponents.
Metric notation
The metric system, besides being a collection of measurement
units for all sorts of physical quantities, is structured around
the concept of scientific notation. The primary difference is
that the powers-of-ten are represented with alphabetical
prefixes instead of by literal powers-of-ten. The following
number line shows some of the more common prefixes and
their respective powers-of-ten:
METRIC PREFIX SCALE
Ti G M k m u n p
tera giga mega kilo (none) milli micro nano pico
10:7. 10° 10° 10? 10° ig? 10° 10-7. 210:%
+tt—t ttt t
107 10° 107% 10%
hecto deca deci centi
h da d c
Looking at this scale, we can see that 2.5 Gigabytes would
mean 2.5 x 102 bytes, or 2.5 billion bytes. Likewise, 3.21
picoamps would mean 3.21 x 10°!4 amps, or 3.21 1/trillionths
of an amp.
Other metric prefixes exist to symbolize powers of ten for
extremely small and extremely large multipliers. On the
extremely small end of the spectrum, femto (f) = 10°), atto (a)
= 10718 zepto (z) = 10°2!, and yocto (y) = 10°24. On the
extremely large end of the spectrum, Peta (P) = 101°, Exa (E)
= 1018 Zetta(Z) = 102!, and Yotta (Y) = 102%.
Because the major prefixes in the metric system refer to
powers of 10 that are multiples of 3 (from "kilo" on up, and
from "milli" on down), metric notation differs from regular
scientific notation in that the mantissa can be anywhere
between 1 and 999, depending on which prefix is chosen. For
example, if a laboratory sample weighs 0.000267 grams,
scientific notation and metric notation would express it
differently:
2.67 x 10% grams (scientific notation)
267 ugrams (metric notation)
The same figure may also be expressed as 0.267 milligrams
(0.267 mg), although it is usually more common to see the
significant digits represented as a figure greater than 1.
In recent years a new style of metric notation for electric
quantities has emerged which seeks to avoid the use of the
decimal point. Since decimal points (".") are easily misread
and/or "lost" due to poor print quality, quantities such as 4.7 k
may be mistaken for 47 k. The new notation replaces the
decimal point with the metric prefix character, so that "4.7 k"
is printed instead as "4k7". Our last figure from the prior
example, "0.267 m", would be expressed in the new notation
as "0m267".
e REVIEW:
e The metric system of notation uses alphabetical prefixes to
represent certain powers-of-ten instead of the lengthier
scientific notation.
Metric prefix conversions
To express a quantity in a different metric prefix that what it
was originally given, all we need to do is skip the decimal
point to the right or to the left as needed. Notice that the
metric prefix "number line" in the previous section was laid out
from larger to smaller, left to right. This layout was purposely
chosen to make it easier to remember which direction you
need to skip the decimal point for any given conversion.
Example problem: express 0.000023 amps in terms of
microamps.
0.000023 amps (has no prefix, just plain unit of amps)
From UNITS to micro on the number line is 6 places (powers of
ten) to the right, so we need to skip the decimal point 6 places
to the right:
0.000023 amps = 23., or 23 microamps (UA)
Example problem: express 304,212 volts in terms of kilovolts.
304,212 volts (has no prefix, just plain unit of volts)
From the (none) place to kilo place on the number line is 3
places (powers of ten) to the left, so we need to skip the
decimal point 3 places to the left:
304,212. = 304.212 kilovolts (kV)
Example problem: express 50.3 Mega-ohms in terms of milli-
ohms.
50.3 M ohms (mega = 10°)
From mega to milli is 9 places (powers of ten) to the right (from
10 to the 6th power to 10 to the -3rd power), so we need to
skip the decimal point 9 places to the right:
50.3 M ohms = 50,300,000,000 milli-ohms (mQ)
e REVIEW:
e Follow the metric prefix number line to know which
direction you skip the decimal point for conversion
purposes.
e Anumber with no decimal point shown has an implicit
decimal point to the immediate right of the furthest right
digit (i.e. for the number 436 the decimal point is to the
right of the 6, as such: 436.)
Hand calculator use
To enter numbers in scientific notation into a hand calculator,
there is usually a button marked "E" or "EE" used to enter the
correct power of ten. For example, to enter the mass of a
proton in grams (1.67 x 10°24 grams) into a hand calculator, |
would enter the following keystrokes:
[1] [.] (6) (7) (CEE) [2] [4] [+/-]
The [+/-] keystroke changes the sign of the power (24) into a
-24. Some calculators allow the use of the subtraction key [-]
to do this, but | prefer the "change sign" [+/-] key because its
more consistent with the use of that key in other contexts.
If | wanted to enter a negative number in scientific notation
into a hand calculator, | would have to be careful how | used
the [+/-] key, lest | change the sign of the power and not the
significant digit value. Pay attention to this example:
Number to be entered: -3.221 x 1071°:
bel! el! ub2is zy: WEE: Wheel EET” (Pa) teal: taza
The first [+/-] keystroke changes the entry from 3.221 to
-3.221; the second [+/-] keystroke changes the power from 15
to -15.
Displaying metric and scientific notation on a hand calculator
is a different matter. It involves changing the display option
from the normal "fixed" decimal point mode to the "scientific"
or "engineering" mode. Your calculator manual will tell you
how to set each display mode.
These display modes tell the calculator how to represent any
number on the numerical readout. The actual value of the
number is not affected in any way by the choice of display
modes -- only how the number appears to the calculator user.
Likewise, the procedure for entering numbers into the
calculator does not change with different display modes either.
Powers of ten are usually represented by a pair of digits in the
upper-right hand corner of the display, and are visible only in
the "scientific" and "engineering" modes.
The difference between "scientific" and "engineering" display
modes is the difference between scientific and metric notation.
In "scientific" mode, the power-of-ten display is set so that the
main number on the display is always a value between 1 and
10 (or -1 and -10 for negative numbers). In "engineering"
mode, the powers-of-ten are set to display in multiples of 3, to
represent the major metric prefixes. All the user has to do is
memorize a few prefix/power combinations, and his or her
calculator will be "speaking" metric!
POWER METRIC PREFIX
VDE sieve atin Yan tae 4 Tera (T)
Od eae ave Giga (G)
OM oeee ie heecteies Mega (M)
Oe a dw Stee a ite Kilo (k)
e REVIEW:
e Use the [EE] key to enter powers of ten.
e Use "scientific" or "engineering" to display powers of ten,
in scientific or metric notation, respectively.
Scientific notation in SPICE
The SPICE circuit simulation computer program uses scientific
notation to display its output information, and can interpret
both scientific notation and metric prefixes in the circuit
description files. If you are going to be able to successfully
interpret the SPICE analyses throughout this book, you must
be able to understand the notation used to express variables
of voltage, current, etc. in the program.
Let's start with a very simple circuit composed of one voltage
source (a battery) and one resistor:
24Vv — 5 Q
To simulate this circuit using SPICE, we first have to designate
node numbers for all the distinct points in the circuit, then list
the components along with their respective node numbers so
the computer knows which component is connected to which,
and how. For a circuit of this simplicity, the use of SPICE seems
like overkill, but it serves the purpose of demonstrating
practical use of scientific notation:
1 1
av — 5 Q
0 0
Typing out a circuit description file, or neti/ist, for this circuit,
we get this:
Simple circuit
vl 10 dc 24
rl 105
.end
The line "v1 1 © de 24" describes the battery, positioned
between nodes 1 and 0, with a DC voltage of 24 volts. The line
"rl 1 0 5" describes the 5 Q resistor placed between nodes 1
and 0.
Using a computer to run a SPICE analysis on this circuit
description file, we get the following results:
node voltage
( 1) 24.0000
voltage source currents
name current
v1 -4,800E+00
total power dissipation 1.15E+02 watts
SPICE tells us that the voltage "at" node number 1 (actually,
this means the voltage between nodes 1 and O, node 0 being
the default reference point for all voltage measurements) is
equal to 24 volts. The current through battery "v1" is
displayed as -4.800E+00 amps. This is SPICE's method of
denoting scientific notation. What its really saying is "-4.800 x
10° amps," or simply -4.800 amps. The negative value for
current here is due to a quirk in SPICE and does not indicate
anything significant about the circuit itself. The "total power
dissipation" is given to us as 1.15E+02 watts, which means
"1.15 x 102 watts," or 115 watts.
Let's modify our example circuit so that it has a 5 kQ (5 kilo-
ohm, or 5,000 ohm) resistor instead of a5 Q resistor and see
what happens.
1 1
mY — 5kQ
0 0
Once again is our circuit description file, or "netlist:"
Simple circuit
vl 10 dc 24
rl 10 5k
.end
The letter "k" following the number 5 on the resistor's line tells
SPICE that it is a figure of 5 kQ, not 5 Q. Let's see what result
we get when we run this through the computer:
node voltage
( 1) 24.0000
voltage source currents
name current
v1 -4,.800E-03
total power dissipation 1.15E-01 watts
The battery voltage, of course, hasn't changed since the first
simulation: its still at 24 volts. The circuit current, on the other
hand, is much less this time because we've made the resistor a
larger value, making it more difficult for electrons to flow.
SPICE tells us that the current this time is equal to -4.800E-03
amps, or -4.800 x 10°3 amps. This is equivalent to taking the
number -4.8 and skipping the decimal point three places to
the left.
Of course, if we recognize that 10-3 is the same as the metric
prefix "milli," we could write the figure as -4.8 milliamps, or
-4.8 MA.
Looking at the "total power dissipation" given to us by SPICE
on this second simulation, we see that it is 1.15E-01 watts, or
1.15 x 107! watts. The power of -1 corresponds to the metric
prefix "deci," but generally we limit our use of metric prefixes
in electronics to those associated with powers of ten that are
multiples of three (ten to the power of... -12, -9, -6, -3, 3, 6, 9,
12, etc.). So, if we want to follow this convention, we must
express this power dissipation figure as 0.115 watts or 115
milliwatts (115 mW) rather than 1.15 deciwatts (1.15 dW).
Perhaps the easiest way to convert a figure from scientific
notation to common metric prefixes is with a scientific
calculator set to the "engineering" or "metric" display mode.
Just set the calculator for that display mode, type any scientific
notation figure into it using the proper keystrokes (See your
owner's manual), press the "equals" or "enter" key, and it
should display the same figure in engineering/metric notation.
Again, I'll be using SPICE as a method of demonstrating circuit
concepts throughout this book. Consequently, it is in your best
interest to understand scientific notation so you can easily
comprehend its output data format.
Contributors
Contributors to this chapter are listed in chronological order of
their contributions, from most recent to first. See Appendix 2
(Contributor List) for dates and contact information.
Jason Starck (June 2000): HTML document formatting, which
led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—|/|4/]l—
—||+]l—
Lessons In Electric Circuits
-- Volume |!
Chapter 5
SERIES AND PARALLEL
CIRCUITS
What are "series" and "parallel" circuits?
Simple series circuits
Simple parallel circuits
Conductance
Power calculations
Correct use of Ohm's Law
Component failure analysis
Building simple resistor circuits
Contributors
What are "Series" and "parallel"
circuits?
Circuits consisting of just one battery and one load resistance
are very simple to analyze, but they are not often found in
practical applications. Usually, we find circuits where more
than two components are connected together.
There are two basic ways in which to connect more than two
circuit components: series and parallel. First, an example of a
series circuit:
Series
Here, we have three resistors (labeled R;, R>, and R3),
connected in a long chain from one terminal of the battery to
the other. (It should be noted that the subscript labeling --
those little numbers to the lower-right of the letter "R" -- are
unrelated to the resistor values in ohms. They serve only to
identify one resistor from another.) The defining characteristic
of a series circuit is that there is only one path for electrons to
flow. In this circuit the electrons flow in a counter-clockwise
direction, from point 4 to point 3 to point 2 to point 1 and
back around to 4.
Now, let's look at the other type of circuit, a parallel
configuration:
Parallel
Again, we have three resistors, but this time they form more
than one continuous path for electrons to flow. There's one
path from 8 to7 to 2 to 1 and back to 8 again. There's
another from 8 to 7 to 6 to 3 to 2 to 1 and back to 8 again.
And then there's a third path from 8 to 7 to 6 to 5 to 4 to 3 to
2 to 1 and back to 8 again. Each individual path (through Rj,
R>, and R3) is called a branch.
The defining characteristic of a parallel circuit is that all
components are connected between the same set of
electrically common points. Looking at the schematic
diagram, we see that points 1, 2, 3, and 4 are all electrically
common. So are points 8, 7, 6, and 5. Note that all resistors as
well as the battery are connected between these two sets of
points.
And, of course, the complexity doesn't stop at simple series
and parallel either! We can have circuits that are a
combination of series and parallel, too:
Series-parallel
In this circuit, we have two loops for electrons to flow through:
one from 6 to 5 to 2 to 1 and back to 6 again, and another
from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how
both current paths go through R, (from point 2 to point 1). In
this configuration, we'd say that R> and R3 are in parallel with
each other, while Rj is in series with the parallel combination
of R> and R3.
This is just a preview of things to come. Don't worry! We'll
explore all these circuit configurations in detail, one ata
time!
The basic idea of a "series" connection is that components are
connected end-to-end in a line to form a single path for
electrons to flow:
Series connection
R, R, R, R,
- 3
VN VV VV
only one path for electrons to flow!
The basic idea of a "parallel" connection, on the other hand, is
that all components are connected across each other's leads.
In a purely parallel circuit, there are never more than two sets
of electrically common points, no matter how many
components are connected. There are many paths for
electrons to flow, but only one voltage across all components:
Parallel connection
These points are electrically common
ee ee
ee
These points are electrically common
Series and parallel resistor configurations have very different
electrical properties. We'll explore the properties of each
configuration in the sections to come.
e REVIEW:
e In a series circuit, all components are connected end-to-
end, forming a single path for electrons to flow.
e In a parallel circuit, all components are connected across
each other, forming exactly two sets of electrically
common points.
e A "branch" in a parallel circuit is a path for electric
current formed by one of the load components (such as a
resistor).
Simple series circuits
Let's start with a series circuit consisting of three resistors and
a single battery:
The first principle to understand about series circuits is that
the amount of current is the same through any component in
the circuit. This is because there is only one path for electrons
to flow in a series circuit, and because free electrons flow
through conductors like marbles in a tube, the rate of flow
(marble speed) at any point in the circuit (tube) at any
specific point in time must be equal.
From the way that the 9 volt battery is arranged, we can tell
that the electrons in this circuit will flow in a counter-
clockwise direction, from point 4 to 3 to 2 to 1 and back to 4.
However, we have one source of voltage and three
resistances. How do we use Ohm's Law here?
An important caveat to Ohm's Law is that all quantities
(voltage, current, resistance, and power) must relate to each
other in terms of the same two points in a circuit. For
instance, with a single-battery, single-resistor circuit, we
could easily calculate any quantity because they all applied
to the same two points in the circuit:
1=—
R
j= volts _ 3mA
3kQ
Since points 1 and 2 are connected together with wire of
negligible resistance, as are points 3 and 4, we can say that
point 1 is electrically common to point 2, and that point 3 is
electrically common to point 4. Since we know we have 9
volts of electromotive force between points 1 and 4 (directly
across the battery), and since point 2 is common to point 1
and point 3 common to point 4, we must also have 9 volts
between points 2 and 3 (directly across the resistor).
Therefore, we can apply Ohm's Law (I = E/R) to the current
through the resistor, because we know the voltage (E) across
the resistor and the resistance (R) of that resistor. All terms (E,
|, R) apply to the same two points in the circuit, to that same
resistor, So we can use the Ohm's Law formula with no
reservation.
However, in circuits containing more than one resistor, we
must be careful in how we apply Ohm's Law. In the three-
resistor example circuit below, we know that we have 9 volts
between points 1 and 4, which is the amount of electromotive
force trying to push electrons through the series combination
of Rj, Ro, and R3. However, we cannot take the value of 9
volts and divide it by 3k, 10k or 5k Q to try to find a current
value, because we don't know how much voltage is across
any one of those resistors, individually.
The figure of 9 volts is a tota/ quantity for the whole circuit,
whereas the figures of 3k, 10k, and 5k Q are individual
quantities for individual resistors. If we were to plug a figure
for total voltage into an Ohm's Law equation with a figure for
individual resistance, the result would not relate accurately to
any quantity in the real circuit.
For R;, Ohm's Law will relate the amount of voltage across R,
with the current through Rj, given R's resistance, 3kQ:
1,1 = E,i= 12,3 kQ)
But, since we don't know the voltage across R, (only the total
voltage supplied by the battery across the three-resistor
series combination) and we don't know the current through
Rj, we can't do any calculations with either formula. The
Same goes for R> and R3: we can apply the Ohm's Law
equations if and only if all terms are representative of their
respective quantities between the same two points in the
circuit.
So what can we do? We know the voltage of the source (9
volts) applied across the series combination of R;, Rz, and R3,
and we know the resistances of each resistor, but since those
quantities aren't in the same context, we can't use Ohm's Law
to determine the circuit current. If only we knew what the
total resistance was for the circuit: then we could calculate
tota/ current with our figure for tota/ voltage (I=E/R).
This brings us to the second principle of series circuits: the
total resistance of any series circuit is equal to the sum of the
individual resistances. This should make intuitive sense: the
more resistors in series that the electrons must flow through,
the more difficult it will be for those electrons to flow. In the
example problem, we had a 3 kQ, 10 kQ, and 5 kQ resistor in
series, giving us a total resistance of 18 kQ:
Ryotal = R, * R, + R;
Riot = 3 kQ + 1OkQ +5kQ
Rectal = 18 kQ
In essence, we've calculated the equivalent resistance of Rj,
Rz, and R3 combined. Knowing this, we could re-draw the
circuit with a single equivalent resistor representing the
series combination of Rj, Rz, and R3:
R, +R, +R;=
18 kQ
Now we have all the necessary information to calculate circuit
current, because we have the voltage between points 1 and 4
(9 volts) and the resistance between points 1 and 4 (18 kQ):
E, tal
Lotal= _
total
9 volts ss
J tal 18 kO L
Knowing that current is equal through all components of a
series circuit (and we just determined the current through the
battery), we can go back to our original circuit schematic and
note the current through each component:
R, 3kQ
Now that we know the amount of current through each
resistor, we can use Ohm's Law to determine the voltage drop
across each one (applying Ohm's Law in its proper context):
Eri = Ie Ri Epo = Ip Ry Eg3 = Ip3 R3
E,, = (500 pA)(3 kQ)=1.5V
E,, = (500 HA)(10 kQ)=5 V
Ep; = (500 pA)(5 kQ)=2.5 V
Notice the voltage drops across each resistor, and how the
sum of the voltage drops (1.5 + 5 + 2.5) is equal to the
battery (supply) voltage: 9 volts. This is the third principle of
series circuits: that the supply voltage is equal to the sum of
the individual voltage drops.
However, the method we just used to analyze this simple
series circuit can be streamlined for better understanding. By
using a table to list all voltages, currents, and resistances in
the circuit, it becomes very easy to see which of those
quantities can be properly related in any Ohm's Law
equation:
R, R» R; Total
E Volts
| Amps
R Ohms
t t t t
Ohm's Ohm's Ohm's Ohm's
Law Law Law Law
The rule with such a table is to apply Ohm's Law only to the
values within each vertical column. For instance, Ep; only
with Ip; and Rj; Eps only with Ip5 and R>; etc. You begin your
analysis by filling in those elements of the table that are
given to you from the beginning:
As you can see from the arrangement of the data, we can't
apply the 9 volts of Ey (total voltage) to any of the resistances
(R , Ro, or R3) in any Ohm's Law formula because they're in
different columns. The 9 volts of battery voltage is not
applied directly across Rj, R>, or R3. However, we can use our
"rules" of series circuits to fill in blank spots on a horizontal
row. In this case, we can use the series rule of resistances to
determine a total resistance from the sum of individual
resistances:
Rule of series
circuits
Rr=R; +R,+R;
Rsk | tok | sk | 18k | Ohms a"
Now, with a value for total resistance inserted into the
rightmost ("Total") column, we can apply Ohm's Law of I=E/R
to total voltage and total resistance to arrive at a total current
of 500 UA:
Then, knowing that the current is shared equally by all
components of a series circuit (another "rule" of series
circuits), we can fill in the currents for each resistor from the
current figure just calculated:
R, R, R, Total
Rule of series
circuits
I;= I; =1,=1,
Finally, we can use Ohm's Law to determine the voltage drop
across each resistor, one column at a time:
R, R, R, Total
E} uw | 5s | 25 | 9 _| Volts
| Amps
R Ohms
t t t
Ohm's Ohm's Ohm's
Law Law Law
Just for fun, we can use a computer to analyze this very same
circuit automatically. It will be a good way to verify our
calculations and also become more familiar with computer
analysis. First, we have to describe the circuit to the computer
in a format recognizable by the software. The SPICE program
we'll be using requires that all electrically unique points ina
circuit be numbered, and component placement is
understood by which of those numbered points, or "nodes,"
they share. For clarity, | numbered the four corners of our
example circuit 1 through 4. SPICE, however, demands that
there be a node zero somewhere in the circuit, so I'll re-draw
the circuit, changing the numbering scheme slightly:
All I've done here is re-numbered the lower-left corner of the
circuit O instead of 4. Now, | can enter several lines of text
into a computer file describing the circuit in terms SPICE will
understand, complete with a couple of extra lines of code
directing the program to display voltage and current data for
our viewing pleasure. This computer file is known as the
netlist in SPICE terminology:
series circuit
vl 1 0
rl 12 3k
r2 2 3 10k
r3 3 0 5k
.dc vl 991
.print dc v(1,2) v(2,3) v(3,0)
.end
Now, all | have to do is run the SPICE program to process the
netlist and output the results:
v1 v(1,2) v(2,3) v(3) i(vl)
9.000E+00 1.500E+00 5.000E+00 2.500E+00 -5.000E-04
This printout is telling us the battery voltage is 9 volts, and
the voltage drops across Rj, Rz, and R3 are 1.5 volts, 5 volts,
and 2.5 volts, respectively. Voltage drops across any
component in SPICE are referenced by the node numbers the
component lies between, so v(1,2) is referencing the voltage
between nodes 1 and 2 in the circuit, which are the points
between which R, is located. The order of node numbers is
important: when SPICE outputs a figure for v(1,2), it regards
the polarity the same way as if we were holding a voltmeter
with the red test lead on node 1 and the black test lead on
node 2.
We also have a display showing current (albeit with a
negative value) at 0.5 milliamps, or 500 microamps. So our
mathematical analysis has been vindicated by the computer.
This figure appears as a negative number in the SPICE
analysis, due to a quirk in the way SPICE handles current
calculations.
In summary, a series circuit is defined as having only one
path for electrons to flow. From this definition, three rules of
series circuits follow: all components share the same current;
resistances add to equal a larger, total resistance; and
voltage drops add to equal a larger, total voltage. All of these
rules find root in the definition of a series circuit. If you
understand that definition fully, then the rules are nothing
more than footnotes to the definition.
e REVIEW:
e Components in a series circuit share the same current:
otal = 1, = Io =~ - + In
e Total resistance in a series circuit is equal to the sum of
the individual resistances: Ryota; = Ry + Ro +... Ra
e Total voltage in a series circuit is equal to the sum of the
individual voltage drops: Ey 44; = E, + Eo +... Ey
Simple parallel circuits
Let's start with a parallel circuit consisting of three resistors
and a single battery:
1 2 3 4
The first principle to understand about parallel circuits is that
the voltage is equal across all components in the circuit. This
is because there are only two sets of electrically common
points in a parallel circuit, and voltage measured between
sets of common points must always be the same at any given
time. Therefore, in the above circuit, the voltage across R; is
equal to the voltage across R» which is equal to the voltage
across R3 which is equal to the voltage across the battery.
This equality of voltages can be represented in another table
for our starting values:
Just as in the case of series circuits, the same caveat for
Ohm's Law applies: values for voltage, current, and resistance
must be in the same context in order for the calculations to
work correctly. However, in the above example circuit, we can
immediately apply Ohm's Law to each resistor to find its
current because we know the voltage across each resistor (9
volts) and the resistance of each resistor:
E Bu Ex;
lei = = Ip a = 13 — =
V
R1— ay = 0.9 mA
10 kQ
leo = oe a = 4.5 mA
2kQ
V
Las ? = 9mA
LkQ
At this point we still don't know what the total current or total
resistance for this parallel circuit is, so we can't apply Ohm's
Law to the rightmost ("Total") column. However, if we think
carefully about what is happening it should become apparent
that the total current must equal the sum of all individual
resistor ("branch") currents:
As the total current exits the negative (-) battery terminal at
point 8 and travels through the circuit, some of the flow splits
off at point 7 to go up through R,, some more splits off at
point 6 to go up through R3>, and the remainder goes up
through R3. Like a river branching into several smaller
streams, the combined flow rates of all streams must equal
the flow rate of the whole river. The same thing is
encountered where the currents through Rj, R>, and R3 join to
flow back to the positive terminal of the battery (+) toward
point 1: the flow of electrons from point 2 to point 1 must
equal the sum of the (branch) currents through Rj, R>, and
R3.
This is the second principle of parallel circuits: the total
circuit current is equal to the sum of the individual branch
currents. Using this principle, we can fill in the I; spot on our
table with the sum of Ipj, Ip>, and Ip3:
R, Total
R, R.
E Volts
Rule of parallel
circuits
Fotal = l; + l, + 1;
Finally, applying Ohm's Law to the rightmost ("Total") column,
we can calculate the total circuit resistance:
R, R, R, Total
14.4m
Please note something very important here. The total circuit
resistance is only 625 Q: /ess than any one of the individual
resistors. In the series circuit, where the total resistance was
the sum of the individual resistances, the total was bound to
be greater than any one of the resistors individually. Here in
the parallel circuit, however, the opposite is true: we say that
the individual resistances diminish rather than add to make
the total. This principle completes our triad of "rules" for
parallel circuits, just as series circuits were found to have
three rules for voltage, current, and resistance.
Mathematically, the relationship between total resistance and
individual resistances in a parallel circuit looks like this:
R
total —
l
-f
R,
2 3
+
-|- —_
as
R,
The same basic form of equation works for any number of
resistors connected together in parallel, just add as many 1/R
terms on the denominator of the fraction as needed to
accommodate all parallel resistors in the circuit.
Just as with the series circuit, we can use computer analysis to
double-check our calculations. First, of course, we have to
describe our example circuit to the computer in terms it can
understand. I'll start by re-drawing the circuit:
Once again we find that the original numbering scheme used
to identify points in the circuit will have to be altered for the
benefit of SPICE. In SPICE, all electrically common points must
share identical node numbers. This is how SPICE knows what's
connected to what, and how. In a simple parallel circuit, all
points are electrically common in one of two sets of points.
For our example circuit, the wire connecting the tops of all the
components will have one node number and the wire
connecting the bottoms of the components will have the
other. Staying true to the convention of including zero as a
node number, | choose the numbers O and 1:
An example like this makes the rationale of node numbers in
SPICE fairly clear to understand. By having all components
share common sets of numbers, the computer "Knows" they're
all connected in parallel with each other.
In order to display branch currents in SPICE, we need to insert
zero-voltage sources in line (in series) with each resistor, and
then reference our current measurements to those sources.
For whatever reason, the creators of the SPICE program made
it so that current could only be calculated through a voltage
source. This is a Somewhat annoying demand of the SPICE
simulation program. With each of these "dummy" voltage
sources added, some new node numbers must be created to
connect them to their respective branch resistors:
NOTE: vr1, vr2, and vr3 are all
"dummy" voltage sources with
values of 0 volts each!!
The dummy voltage sources are all set at 0 volts so as to have
no impact on the operation of the circuit. The circuit
description file, or netlist, looks like this:
Parallel circuit
vl 10
rl 2 0 10k
r2 3 0 2k
r3 4 0 1k
vrl 12 dc 0
vr2 13 dc 0
vr3 14dc 0
.dc vl 991
.print dc v(2,0) v(3,0) v(4,0)
.print dc i(vrl) i(vr2) i(vr3)
.end
Running the computer analysis, we get these results (I've
annotated the printout with descriptive labels):
v1 v(2) v(3) v(4)
9.000E+00 9.000E+00 9.000E+00 9.000E+00
battery Rl voltage R2 voltage R3 voltage
voltage
vl i(vrl1) i(vr2) i(vr3)
9.000E+00 9.000E-04 4.500E-03 9.000E-03
battery Rl current R2 current = R3 current
voltage
These values do indeed match those calculated through
Ohm's Law earlier: 0.9 mA for Ip;, 4.5 MA for Ip5, and 9 mA for
Ip3. Being connected in parallel, of course, all resistors have
the same voltage dropped across them (9 volts, same as the
battery).
In summary, a parallel circuit is defined as one where all
components are connected between the same set of
electrically common points. Another way of saying this is that
all components are connected across each other's terminals.
From this definition, three rules of parallel circuits follow: all
components share the same voltage; resistances diminish to
equal a smaller, total resistance; and branch currents add to
equal a larger, total current. Just as in the case of series
circuits, all of these rules find root in the definition of a
parallel circuit. If you understand that definition fully, then
the rules are nothing more than footnotes to the definition.
¢ REVIEW:
e Components in a parallel circuit share the same voltage:
Etotal = EF) = EQ =... E,
e Total resistance in a parallel circuit is /ess than any of the
individual resistances: Rota; = 1 / (1/Ry + 1/Ro +... 1/R,)
e Total current in a parallel circuit is equal to the sum of the
individual branch currents: lqo44; = 14 + lo +... Ip-
Conductance
When students first see the parallel resistance equation, the
natural question to ask is, "Where did that thing come from?"
It is truly an odd piece of arithmetic, and its origin deserves a
good explanation.
Resistance, by definition, is the measure of friction a
component presents to the flow of electrons through it.
Resistance is symbolized by the capital letter "R" and is
measured in the unit of "ohm." However, we can also think of
this electrical property in terms of its inverse: how easy it is
for electrons to flow through a component, rather than how
difficult. lf resistance is the word we use to symbolize the
measure of how difficult it is for electrons to flow, then a good
word to express how easy it is for electrons to flow would be
conductance.
Mathematically, conductance is the reciprocal, or inverse, of
resistance:
Conductance = ——
Resistance
The greater the resistance, the less the conductance, and vice
versa. This should make intuitive sense, resistance and
conductance being opposite ways to denote the same
essential electrical property. If two components’ resistances
are compared and it is found that component "A" has one-half
the resistance of component "B," then we could alternatively
express this relationship by saying that component "A" is
twice as conductive as component "B." If component "A" has
but one-third the resistance of component "B," then we could
say it is three times more conductive than component "B,"
and so on.
Carrying this idea further, a symbol and unit were created to
represent conductance. The symbol is the capital letter "G"
and the unit is the mho, which is "ohm" spelled backwards
(and you didn't think electronics engineers had any sense of
humor!). Despite its appropriateness, the unit of the mho was
replaced in later years by the unit of siemens (abbreviated by
the capital letter "S"). This decision to change unit names is
reminiscent of the change from the temperature unit of
degrees Centigrade to degrees Celsius, or the change from
the unit of frequency c.p.s. (cycles per second) to Hertz. If
you're looking for a pattern here, Siemens, Celsius, and Hertz
are all surnames of famous scientists, the names of which,
sadly, tell us less about the nature of the units than the units'
original designations.
As a footnote, the unit of siemens is never expressed without
the last letter "s." In other words, there is no such thing asa
unit of "siemen" as there is in the case of the "ohm" or the
“mho." The reason for this is the proper spelling of the
respective scientists' surnames. The unit for electrical
resistance was named after someone named "Ohm," whereas
the unit for electrical conductance was named after someone
named "Siemens," therefore it would be improper to
"singularize" the latter unit as its final "s" does not denote
plurality.
Back to our parallel circuit example, we should be able to see
that multiple paths (branches) for current reduces total
resistance for the whole circuit, as electrons are able to flow
easier through the whole network of multiple branches than
through any one of those branch resistances alone. In terms
of resistance, additional branches result in a lesser total
(current meets with less opposition). In terms of conductance,
however, additional branches results in a greater total
(electrons flow with greater conductance):
Total parallel resistance is /ess than any one of the individual
branch resistances because parallel resistors resist less
together than they would separately:
/
J
total
Piota) (Ss less than R,, Ro, R3, or R, individually
Total parallel conductance is greater than any of the
individual branch conductances because parallel resistors
conduct better together than they would separately:
/
=
Grotal
\
G,ota) ‘S greater than G,, Go, Gz, or G, individually
To be more precise, the total conductance in a parallel circuit
is equal to the sum of the individual conductances:
=G,+G,+G,+G
ap
Giotal 4
If we know that conductance is nothing more than the
mathematical reciprocal (1/x) of resistance, we can translate
each term of the above formula into resistance by
substituting the reciprocal of each respective conductance:
I I l |
— cae
R, R; R,
& |
+
a
Ristal R,
Solving the above equation for total resistance (instead of the
reciprocal of total resistance), we can invert (reciprocate)
both sides of the equation:
l
Riotal =
ae ee ee ee
R, RR, R,
3
+
1
R,
So, we arrive at our cryptic resistance formula at last!
Conductance (G) is seldom used as a practical measurement,
and so the above formula is a common one to see in the
analysis of parallel circuits.
e REVIEW:
e Conductance is the opposite of resistance: the measure of
how easy it is for electrons to flow through something.
e Conductance is symbolized with the letter "G" and is
measured in units of mhos or Siemens.
e Mathematically, conductance equals the reciprocal of
resistance: G = 1/R
Power calculations
When calculating the power dissipation of resistive
components, use any one of the three power equations to
derive the answer from values of voltage, current, and/or
resistance pertaining to each component:
Power equations
PIE p-_—= P= ER
R
This is easily managed by adding another row to our familiar
table of voltages, currents, and resistances:
R, R, R, Total
Volts
Amps
Ohms
Watts
TD - Mm
Power for any particular table column can be found by the
appropriate Ohm's Law equation (appropriate based on what
figures are present for E, |, and R in that column).
An interesting rule for total power versus individual power is
that it is additive for any configuration of circuit: series,
parallel, series/parallel, or otherwise. Power is a measure of
rate of work, and since power dissipated must equal the total
power applied by the source(s) (as per the Law of
Conservation of Energy in physics), circuit configuration has
no effect on the mathematics.
e REVIEW:
e Power is additive in any configuration of resistive circuit:
Paar ha Pg tase Re
Correct use of Ohm's Law
One of the most common mistakes made by beginning
electronics students in their application of Ohm's Laws is
mixing the contexts of voltage, current, and resistance. In
other words, a student might mistakenly use a value for |
through one resistor and the value for E across a set of
interconnected resistors, thinking that they'll arrive at the
resistance of that one resistor. Not so! Remember this
important rule: The variables used in Ohm's Law equations
must be common to the same two points in the circuit under
consideration. | cannot overemphasize this rule. This is
especially important in series-parallel combination circuits
where nearby components may have different values for both
voltage drop and current.
When using Ohm's Law to calculate a variable pertaining toa
single component, be sure the voltage you're referencing is
solely across that single component and the current you're
referencing is solely through that single component and the
resistance you're referencing is solely for that single
component. Likewise, when calculating a variable pertaining
to a set of components in a circuit, be sure that the voltage,
current, and resistance values are specific to that complete
set of components only! A good way to remember this is to
pay close attention to the two points terminating the
component or set of components being analyzed, making
sure that the voltage in question is across those two points,
that the current in question is the electron flow from one of
those points all the way to the other point, that the resistance
in question is the equivalent of a single resistor between
those two points, and that the power in question is the total
power dissipated by all components between those two
points.
The "table" method presented for both series and parallel
circuits in this chapter is a good way to keep the context of
Ohm's Law correct for any kind of circuit configuration. Ina
table like the one shown below, you are only allowed to apply
an Ohm's Law equation for the values of a single vertical
column at a time:
R, R R, Total
Volts
Amps
Ohms
Watts
vDVaD —- Mm
f of ff
Ohm's Ohm's Ohm's Ohm's
Law Law Law Law
Deriving values horizontally across columns is allowable as
per the principles of series and parallel circuits:
For series circuits:
R, R, R, Total
Feoai = E, + E, + EB;
Lora = 1, =L=1;
Rioat = R, + Ry + R;
Prowat = Py +P, +P
tota
For parallel circuits:
Eotal =E, =E, =E,
Liotal — 1, 7 1, + 1,
l
l l l
—+ +
R, RR, R;
Riotal =
Pista = P, + P, + P;
Not only does the "table" method simplify the management of
all relevant quantities, it also facilitates cross-checking of
answers by making it easy to solve for the original unknown
variables through other methods, or by working backwards to
solve for the initially given values from your solutions. For
example, if you have just solved for all unknown voltages,
currents, and resistances in a circuit, you can check your work
by adding a row at the bottom for power calculations on each
resistor, seeing whether or not all the individual power values
add up to the total power. If not, then you must have made a
mistake somewhere! While this technique of "cross-checking"
your work is nothing new, using the table to arrange all the
data for the cross-check(s) results in a minimum of confusion.
e REVIEW:
e Apply Ohm's Law to vertical columns in the table.
e Apply rules of series/parallel to horizontal rows in the
table.
e Check your calculations by working "backwards" to try to
arrive at originally given values (from your first calculated
answers), or by solving for a quantity using more than
one method (from different given values).
Component failure analysis
The job of a technician frequently entails "troubleshooting"
(locating and correcting a problem) in malfunctioning circuits.
Good troubleshooting is a demanding and rewarding effort,
requiring a thorough understanding of the basic concepts, the
ability to formulate hypotheses (proposed explanations of an
effect), the ability to judge the value of different hypotheses
based on their probability (how likely one particular cause
may be over another), and a sense of creativity in applying a
solution to rectify the problem. While it is possible to distill
these skills into a scientific methodology, most practiced
troubleshooters would agree that troubleshooting involves a
touch of art, and that it can take years of experience to fully
develop this art.
An essential skill to have is a ready and intuitive
understanding of how component faults affect circuits in
different configurations. We will explore some of the effects of
component faults in both series and parallel circuits here,
then to a greater degree at the end of the "Series-Parallel
Combination Circuits" chapter.
Let's start with a simple series circuit:
R, R, R,
< 3
100 Q 300 2 50 2
With all components in this circuit functioning at their proper
values, we can mathematically determine all currents and
voltage drops:
R, R, R; Total
Now let us suppose that R> fails shorted. Shorted means that
the resistor now acts like a straight piece of wire, with little or
no resistance. The circuit will behave as though a "jumper"
wire were connected across R> (in case you were wondering,
"jumper wire" is a common term for a temporary wire
connection in a circuit). What causes the shorted condition of
R> is no matter to us in this example; we only care about its
effect upon the circuit:
jumper wire
100 Q 300 22 50 2
With R> shorted, either by a jumper wire or by an internal
resistor failure, the total circuit resistance will decrease. Since
the voltage output by the battery is a constant (at least in our
ideal simulation here), a decrease in total circuit resistance
means that total circuit current must increase:
R, R, Total
Volts
ot
Amps
Ce Ohms
Shorted
resistor
As the circuit current increases from 20 milliamps to 60
milliamps, the voltage drops across R, and R3 (which haven't
changed resistances) increase as well, so that the two
resistors are dropping the whole 9 volts. Rz, being bypassed
by the very low resistance of the jumper wire, is effectively
eliminated from the circuit, the resistance from one lead to
the other having been reduced to zero. Thus, the voltage drop
across R>, even with the increased total current, is zero volts.
On the other hand, if R> were to fail "open" -- resistance
increasing to nearly infinite levels -- it would also create wide-
reaching effects in the rest of the circuit:
Open
resistor
With R> at infinite resistance and total resistance being the
sum of all individual resistances in a series circuit, the total
current decreases to zero. With zero circuit Current, there is no
electron flow to produce voltage drops across R, or R3. Ro, on
the other hand, will manifest the full supply voltage across its
terminals.
We can apply the same before/after analysis technique to
parallel circuits as well. First, we determine what a "healthy"
parallel circuit should behave like.
Volts
Amps
Ohms
Supposing that R> opens in this parallel circuit, here's what
the effects will be:
R; Total
R{_ 9 | © | 180 | © | Ohms
t
Open
resistor
Notice that in this parallel circuit, an open branch only affects
the current through that branch and the circuit's total current.
Total voltage -- being shared equally across all components in
a parallel circuit, will be the same for all resistors. Due to the
fact that the voltage source's tendency is to hold voltage
constant, its voltage will not change, and being in parallel
with all the resistors, it will hold all the resistors' voltages the
same as they were before: 9 volts. Being that voltage is the
only common parameter in a parallel circuit, and the other
resistors haven't changed resistance value, their respective
branch currents remain unchanged.
This is what happens in a household lamp circuit: all lamps
get their operating voltage from power wiring arranged ina
parallel fashion. Turning one lamp on and off (one branch in
that parallel circuit closing and opening) doesn't affect the
operation of other lamps in the room, only the current in that
one lamp (branch circuit) and the total current powering all
the lamps in the room:
In an ideal case (with perfect voltage sources and zero-
resistance connecting wire), shorted resistors in a simple
parallel circuit will also have no effect on what's happening in
other branches of the circuit. In real life, the effect is not quite
the same, and we'll see why in the following example:
R, "shorted" with a jumper wire
Shorted
resistor
A shorted resistor (resistance of 0 QO) would theoretically draw
infinite current from any finite source of voltage (I=E/0O). In
this case, the zero resistance of R> decreases the circuit total
resistance to zero Q as well, increasing total current to a value
of infinity. As long as the voltage source holds steady at 9
volts, however, the other branch currents (Ip; and Ip3) will
remain unchanged.
The critical assumption in this "perfect" scheme, however, is
that the voltage supply will hold steady at its rated voltage
while supplying an infinite amount of current to a short-circuit
load. This is simply not realistic. Even if the short has a small
amount of resistance (as opposed to absolutely zero
resistance), no rea/ voltage source could arbitrarily supply a
huge overload current and maintain steady voltage at the
same time. This is primarily due to the internal resistance
intrinsic to all electrical power sources, stemming from the
inescapable physical properties of the materials they're
constructed of:
R
internal
Battery ns
9V —
‘|
These internal resistances, small as they may be, turn our
simple parallel circuit into a series-parallel combination
circuit. Usually, the internal resistances of voltage sources are
low enough that they can be safely ignored, but when high
currents resulting from shorted components are encountered,
their effects become very noticeable. In this case, a shorted
R> would result in almost all the voltage being dropped across
the internal resistance of the battery, with almost no voltage
left over for resistors R,, Rz, and R3:
R internal
Battery Rs
aE cl 180 Q
R, "shorted" with a jumper wire
R, R, R, Total
E Volts
| Amps
R Ohms
1 Supp! ea e
Shorted decrease due to
resistor voltage drop across
internal resistance
Suffice it to say, intentional direct short-circuits across the
terminals of any voltage source is a bad idea. Even if the
resulting high current (heat, flashes, sparks) causes no harm
to people nearby, the voltage source will likely sustain
damage, unless it has been specifically designed to handle
short-circuits, which most voltage sources are not.
Eventually in this book | will lead you through the analysis of
circuits without the use of any numbers, that is, analyzing the
effects of component failure in a circuit without knowing
exactly how many volts the battery produces, how many
ohms of resistance is in each resistor, etc. This section serves
as an introductory step to that kind of analysis.
Whereas the normal application of Ohm's Law and the rules of
series and parallel circuits is performed with numerical
quantities ("quantitative"), this new kind of analysis without
precise numerical figures is something | like to call qualitative
analysis. In other words, we will be analyzing the qualities of
the effects in a circuit rather than the precise quantities. The
result, for you, will be a much deeper intuitive understanding
of electric circuit operation.
e REVIEW:
e To determine what would happen in a circuit if a
component fails, re-draw that circuit with the equivalent
resistance of the failed component in place and re-
calculate all values.
e The ability to intuitively determine what will happen toa
circuit with any given component fault is a cruci/a/ skill for
any electronics troubleshooter to develop. The best way
to learn is to experiment with circuit calculations and real-
life circuits, paying close attention to what changes with a
fault, what remains the same, and why!
e A shorted component is one whose resistance has
dramatically decreased.
e An open component is one whose resistance has
dramatically increased. For the record, resistors tend to
fail open more often than fail shorted, and they almost
never fail unless physically or electrically overstressed
(physically abused or overheated).
Building simple resistor circuits
In the course of learning about electricity, you will want to
construct your own circuits using resistors and batteries.
Some options are available in this matter of circuit assembly,
some easier than others. In this section, | will explore a couple
of fabrication techniques that will not only help you build the
circuits shown in this chapter, but also more advanced
circuits.
If all we wish to construct is a simple single-battery, single-
resistor circuit, we may easily use a//igator clip jumper wires
like this:
Schematic
diagram
Real circuit using jumper wires
Resistor
Battery
Jumper wires with "alligator" style spring clips at each end
provide a safe and convenient method of electrically joining
components together.
If we wanted to build a simple series circuit with one battery
and three resistors, the same "point-to-point" construction
technique using jumper wires could be applied:
Schematic
diagram
Real circuit using jumper wires
Battery
This technique, however, proves impractical for circuits much
more complex than this, due to the awkwardness of the
jumper wires and the physical fragility of their connections. A
more common method of temporary construction for the
hobbyist is the so/derless breadboard, a device made of
plastic with hundreds of spring-loaded connection sockets
joining the inserted ends of components and/or 22-gauge
solid wire pieces. A photograph of a real breadboard is shown
here, followed by an illustration showing a simple series
circuit constructed on one:
Schematic
diagram
Underneath each hole in the breadboard face is a metal
spring clip, designed to grasp any inserted wire or component
lead. These metal spring clips are joined underneath the
breadboard face, making connections between inserted leads.
The connection pattern joins every five holes along a vertical
column (as shown with the long axis of the breadboard
situated horizontally):
Lines show common connections
underneath board between holes
HHAAHHHLHHHHLLETIIILL
HHddHHHHHHHHLLLLLLLLILL
Thus, when a wire or component lead is inserted into a hole
on the breadboard, there are four more holes in that column
providing potential connection points to other wires and/or
component leads. The result is an extremely flexible platform
for constructing temporary circuits. For example, the three-
resistor circuit just shown could also be built on a breadboard
like this:
Schematic
diagram
Battery
A parallel circuit is also easy to construct on a solderless
breadboard:
Schematic
diagram
Breadboards have their limitations, though. First and
foremost, they are intended for temporary construction only.
If you pick up a breadboard, turn it upside-down, and shake it,
any components plugged into it are sure to loosen, and may
fall out of their respective holes. Also, breadboards are limited
to fairly low-current (less than 1 amp) circuits. Those spring
clips have a small contact area, and thus cannot support high
currents without excessive heating.
For greater permanence, one might wish to choose soldering
or wire-wrapping. These techniques involve fastening the
components and wires to some structure providing a secure
mechanical location (Such as a phenolic or fiberglass board
with holes drilled in it, much like a breadboard without the
intrinsic spring-clip connections), and then attaching wires to
the secured component leads. Soldering is a form of low-
temperature welding, using a tin/lead or tin/silver alloy that
melts to and electrically bonds copper objects. Wire ends
soldered to component leads or to small, copper ring "pads"
bonded on the surface of the circuit board serve to connect
the components together. In wire wrapping, a small-gauge
wire is tightly wrapped around component leads rather than
soldered to leads or copper pads, the tension of the wrapped
wire providing a sound mechanical and electrical junction to
connect components together.
An example of a printed circuit board, or PCB, intended for
hobbyist use is shown in this photograph:
This board appears copper-side-up: the side where all the
soldering is done. Each hole is ringed with a small layer of
copper metal for bonding to the solder. All holes are
independent of each other on this particular board, unlike the
holes on a solderless breadboard which are connected
together in groups of five. Printed circuit boards with the
same 5-hole connection pattern as breadboards can be
purchased and used for hobby circuit construction, though.
Production printed circuit boards have traces of copper laid
down on the phenolic or fiberglass substrate material to form
pre-engineered connection pathways which function as wires
in a circuit. An example of such a board is shown here, this
unit actually a "power supply" circuit designed to take 120
volt alternating current (AC) power from a household wall
socket and transform it into low-voltage direct current (DC). A
resistor appears on this board, the fifth component counting
up from the bottom, located in the middle-right area of the
board.
A view of this board's underside reveals the copper "traces"
connecting components together, as well as the silver-colored
deposits of solder bonding the component leads to those
traces:
A soldered or wire-wrapped circuit is considered permanent:
that is, it is unlikely to fall apart accidently. However, these
construction techniques are sometimes considered too
permanent. If anyone wishes to replace a component or
change the circuit in any substantial way, they must invest a
fair amount of time undoing the connections. Also, both
soldering and wire-wrapping require specialized tools which
may not be immediately available.
An alternative construction technique used throughout the
industrial world is that of the terminal strip. Terminal strips,
alternatively called barrier strips or terminal blocks, are
comprised of a length of nonconducting material with several
small bars of metal embedded within. Each metal bar has at
least one machine screw or other fastener under which a wire
or component lead may be secured. Multiple wires fastened
by one screw are made electrically common to each other, as
are wires fastened to multiple screws on the same bar. The
following photograph shows one style of terminal strip, with a
few wires attached.
Another, smaller terminal strip is shown in this next
photograph. This type, sometimes referred to as a "European"
style, has recessed screws to help prevent accidental shorting
between terminals by a screwdriver or other metal object:
In the following illustration, a single-battery, three-resistor
circuit is shown constructed on a terminal strip:
Series circuit constructed on a
terminal strip
@| |S! |S | S| |S |S |S) |S] |S) |S! |S) |S |S] |S |e
@} |O} |S] |S] |S} [S| |S} |S] |S} |S} |S} |S} |S} [S| |e
If the terminal strip uses machine screws to hold the
component and wire ends, nothing but a screwdriver is
needed to secure new connections or break old connections.
Some terminal strips use spring-loaded clips -- similar to a
breadboard's except for increased ruggedness -- engaged and
disengaged using a screwdriver as a push tool (no twisting
involved). The electrical connections established by a
terminal strip are quite robust, and are considered suitable for
both permanent and temporary construction.
One of the essential skills for anyone interested in electricity
and electronics is to be able to "translate" a schematic
diagram to a real circuit layout where the components may
not be oriented the same way. Schematic diagrams are
usually drawn for maximum readability (excepting those few
noteworthy examples sketched to create maximum
confusion!), but practical circuit construction often demands
a different component orientation. Building simple circuits on
terminal strips is one way to develop the spatial-reasoning
Skill of "stretching" wires to make the same connection paths.
Consider the case of a single-battery, three-resistor parallel
circuit constructed on a terminal strip:
Schematic diagram
Real circuit using a terminal strip
@} |O| | S| |S |S |e |e@
@} |O| |S] |S} |S} |e} |e
we
@
Progressing from a nice, neat, schematic diagram to the real
circuit -- especially when the resistors to be connected are
physically arranged in a /inear fashion on the terminal strip --
is not obvious to many, so I'll outline the process step-by-
step. First, start with the clean schematic diagram and all
components secured to the terminal strip, with no connecting
wires:
Schematic diagram
Real circuit using a terminal strip
@| |O| |S} |S} |S) |S} |S} || |S} |S} |S) | S| |S} |e) |@
A, /
Z)
Next, trace the wire connection from one side of the battery
to the first component in the schematic, securing a
connecting wire between the same two points on the real
circuit. | find it helpful to over-draw the schematic's wire with
another line to indicate what connections I've made in real
life:
Schematic diagram
Real circuit using a terminal strip
@
Q| |@| | S| |S! |S! |S (S| |S |S |e
@| |@} | S| |S} |S} |S} [S| |S} [S| |e
Continue this process, wire by wire, until all connections in
the schematic diagram have been accounted for. It might be
helpful to regard common wires in a SPICE-like fashion: make
all connections to a common wire in the circuit as one step,
making sure each and every component with a connection to
that wire actually has a connection to that wire before
proceeding to the next. For the next step, I'll show how the
top sides of the remaining two resistors are connected
together, being common with the wire secured in the previous
step:
Schematic diagram
Real circuit using a terminal strip
@} |O| |S} |@| |e
With the top sides of all resistors (as shown in the schematic)
connected together, and to the battery's positive (+)
terminal, all we have to do now is connect the bottom sides
together and to the other side of the battery:
Schematic diagram
Real circuit using a terminal strip
Typically in industry, all wires are labeled with number tags,
and electrically common wires bear the same tag number,
just as they do ina SPICE simulation. In this case, we could
label the wires 1 and 2:
Common wire numbers representing
electrically common points
Another industrial convention is to modify the schematic
diagram slightly so as to indicate actual wire connection
points on the terminal strip. This demands a labeling system
for the strip itself: a "TB" number (terminal block number) for
the strip, followed by another number representing each
metal bar on the strip.
Terminal strip bars labeled and
connection points referenced in diagram
TB1
2
@
15
@
2
This way, the schematic may be used as a "map" to locate
points in a real circuit, regardless of how tangled and complex
the connecting wiring may appear to the eyes. This may seem
excessive for the simple, three-resistor circuit shown here, but
such detail is absolutely necessary for construction and
maintenance of large circuits, especially when those circuits
may span a great physical distance, using more than one
terminal strip located in more than one panel or box.
e REVIEW:
e A solderless breadboard is a device used to quickly
assemble temporary circuits by plugging wires and
components into electrically common spring-clips
arranged underneath rows of holes in a plastic board.
e Soldering is a low-temperature welding process utilizing a
lead/tin or tin/silver alloy to bond wires and component
leads together, usually with the components secured to a
fiberglass board.
e Wire-wrapping is an alternative to soldering, involving
small-gauge wire tightly wrapped around component
leads rather than a welded joint to connect components
together.
e A terminal strip, also Known as a barrier strip or terminal
block is another device used to mount components and
wires to build circuits. Screw terminals or heavy spring
clips attached to metal bars provide connection points for
the wire ends and component leads, these metal bars
mounted separately to a piece of nonconducting material
such as plastic, bakelite, or ceramic.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Jason Starck (June 2000): HTML document formatting, which
led to a much better-looking second edition.
Ron LaPlante (October 1998): helped create "table" method
of series and parallel circuit analysis.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—| | 4/l—
—/ | 4]
Lessons In Electric Circuits
-- Volume |
Chapter 6
DIVIDER CIRCUITS AND
KIRCHHOFF'S LAWS
Voltage divider circuits
Kirchhoff's Voltage Law (KVL)
Current divider circuits
Kirchhoff's Current Law (KCL)
Contributors
Voltage divider circuits
Let's analyze a simple series circuit, determining the voltage
drops across individual resistors:
From the given values of individual resistances, we can
determine a total circuit resistance, knowing that resistances
add in series:
From here, we can use Ohm's Law (I=E/R) to determine the
total current, which we know will be the same as each
resistor current, currents being equal in all parts of a series
circuit:
R, R, R, Total
E pts | Vorts
| Amps
R| 5k 10k Ohms
Now, knowing that the circuit current is 2 mA, we can use
Ohm's Law (E=IR) to calculate voltage across each resistor:
R, R, R, Total
It should be apparent that the voltage drop across each
resistor is proportional to its resistance, given that the
current is the same through all resistors. Notice how the
voltage across R> is double that of the voltage across Rj, just
as the resistance of R> is double that of Rj.
If we were to change the total voltage, we would find this
proportionality of voltage drops remains constant:
R, R, R, Total
22.5k
The voltage across R; is still exactly twice that of R,'s drop,
despite the fact that the source voltage has changed. The
proportionality of voltage drops (ratio of one to another) is
strictly a function of resistance values.
With a little more observation, it becomes apparent that the
voltage drop across each resistor is also a fixed proportion of
the supply voltage. The voltage across Rj, for example, was
10 volts when the battery supply was 45 volts. When the
battery voltage was increased to 180 volts (4 times as
much), the voltage drop across R, also increased by a factor
of 4 (from 10 to 40 volts). The ratio between R,'s voltage
drop and total voltage, however, did not change:
Eo, 7 10 V a 40 V ~ 922277
Bea 45 V is0V—~™
Likewise, none of the other voltage drop ratios changed with
the increased supply voltage either:
Ep 20 V 80 V
= —— = ———_ = 044444
Baa 45 V 180 V
E 2 a .
oe A15Vv Ss 0.33333
Bos 45 V 180 V
For this reason a series circuit is often called a vo/tage
divider for its ability to proportion -- or divide -- the total
voltage into fractional portions of constant ratio. With a little
bit of algebra, we can derive a formula for determining
series resistor voltage drop given nothing more than total
voltage, individual resistance, and total resistance:
Voltage drop across any resistor E.=1_R.
; ere Evotal
Current in a series circuit Lowi = =
Riota
. . Etat . . .
. .. Substituting ——— for I, inthe first equation . ..
total
: : Estat
Voltage drop across any series resistor E =
1 R iT
a? | come
The ratio of individual resistance to total resistance is the
same as the ratio of individual voltage drop to total supply
voltage in a voltage divider circuit. This is known as the
voltage divider formula, and it is a short-cut method for
determining voltage drop in a series circuit without going
through the current calculation(s) of Ohm's Law.
Using this formula, we can re-analyze the example circuit's
voltage drops in fewer steps:
Eg, = 45 V ———=10V
22.5 kQ
- 22.5 kQ
fay ts ayy
5 kQ
Voltage dividers find wide application in electric meter
circuits, where specific combinations of series resistors are
used to "divide" a voltage into precise proportions as part of
a voltage measurement device.
voltage \
voltage
One device frequently used as a voltage-dividing
component is the potentiometer, which is a resistor with a
movable element positioned by a manual knob or lever. The
movable element, typically called a wiper, makes contact
with a resistive strip of material (commonly called the
slidewire if made of resistive metal wire) at any point
selected by the manual control:
1
Potentiometer
wiper contact
2
The wiper contact is the left-facing arrow symbol drawn in
the middle of the vertical resistor element. As it is moved up,
it contacts the resistive strip closer to terminal 1 and further
away from terminal 2, lowering resistance to terminal 1 and
raising resistance to terminal 2. As it is moved down, the
opposite effect results. The resistance as measured between
terminals 1 and 2 is constant for any wiper position.
1
| less resistance _
more resistance
less resistance
~—
2
a —— a ance
Shown here are internal illustrations of two potentiometer
types, rotary and linear:
Terminals
f\\
Rotary potentiometer
construction
Wiper
Resistive strip
Linear potentiometer construction
Wiper oa .
Resistive strip
\\ 7
Terminals
Some linear potentiometers are actuated by straight-line
motion of a lever or slide button. Others, like the one
depicted in the previous illustration, are actuated by a turn-
screw for fine adjustment ability. The latter units are
sometimes referred to as trimpots, because they work well
for applications requiring a variable resistance to be
"trimmed" to some precise value. It should be noted that not
all linear potentiometers have the same terminal
assignments as shown in this illustration. With some, the
wiper terminal is in the middle, between the two end
terminals.
The following photograph shows a real, rotary potentiometer
with exposed wiper and slidewire for easy viewing. The shaft
which moves the wiper has been turned almost fully
clockwise so that the wiper is nearly touching the left
terminal end of the slidewire:
Here is the same potentiometer with the wiper shaft moved
almost to the full-counterclockwise position, so that the
wiper is near the other extreme end of travel:
If a constant voltage is applied between the outer terminals
(across the length of the slidewire), the wiper position will
tap off a fraction of the applied voltage, measurable
between the wiper contact and either of the other two
terminals. The fractional value depends entirely on the
physical position of the wiper:
Using a potentiometer as a variable voltage divider
: : : : 7
“A
Just like the fixed voltage divider, the potentiometer's
voltage division ratio is strictly a function of resistance and
not of the magnitude of applied voltage. In other words, if
the potentiometer knob or lever is moved to the 50 percent
(exact center) position, the voltage dropped between wiper
and either outside terminal would be exactly 1/2 of the
applied voltage, no matter what that voltage happens to be,
or what the end-to-end resistance of the potentiometer is. In
other words, a potentiometer functions as a variable voltage
divider where the voltage division ratio is set by wiper
position.
This application of the potentiometer is a very useful means
of obtaining a variable voltage from a fixed-voltage source
such as a battery. If a circuit you're building requires a
certain amount of voltage that is less than the value of an
available battery's voltage, you may connect the outer
terminals of a potentiometer across that battery and "dial
up" whatever voltage you need between the potentiometer
wiper and one of the outer terminals for use in your circuit:
Adjust potentiometer
to obtain desired
/ roltag e
Battery —
Circuit requiring
less voltage than
what the battery
provides
When used in this manner, the name potentiometer makes
perfect sense: they meter (control) the potential (voltage)
applied across them by creating a variable voltage-divider
ratio. This use of the three-terminal potentiometer as a
variable voltage divider is very popular in circuit design.
Shown here are several small potentiometers of the kind
commonly used in consumer electronic equipment and by
hobbyists and students in constructing circuits:
The smaller units on the very left and very right are
designed to plug into a solderless breadboard or be soldered
into a printed circuit board. The middle units are designed to
be mounted on a flat panel with wires soldered to each of
the three terminals.
Here are three more potentiometers, more specialized than
the set just shown:
5 000 rly
Lineanety TOL 201%
S WATTS Of AMPERES
HeliporT
The large "Helipot" unit is a laboratory potentiometer
designed for quick and easy connection to a circuit. The unit
in the lower-left corner of the photograph is the same type of
potentiometer, just without a case or 10-turn counting dial.
Both of these potentiometers are precision units, using
multi-turn helical-track resistance strips and wiper
mechanisms for making small adjustments. The unit on the
lower-right is a panel-mount potentiometer, designed for
rough service in industrial applications.
e REVIEW:
e Series circuits proportion, or divide, the total supply
voltage among individual voltage drops, the proportions
being strictly dependent upon resistances: Epa, = Erptal
(Ry / Rtotal)
e A potentiometer is a variable-resistance component with
three connection points, frequently used as an
adjustable voltage divider.
Kirchhoff's Voltage Law (KVL)
Let's take another look at our example series circuit, this
time numbering the points in the circuit for voltage
reference:
If we were to connect a voltmeter between points 2 and 1,
red test lead to point 2 and black test lead to point 1, the
meter would register +45 volts. Typically the "+" sign is not
shown, but rather implied, for positive readings in digital
meter displays. However, for this lesson the polarity of the
voltage reading is very important and so | will show positive
numbers explicitly:
When a voltage is specified with a double subscript (the
characters "2-1" in the notation "E>_;"), it means the voltage
at the first point (2) as measured in reference to the second
point (1). A voltage specified as "E.g" would mean the
voltage as indicated by a digital meter with the red test lead
on point "c" and the black test lead on point "d": the voltage
at "c" in reference to "d".
The meaning of
Fea
[O* | [cm@l
Black Red
d c
If we were to take that same voltmeter and measure the
voltage drop across each resistor, stepping around the
circuit in a clockwise direction with the red test lead of our
meter on the point ahead and the black test lead on the
point behind, we would obtain the following readings:
E,., = -10 V
E,;=-20V
We should already be familiar with the general principle for
series circuits stating that individual voltage drops add up to
the total applied voltage, but measuring voltage drops in
this manner and paying attention to the polarity
(mathematical sign) of the readings reveals another facet of
this principle: that the voltages measured as such all add up
to zero:
E,,= +#45V_ voltage from point 2to point 1
32= -l0V_ voltage from point 3to point 2
E,;= -20V__ voltage from point 4to point 3
+ E,,= -15V_ voltage from point 1to point 4
OV
This principle is known as Kirchhoff's Voltage Law
(discovered in 1847 by Gustav R. Kirchhoff, a German
physicist), and it can be stated as such:
"The algebraic sum of all voltages in a loop must
equal zero"
By algebraic, | mean accounting for signs (polarities) as well
as magnitudes. By /oop, | mean any path traced from one
point in a circuit around to other points in that circuit, and
finally back to the initial point. In the above example the
loop was formed by following points in this order: 1-2-3-4-1.
It doesn't matter which point we start at or which direction
we proceed in tracing the loop; the voltage sum will still
equal zero. To demonstrate, we can tally up the voltages in
loop 3-2-1-4-3 of the same circuit:
+10 V_ voltage from point 2to point 3
12= -45V_ voltage from point 1to point 2
+15 V_ voltage from point 4to point 1
34= +20V_ voltage from point 3to point 4
OV
m™
un
oi
This may make more sense if we re-draw our example series
circuit so that all components are represented in a straight
line:
current
SkKQ°> tok 5KQ sy
current
It's still the same series circuit, just with the components
arranged in a different form. Notice the polarities of the
resistor voltage drops with respect to the battery: the
battery's voltage is negative on the left and positive on the
right, whereas all the resistor voltage drops are oriented the
other way: positive on the left and negative on the right.
This is because the resistors are resisting the flow of
electrons being pushed by the battery. In other words, the
“oush" exerted by the resistors against the flow of electrons
must be in a direction opposite the source of electromotive
force.
Here we see what a digital voltmeter would indicate across
each component in this circuit, black lead on the left and red
lead on the right, as laid out in horizontal fashion:
current
2 2
kQ | “10kQ
-10V -20 V -15V +45V
E35 E, ; E.4 E, l
If we were to take that same voltmeter and read voltage
across combinations of components, starting with only R, on
the left and progressing across the whole string of
components, we will see how the voltages add algebraically
(to zero):
current
E,,
The fact that series voltages add up should be no mystery,
but we notice that the po/arity of these voltages makes a lot
of difference in how the figures add. While reading voltage
across Rj, R,--R>, and Rj--R>--R3 (I'm using a "double-dash"
symbol "--" to represent the series connection between
resistors R;, Ro, and R3), we see how the voltages measure
successively larger (albeit negative) magnitudes, because
the polarities of the individual voltage drops are in the same
orientation (positive left, negative right). The sum of the
voltage drops across Rj, Rz, and R3 equals 45 volts, which is
the same as the battery's output, except that the battery's
polarity is opposite that of the resistor voltage drops
(negative left, positive right), So we end up with 0 volts
measured across the whole string of components.
That we should end up with exactly 0 volts across the whole
string should be no mystery, either. Looking at the circuit,
we can see that the far left of the string (left side of Ry: point
number 2) is directly connected to the far right of the string
(right side of battery: point number 2), as necessary to
complete the circuit. Since these two points are directly
connected, they are electrically common to each other. And,
as such, the voltage between those two electrically common
points must be zero.
Kirchhoff's Voltage Law (Sometimes denoted as KVL for
Short) will work for any circuit configuration at all, not just
simple series. Note how it works for this parallel circuit:
Being a parallel circuit, the voltage across every resistor is
the same as the supply voltage: 6 volts. Tallying up voltages
around loop 2-3-4-5-6-7 -2, we get:
E,. 0V_ voltage from point 3to point 2
E,;= 0V_ voltage from point 4to point 3
E -6 V___ voltage from point 5to point 4
E 0V_ voltage from point 6to point 5
= OV _ voltage from point 7to point 6
+E,,=+6V_ voltage from point 2to point 7
E,,= OV
Note how | label the final (sum) voltage as E3_5. Since we
began our loop-stepping sequence at point 2 and ended at
point 2, the algebraic sum of those voltages will be the same
as the voltage measured between the same point (E>_5),
which of course must be zero.
The fact that this circuit is parallel instead of series has
nothing to do with the validity of Kirchhoff's Voltage Law. For
that matter, the circuit could be a "black box" -- its
component configuration completely hidden from our view,
with only a set of exposed terminals for us to measure
voltage between -- and KVL would still hold true:
Try any order of steps from any terminal in the above
diagram, stepping around back to the original terminal, and
you'll find that the algebraic sum of the voltages a/ways
equals zero.
Furthermore, the "loop" we trace for KVL doesn't even have
to be a real current path in the closed-circuit sense of the
word. All we have to do to comply with KVL is to begin and
end at the same point in the circuit, tallying voltage drops
and polarities as we go between the next and the last point.
Consider this absurd example, tracing "loop" 2-3-6-3-2 in the
same parallel resistor circuit:
E;,= 0V_ voltage from point 3to point 2
E,;= -6V__ voltage from point 6to point 3
E,;,= +6V voltage from point 3to point 6
+E,;= 0V_ voltage from point 2to point 3
E,2.= OV
KVL can be used to determine an unknown voltage in a
complex circuit, where all other voltages around a particular
"loop" are known. Take the following complex circuit
(actually two series circuits joined by a single wire at the
bottom) as an example:
To make the problem simpler, I've omitted resistance values
and simply given voltage drops across each resistor. The two
series circuits share a common wire between them (wire 7 -8-
9-10), making voltage measurements between the two
circuits possible. If we wanted to determine the voltage
between points 4 and 3, we could set up a KVL equation
with the voltage between those points as the unknown:
E,;+E,,+E,,+E;,=0
E,;+124+0+20=0
E,,;+32=0
E,;=-32V
7 8 9 10
Measuring voltage from point 4 to point 3 (unknown amount)
E
43
F 8 9 10
Measuring voltage from point 9 to point 4 (+12 volts)
E,,+ 12
7 8 9 10
Measuring voltage from point 8 to point 9 (0 volts)
E,,;+12+0
7 8 9 10
Measuring voltage from point 3 to point 8 (+20 volts)
E,,+12+0+20=0
Stepping around the loop 3-4-9-8-3, we write the voltage
drop figures as a digital voltmeter would register them,
measuring with the red test lead on the point ahead and
black test lead on the point behind as we progress around
the loop. Therefore, the voltage from point 9 to point 4 isa
positive (+) 12 volts because the "red lead" is on point 9
and the "black lead" is on point 4. The voltage from point 3
to point 8 is a positive (+) 20 volts because the "red lead" is
on point 3 and the "black lead" is on point 8. The voltage
from point 8 to point 9 is zero, of course, because those two
points are electrically common.
Our final answer for the voltage from point 4 to point 3 isa
negative (-) 32 volts, telling us that point 3 is actually
positive with respect to point 4, precisely what a digital
voltmeter would indicate with the red lead on point 4 and
the black lead on point 3:
E,;=-32
3
In other words, the initial placement of our "meter leads" in
this KVL problem was "backwards." Had we generated our
KVL equation starting with E3_, instead of E,.3, stepping
around the same loop with the opposite meter lead
orientation, the final answer would have been E3.4 = +32
volts:
It is important to realize that neither approach is "wrong." In
both cases, we arrive at the correct assessment of voltage
between the two points, 3 and 4: point 3 is positive with
respect to point 4, and the voltage between them is 32 volts.
e REVIEW:
e Kirchhoff's Voltage Law (KVL): "The algebraic sum of all
voltages in a loop must equal zero"
Current divider circuits
Let's analyze a simple parallel circuit, determining the
branch currents through individual resistors:
Knowing that voltages across all components in a parallel
circuit are the same, we can fill in our
voltage/current/resistance table with 6 volts across the top
row:
Using Ohm's Law (I=E/R) we can calculate each branch
current:
Knowing that branch currents add up in parallel circuits to
equal the total current, we can arrive at total current by
summing 6 mA, 2 mA, and 3 mA:
R, R, R, Total
The final step, of course, is to figure total resistance. This
can be done with Ohm's Law (R=E/I) in the "total" column,
or with the parallel resistance formula from individual
resistances. Either way, we'll get the same answer:
R, R, R, Total
Volts
3m Amps
Ohms
Once again, it should be apparent that the current through
each resistor is related to its resistance, given that the
voltage across all resistors is the same. Rather than being
directly proportional, the relationship here is one of inverse
proportion. For example, the current through R, is twice as
much as the current through R3, which has twice the
resistance of R,.
If we were to change the supply voltage of this circuit, we
find that (Surprise!) these proportional ratios do not change:
R, R, R, Total
Volts
Amps
24 4
é 8m 2
Ohms
The current through R; is still exactly twice that of R3,
despite the fact that the source voltage has changed. The
proportionality between different branch currents is strictly
a function of resistance.
Also reminiscent of voltage dividers is the fact that branch
currents are fixed proportions of the total current. Despite
the fourfold increase in supply voltage, the ratio between
any branch current and the total current remains
unchanged:
l }
Rl 7 6mA = 24 mA — 0.54545
Loti llmA 44 mA
Tiss
R2 = 2 mA = 8mA — 0.18182
Luis! ll mA 44 mA
los ’ on i
R3 = 3 mA 7 12 mA ~ 027273
i ll mA 44 mA
For this reason a parallel circuit is often called a current
divider for its ability to proportion -- or divide -- the total
Current into fractional parts. With a little bit of algebra, we
can derive a formula for determining parallel resistor current
given nothing more than total current, individual resistance,
and total resistance:
Current through any resistor Le —
Voltage in a parallel circuit Boob -=1L Bi
... Substituting Those Rioras fOr E,, in the first equation .
Total Riotal
Current through any paralle/resistor 1, = R
T
Ph)? | ere
The ratio of total resistance to individual resistance is the
same ratio as individual (branch) current to total current.
This is Known as the current divider formula, and it is a
short-cut method for determining branch currents in a
parallel circuit when the total current is known.
Using the original parallel circuit as an example, we can re-
calculate the branch currents using this formula, if we start
by knowing the total current and total resistance:
I, = 11 ey cake = 6mA
LkQ
L,= 11 mA 245 2 = 2mA
~ 3kQ
L,= 11 mA 245 2 = 3mA
2kQ
If you take the time to compare the two divider formulae,
you'll see that they are remarkably similar. Notice, however,
that the ratio in the voltage divider formula is R, (individual
resistance) divided by Ryota;, and how the ratio in the current
divider formula is Ryp¢4; divided by R,:
Voltage divider Current divider
formula formula
It is quite easy to confuse these two equations, getting the
resistance ratios backwards. One way to help remember the
proper form is to keep in mind that both ratios in the voltage
and current divider equations must equal less than one.
After all these are divider equations, not multiplier
equations! If the fraction is upside-down, it will provide a
ratio greater than one, which is incorrect. Knowing that total
resistance in a series (voltage divider) circuit is always
greater than any of the individual resistances, we know that
the fraction for that formula must be R,, over Ryotal-
Conversely, knowing that total resistance in a parallel
(current divider) circuit is always less then any of the
individual resistances, we know that the fraction for that
formula must be Ryp¢,; over R,.
Current divider circuits also find application in electric meter
circuits, where a fraction of a measured current is desired to
be routed through a sensitive detection device. Using the
current divider formula, the proper shunt resistor can be
sized to proportion just the right amount of current for the
device in any given instance:
Lotal —_> R shunt —_ Liotal
fraction of total
current
sensitive device
e REVIEW:
e Parallel circuits proportion, or "divide," the total circuit
current among individual branch currents, the
proportions being strictly dependent upon resistances:
In = Irotar (Rtotat / Rn)
Kirchhoff s Current Law (KCL)
Let's take a closer look at that last parallel example circuit:
Solving for all values of voltage and current in this circuit:
R, R, R, Total
At this point, we know the value of each branch current and
of the total current in the circuit. We know that the total
Current in a parallel circuit must equal the sum of the branch
currents, but there's more going on in this circuit than just
that. Taking a look at the currents at each wire junction point
(node) in the circuit, we should be able to see something
else:
Iki + Iho + Ih Tho + Ip3
i 2 —
At each node on the negative "rail" (wire 8-7-6-5) we have
current splitting off the main flow to each successive branch
resistor. At each node on the positive "rail" (wire 1-2-3-4) we
have current merging together to form the main flow from
each successive branch resistor. This fact should be fairly
obvious if you think of the water pipe circuit analogy with
every branch node acting as a "tee" fitting, the water flow
splitting or merging with the main piping as it travels from
the output of the water pump toward the return reservoir or
sump.
If we were to take a closer look at one particular "tee" node,
such as node 3, we see that the current entering the node is
equal in magnitude to the current exiting the node:
Igo + Ip; Ip3
—_—
From the right and from the bottom, we have two currents
entering the wire connection labeled as node 3. To the left,
we have a single current exiting the node equal in
magnitude to the sum of the two currents entering. To refer
to the plumbing analogy: so long as there are no leaks in the
piping, what flow enters the fitting must also exit the fitting.
This holds true for any node ("fitting"), no matter how many
flows are entering or exiting. Mathematically, we can
express this general relationship as such:
1 =1
exiting entering
Mr. Kirchhoff decided to express it in a slightly different form
(though mathematically equivalent), calling it Kirchhoff's
Current Law (KCL):
Lantering + (-lexiting) =0
Summarized in a phrase, Kirchhoff's Current Law reads as
such:
"The algebraic sum of all currents entering and
exiting a node must equal zero"
That is, if we assign a mathematical sign (polarity) to each
current, denoting whether they enter (+) or exit (-) a node,
we can add them together to arrive at a total of zero,
guaranteed.
Taking our example node (number 3), we can determine the
magnitude of the current exiting from the left by setting up
a KCL equation with that current as the unknown value:
1,+1,+1=0
2mA+3mA+1=0
... Solving for/...
-2mA-3mA
l=
I=-5mA
The negative (-) sign on the value of 5 milliamps tells us that
the current is exiting the node, as opposed to the 2 milliamp
and 3 milliamp currents, which must both be positive (and
therefore entering the node). Whether negative or positive
denotes current entering or exiting is entirely arbitrary, so
long as they are opposite signs for opposite directions and
we stay consistent in our notation, KCL will work.
Together, Kirchhoff's Voltage and Current Laws are a
formidable pair of tools useful in analyzing electric circuits.
Their usefulness will become all the more apparent in a later
chapter ("Network Analysis"), but suffice it to say that these
Laws deserve to be memorized by the electronics student
every bit as much as Ohm's Law.
e REVIEW:
e Kirchhoff's Current Law (KCL): “The algebraic sum of all
currents entering and exiting a node must equal zero"
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Ron LaPlante (October 1998): helped create "table"
method of series and parallel circuit analysis.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+4]l—
—||+]l—
Lessons In Electric Circuits
-- Volume |!
Chapter 7
SERIES-PARALLEL
COMBINATION CIRCUITS
What is a series-parallel circuit?
Analysis technique
Re-drawing complex schematics
Component failure analysis
Building series-parallel resistor circuits
Contributors
What is a series-parallel circuit?
With simple series circuits, all components are connected
end-to-end to form only one path for electrons to flow through
the circuit:
Series
With simple parallel circuits, all components are connected
between the same two sets of electrically common points,
creating multiple paths for electrons to flow from one end of
the battery to the other:
Parallel
With each of these two basic circuit configurations, we have
specific sets of rules describing voltage, current, and
resistance relationships.
Series Circuits:
Voltage drops add to equal total voltage.
All components share the same (equal) current.
Resistances add to equal total resistance.
Parallel Circuits:
All components share the same (equal) voltage.
Branch currents add to equal total current.
Resistances diminish to equal total resistance.
However, if circuit components are series-connected in some
parts and parallel in others, we won't be able to apply a single
set of rules to every part of that circuit. Instead, we will have
to identify which parts of that circuit are series and which
parts are parallel, then selectively apply series and parallel
rules as necessary to determine what is happening. Take the
following circuit, for instance:
A series-parallel combination circuit
24V
This circuit is neither simple series nor simple parallel. Rather,
it contains elements of both. The current exits the bottom of
the battery, splits up to travel through R3 and Rg, rejoins,
then splits up again to travel through R, and R3>, then rejoins
again to return to the top of the battery. There exists more
than one path for current to travel (not series), yet there are
more than two sets of electrically common points in the
circuit (not parallel).
Because the circuit is a combination of both series and
parallel, we cannot apply the rules for voltage, current, and
resistance "across the table" to begin analysis like we could
when the circuits were one way or the other. For instance, if
the above circuit were simple series, we could just add up Rj
through R, to arrive at a total resistance, solve for total
current, and then solve for all voltage drops. Likewise, if the
above circuit were simple parallel, we could just solve for
branch currents, add up branch currents to figure the total
current, and then calculate total resistance from total voltage
and total current. However, this circuit's solution will be more
complex.
The table will still help us manage the different values for
series-parallel combination circuits, but we'll have to be
careful how and where we apply the different rules for series
and parallel. Ohm's Law, of course, still works just the same
for determining values within a vertical column in the table.
If we are able to identify which parts of the circuit are series
and which parts are parallel, we can analyze it in stages,
approaching each part one at a time, using the appropriate
rules to determine the relationships of voltage, current, and
resistance. The rest of this chapter will be devoted to showing
you techniques for doing this.
e REVIEW:
e The rules of series and parallel circuits must be applied
selectively to circuits containing both types of
interconnections.
Analysis technique
The goal of series-parallel resistor circuit analysis is to be able
to determine all voltage drops, currents, and power
dissipations in a circuit. The general strategy to accomplish
this goal is as follows:
e Step 1: Assess which resistors in a circuit are connected
together in simple series or simple parallel.
e Step 2: Re-draw the circuit, replacing each of those series
or parallel resistor combinations identified in step 1 with a
single, equivalent-value resistor. If using a table to
manage variables, make a new table column for each
resistance equivalent.
e Step 3: Repeat steps 1 and 2 until the entire circuit is
reduced to one equivalent resistor.
e Step 4: Calculate total current from total voltage and total
resistance (I=E/R).
e Step 5: Taking total voltage and total current values, go
back to last step in the circuit reduction process and
insert those values where applicable.
e Step 6: From known resistances and total voltage / total
current values from step 5, use Ohm's Law to calculate
unknown values (voltage or current) (E=IR or I=E/R).
e Step 7: Repeat steps 5 and 6 until all values for voltage
and current are known in the original circuit
configuration. Essentially, you will proceed step-by-step
from the simplified version of the circuit back into its
original, complex form, plugging in values of voltage and
current where appropriate until all values of voltage and
Current are known.
e Step 8: Calculate power dissipations from known voltage,
current, and/or resistance values.
This may sound like an intimidating process, but its much
easier understood through example than through description.
A series-parallel combination circuit
In the example circuit above, R; and R> are connected ina
simple parallel arrangement, as are R3 and Ry. Having been
identified, these sections need to be converted into
equivalent single resistors, and the circuit re-drawn:
71.429Q SR, /R,
24V —
127.272 <SR,//R,
The double slash (//) symbols represent "parallel" to show that
the equivalent resistor values were calculated using the
1/(1/R) formula. The 71.429 O resistor at the top of the circuit
is the equivalent of R; and R> in parallel with each other. The
127.27 Q resistor at the bottom is the equivalent of R3 and Ry
in parallel with each other.
Our table can be expanded to include these resistor
equivalents in their own columns:
It should be apparent now that the circuit has been reduced
to a simple series configuration with only two (equivalent)
resistances. The final step in reduction is to add these two
resistances to come up with a total circuit resistance. When
we add those two equivalent resistances, we get a resistance
of 198.70 Q. Now, we can re-draw the circuit as a single
equivalent resistance and add the total resistance figure to
the rightmost column of our table. Note that the "Total"
column has been relabeled (Rj//R>--R3//R4) to indicate how it
relates electrically to the other columns of figures. The "--"
symbol is used here to represent "series," just as the "//"
symbol is used to represent "parallel."
24V = 198.702 SR,/R, —- R;//R,
R, Re
R3 Ra
R, R. Rs R, R,//R, R3l/Ry Total
Now, total circuit current can be determined by applying
Ohm's Law (I=E/R) to the "Total" column in the table:
R, // Re
R3 “Ry
R, Re R3 Rg R,//Rz Rgl Ry Total
Volts
Amps
Ohms
Back to our equivalent circuit drawing, our total current value
of 120.78 milliamps is shown as the only current here:
<—_—_—
1= 120.78 mA
24V 198.702 SRR, —- RMR,
1= 120.78 mA
—_
Now we start to work backwards in our progression of circuit
re-drawings to the original configuration. The next step is to
go to the circuit where R,//R>z and R3//R, are in series:
ja
1= 120.78 mA
71.4292 SR,/R,
ZAV — 1= 120.78 mA
1= 120.78 mA
oe
Since R;//R>z and R3//Ry are in series with each other, the
current through those two sets of equivalent resistances must
be the same. Furthermore, the current through them must be
the same as the total current, so we can fill in our table with
the appropriate current values, simply copying the current
figure from the Total column to the Rj//R>z and R3//Ry
columns:
R, Re
R3 Ry
R, R. Rs Ry R,//R, Re//R, Total
an OO Volts
Amps
Now, knowing the current through the equivalent resistors
R,//Rz and R3//R4, we can apply Ohm's Law (E=IR) to the two
right vertical columns to find voltage drops across them:
——
1= 120.78mA
oe ie
71.429Q SR,/R,; 8.6275 V
24V 1= 120.78 mA
— ————
127.272 SRR, 15.373 V
1= 120.78 mA
—-
R, Re
R3 Ry
R, Rp Rs R, RW Re Rel/R, Total
8.6275 18973 |__| Volts
Amps
L429 Ohms
Because we know R;j//R> and R3//R, are parallel resistor
equivalents, and we know that voltage drops in parallel
circuits are the same, we can transfer the respective voltage
drops to the appropriate columns on the table for those
individual resistors. In other words, we take another step
backwards in our drawing sequence to the original
configuration, and complete the table accordingly:
—<——_——
1= 120.78 mA
=
2502
<—__/-
100 Q
24V
te
2009 [15.373 V
<—__/-
350 Q
1= 120.78 mA
ed
R 1 if R 2
R5 if Ry
R, R2 R3 Ry R, If R Rs I Ry Total
18.37
12 .
Volts
20.78m_| Amps
Ohms
Finally, the original section of the table (columns R, through
Ry) is complete with enough values to finish. Applying Ohm's
Law to the remaining vertical columns (I=E/R), we can
determine the currents through Rj, R>, R3, and Ry
individually:
R, Re
R3// Ra
R, Re Rs R, RvR, Rs/R, Total
Volts
Amps
Ohms
Having found all voltage and current values for this circuit,
we can show those values in the schematic diagram as such:
—
1= 120.78 mA
La.
250Q [8.6275 V
R,
1002 SR, y
| 34.510 mA! [«———“
24V — £86275 mA
= te
200 2 15.373 V
76.363 mA| [~——“__
43.922 mA |
1= 120.78 mA
As a final check of our work, we can see if the calculated
current values add up as they should to the total. Since R;
and R> are in parallel, their combined currents should add up
to the total of 120.78 mA. Likewise, since R3 and Ry are in
parallel, their combined currents should also add up to the
total of 120.78 mA. You can check for yourself to verify that
these figures do add up as expected.
A computer simulation can also be used to verify the accuracy
of these figures. The following SPICE analysis will show all
resistor voltages and currents (note the current-sensing vil,
vi2,... "dummy" voltage sources in series with each resistor
in the netlist, necessary for the SPICE computer program to
track current through each path). These voltage sources will
be set to have values of zero volts each so they will not affect
the circuit in any way.
1 1
24V —
NOTE: voltage sources vil,
vi2, vi3, and vi4 are "dummy"
sources set at zero volts each.
series-parallel circuit
vl 10
vil 1 2 dc
vi2 13 dc
rl 2 4 100
r2 3 4 250
vi3 4 5 dc
vi4 4 6 dc
r3.5 0 350
r4 6 0 200
.dc vl 24 24 1
0
0
(oo)
.print dc v(2,4) v(3,4) v(5,0) v(6,0)
.print dc i(vil) i(vi2) i(vi3) i(vi4)
.end
I've annotated SPICE's output figures to make them more
readable, denoting which voltage and current figures belong
to which resistors.
vl v(2,4) v(3,4) v(5) v(6)
2.400E+01 8.627E+00 8.627E+00 1.537E+01 1.537E+01
Battery Rl voltage R2 voltage R3 voltage R4 voltage
voltage
vl i(vil) i(vi2) i(vi3) i(vi4)
2.400E+01 8.627E-02 3.451E-02 4.392E-02 7 .686E-02
Battery Rl current R2 current R3 current R4 current
voltage
As you can see, all the figures do agree with the our
calculated values.
e REVIEW:
e To analyze a series-parallel combination circuit, follow
these steps:
e Reduce the original circuit to a single equivalent resistor,
re-drawing the circuit in each step of reduction as simple
series and simple parallel parts are reduced to single,
equivalent resistors.
¢ Solve for total resistance.
e Solve for total current (I=E/R).
e Determine equivalent resistor voltage drops and branch
currents one stage at a time, working backwards to the
Original circuit configuration again.
Re-drawing complex schematics
Typically, complex circuits are not arranged in nice, neat,
clean schematic diagrams for us to follow. They are often
drawn in such a way that makes it difficult to follow which
components are in series and which are in parallel with each
other. The purpose of this section is to show you a method
useful for re-drawing circuit schematics in a neat and orderly
fashion. Like the stage-reduction strategy for solving series-
parallel combination circuits, it is a method easier
demonstrated than described.
Let's start with the following (convoluted) circuit diagram.
Perhaps this diagram was originally drawn this way by a
technician or engineer. Perhaps it was sketched as someone
traced the wires and connections of a real circuit. In any case,
here it is in all its ugliness:
With electric circuits and circuit diagrams, the length and
routing of wire connecting components in a circuit matters
little. (Actually, in some AC circuits it becomes critical, and
very long wire lengths can contribute unwanted resistance to
both AC and DC circuits, but in most cases wire length is
irrelevant.) What this means for us is that we can lengthen,
shrink, and/or bend connecting wires without affecting the
operation of our circuit.
The strategy | have found easiest to apply is to start by
tracing the current from one terminal of the battery around to
the other terminal, following the loop of components closest
to the battery and ignoring all other wires and components
for the time being. While tracing the path of the loop, mark
each resistor with the appropriate polarity for voltage drop.
In this case, I'll begin my tracing of this circuit at the negative
terminal of the battery and finish at the positive terminal, in
the same general direction as the electrons would flow. When
tracing this direction, | will mark each resistor with the
polarity of negative on the entering side and positive on the
exiting side, for that is how the actual polarity will be as
electrons (negative in charge) enter and exit a resistor:
Polarity of voltage drop
> aaah
a en er od
Direction of electron flow
Any components encountered along this short loop are drawn
vertically in order:
Now, proceed to trace any loops of components connected
around components that were just traced. In this case, there's
a loop around R, formed by R3, and another loop around R3
formed by R,:
R, loops aroundR
R, loops aroundR,
Tracing those loops, | draw R> and R, in parallel with R, and
R3 (respectively) on the vertical diagram. Noting the polarity
of voltage drops across R3 and Rj, | mark Ry and R;> likewise:
Now we have a circuit that is very easily understood and
analyzed. In this case, it is identical to the four-resistor series-
parallel configuration we examined earlier in the chapter.
Let's look at another example, even uglier than the one
before:
The first loop I'll trace is from the negative (-) side of the
battery, through Re, through Rj, and back to the positive (+)
end of the battery:
Re-drawing vertically and keeping track of voltage drop
polarities along the way, our equivalent circuit starts out
looking like this:
Next, we can proceed to follow the next loop around one of
the traced resistors (R¢), in this case, the loop formed by Rs
and R;. As before, we start at the negative end of Re and
proceed to the positive end of Re, marking voltage drop
polarities across R7 and Rs as we go:
R,; andR,;
loop around
Re
Now we add the Rs--R7 loop to the vertical drawing. Notice
how the voltage drop polarities across R7 and Rs correspond
with that of Re, and how this is the same as what we found
tracing Rz and Rs in the original circuit:
We repeat the process again, identifying and tracing another
loop around an already-traced resistor. In this case, the R3--R,
loop around Rs looks like a good loop to trace next:
R; andR,
loop around
Rs
Adding the R3--R, loop to the vertical drawing, marking the
correct polarities as well:
With only one remaining resistor left to trace, then next step
is obvious: trace the loop formed by R> around R3:
R, loops arounoR;
Adding R> to the vertical drawing, and we're finished! The
result is a diagram that's very easy to understand compared
to the original:
This simplified layout greatly eases the task of determining
where to start and how to proceed in reducing the circuit
down to a single equivalent (total) resistance. Notice how the
circuit has been re-drawn, all we have to do is start from the
right-hand side and work our way left, reducing simple-series
and simple-parallel resistor combinations one group at a time
until we're done.
In this particular case, we would start with the simple parallel
combination of R> and R3, reducing it to a single resistance.
Then, we would take that equivalent resistance (R>//R3) and
the one in series with it (R,), reducing them to another
equivalent resistance (R>//R3--R,). Next, we would proceed to
calculate the parallel equivalent of that resistance (R>//R3--
Ry) with Rs, then in series with R7, then in parallel with Re,
then in series with R; to give us a grand total resistance for
the circuit as a whole.
From there we could calculate total current from total voltage
and total resistance (I=E/R), then "expand" the circuit back
into its original form one stage at a time, distributing the
appropriate values of voltage and current to the resistances
as we go.
REVIEW:
Wires in diagrams and in real circuits can be lengthened,
shortened, and/or moved without affecting circuit
operation.
To simplify a convoluted circuit schematic, follow these
steps:
Trace current from one side of the battery to the other,
following any single path ("loop") to the battery.
Sometimes it works better to start with the loop
containing the most components, but regardless of the
path taken the result will be accurate. Mark polarity of
voltage drops across each resistor as you trace the loop.
Draw those components you encounter along this loop in
a vertical schematic.
Mark traced components in the original diagram and
trace remaining loops of components in the circuit. Use
polarity marks across traced components as guides for
what connects where. Document new components in
loops on the vertical re-draw schematic as well.
Repeat last step as often as needed until all components
in original diagram have been traced.
Component failure analysis
“l consider that | understand an equation when | can
predict the properties of its solutions, without actually
solving it."
P.A.M Dirac, physicist
There is a lot of truth to that quote from Dirac. With a little
modification, | can extend his wisdom to electric circuits by
saying, "| consider that | understand a circuit when | can
predict the approximate effects of various changes made to it
without actually performing any calculations."
At the end of the series and parallel circuits chapter, we
briefly considered how circuits could be analyzed ina
qualitative rather than quantitative manner. Building this skill
is an important step towards becoming a proficient
troubleshooter of electric circuits. Once you have a thorough
understanding of how any particular failure will affect a
circuit (i.e. you don't have to perform any arithmetic to
predict the results), it will be much easier to work the other
way around: pinpointing the source of trouble by assessing
how a circuit is behaving.
Also shown at the end of the series and parallel circuits
chapter was how the table method works just as well for
aiding failure analysis as it does for the analysis of healthy
circuits. We may take this technique one step further and
adapt it for total qualitative analysis. By "qualitative" | mean
working with symbols representing "increase," "decrease,"
and "same" instead of precise numerical figures. We can still
use the principles of series and parallel circuits, and the
concepts of Ohm's Law, we'll just use symbolic qualities
instead of numerical quantities. By doing this, we can gain
more of an intuitive "feel" for how circuits work rather than
leaning on abstract equations, attaining Dirac's definition of
“understanding.”
Enough talk. Let's try this technique on a real circuit example
and see how it works:
This is the first "convoluted" circuit we straightened out for
analysis in the last section. Since you already know how this
particular circuit reduces to series and parallel sections, I'll
Skip the process and go Straight to the final form:
R3 and Ry are in parallel with each other; so are R; and R>.
The parallel equivalents of R3//R4 and R,//R> are in series with
each other. Expressed in symbolic form, the total resistance
for this circuit is as follows:
Rtotal = (Ry//R2)--(R3//Ra)
First, we need to formulate a table with all the necessary rows
and columns for this circuit:
R, R2 R; R, R,/R, Rel/R, Total
E Volts
| Amps
R Ohms
Next, we need a failure scenario. Let's suppose that resistor
R> were to fail shorted. We will assume that all other
components maintain their original values. Because we'll be
analyzing this circuit qualitatively rather than quantitatively,
we won't be inserting any real numbers into the table. For any
quantity unchanged after the component failure, we'll use the
word "same" to represent "no change from before." For any
quantity that has changed as a result of the failure, we'll use
a down arrow for "decrease" and an up arrow for "increase."
As usual, we start by filling in the spaces of the table for
individual resistances and total voltage, our "given" values:
The only "given" value different from the normal state of the
circuit is Ro, which we said was failed shorted (abnormally low
resistance). All other initial values are the same as they were
before, as represented by the "same" entries. All we have to
do now is work through the familiar Ohm's Law and series-
parallel principles to determine what will happen to all the
other circuit values.
First, we need to determine what happens to the resistances
of parallel subsections Rj//Rz and R3//Rq. If neither R3 nor Ry
have changed in resistance value, then neither will their
parallel combination. However, since the resistance of R>z has
decreased while R; has stayed the same, their parallel
combination must decrease in resistance as well:
Now, we need to figure out what happens to the total
resistance. This part is easy: when we're dealing with only one
component change in the circuit, the change in total
resistance will be in the same direction as the change of the
failed component. This is not to say that the magnitude of
change between individual component and total circuit will
be the same, merely the direction of change. In other words, if
any single resistor decreases in value, then the total circuit
resistance must also decrease, and vice versa. In this case,
since R> is the only failed component, and its resistance has
decreased, the total resistance must decrease:
Now we can apply Ohm's Law (qualitatively) to the Total
column in the table. Given the fact that total voltage has
remained the same and total resistance has decreased, we
can conclude that total current must increase (I=E/R).
In case you're not familiar with the qualitative assessment of
an equation, it works like this. First, we write the equation as
solved for the unknown quantity. In this case, we're trying to
solve for current, given voltage and resistance:
L=——
R
Now that our equation is in the proper form, we assess what
change (if any) will be experienced by "I," given the
change(s) to "E" and "R":
(same)
Y
If the denominator of a fraction decreases in value while the
numerator stays the same, then the overall value of the
fraction must increase:
=
R
\ el,
RY
Therefore, Ohm's Law (l=E/R) tells us that the current (1) will
increase. We'll mark this conclusion in our table with an “up"
arrow:
With all resistance places filled in the table and all quantities
determined in the Total column, we can proceed to determine
the other voltages and currents. Knowing that the total
resistance in this table was the result of R//R> and R3//R, in
series, we know that the value of total current will be the
same as that in R,//R> and R3//R, (because series components
share the same current). Therefore, if total current increased,
then current through R,//Rz and R3//R4 must also have
increased with the failure of R>:
Fundamentally, what we're doing here with a qualitative
usage of Ohm's Law and the rules of series and parallel
circuits is no different from what we've done before with
numerical figures. In fact, its a lot easier because you don't
have to worry about making an arithmetic or calculator
keystroke error in a calculation. Instead, you're just focusing
on the principles behind the equations. From our table above,
we can see that Ohm's Law should be applicable to the R,//R>
and R3//R, columns. For R3//Ry, we figure what happens to
the voltage, given an increase in current and no change in
resistance. Intuitively, we can see that this must result in an
increase in voltage across the parallel combination of R3//R,:
But how do we apply the same Ohm's Law formula (E=IR) to
the R,//R> column, where we have resistance decreasing and
current increasing? It's easy to determine if only one variable
is changing, as it was with R3//Ry, but with two variables
moving around and no definite numbers to work with, Ohm's
Law isn't going to be much help. However, there is another
rule we can apply horizontally to determine what happens to
the voltage across R,//R>: the rule for voltage in series
circuits. If the voltages across R;//R> and R3//R, add up to
equal the total (battery) voltage and we know that the R3//Ry
voltage has increased while total voltage has stayed the
same, then the voltage across R,//R> must have decreased
with the change of R,'s resistance value:
Now we're ready to proceed to some new columns in the
table. Knowing that R3 and Ry comprise the parallel
subsection R3//Ry, and knowing that voltage is shared equally
between parallel components, the increase in voltage seen
across the parallel combination R3//R4 must also be seen
across R3 and R, individually:
The same goes for R; and R>. The voltage decrease seen
across the parallel combination of R; and R> will be seen
across R, and R> individually:
Applying Ohm's Law vertically to those columns with
unchanged ("same") resistance values, we can tell what the
current will do through those components. Increased voltage
across an unchanged resistance leads to increased current.
Conversely, decreased voltage across an unchanged
resistance leads to decreased current:
Once again we find ourselves in a position where Ohm's Law
can't help us: for R>, both voltage and resistance have
decreased, but without knowing how much each one has
changed, we can't use the I=E/R formula to qualitatively
determine the resulting change in current. However, we can
still apply the rules of series and parallel circuits horizontally.
We know that the current through the R,//R> parallel
combination has increased, and we also know that the current
through R, has decreased. One of the rules of parallel circuits
is that total current is equal to the sum of the individual
branch currents. In this case, the current through R,//R> is
equal to the current through R, added to the current through
R>. If current through R,//R> has increased while current
through R, has decreased, current through Rz must have
increased:
And with that, our table of qualitative values stands
completed. This particular exercise may look laborious due to
all the detailed commentary, but the actual process can be
performed very quickly with some practice. An important
thing to realize here is that the general procedure is little
different from quantitative analysis: start with the known
values, then proceed to determining total resistance, then
total current, then transfer figures of voltage and current as
allowed by the rules of series and parallel circuits to the
appropriate columns.
A few general rules can be memorized to assist and/or to
check your progress when proceeding with such an analysis:
e For any single component failure (open or shorted), the
total resistance will always change in the same direction
(either increase or decrease) as the resistance change of
the failed component.
e When a component fails shorted, its resistance always
decreases. Also, the current through it will increase, and
the voltage across it may drop. | say "may" because in
some cases it will remain the same (case in point: a
simple parallel circuit with an ideal power source).
e When a component fails open, its resistance always
increases. The current through that component will
decrease to zero, because it is an incomplete electrical
path (no continuity). This may result in an increase of
voltage across it. The same exception stated above
applies here as well: in a simple parallel circuit with an
ideal voltage source, the voltage across an open-failed
component will remain unchanged.
Building series-parallel resistor
circuits
Once again, when building battery/resistor circuits, the
student or hobbyist is faced with several different modes of
construction. Perhaps the most popular is the so/derless
breadboard: a platform for constructing temporary circuits by
plugging components and wires into a grid of interconnected
points. A breadboard appears to be nothing but a plastic
frame with hundreds of small holes in it. Underneath each
hole, though, is a spring clip which connects to other spring
clips beneath other holes. The connection pattern between
holes is simple and uniform:
Lines show common connections
underneath board between holes
HHddHHHHHHHHLITIIItL
HHddHHHHHHHHHLLILILLILL
Suppose we wanted to construct the following series-parallel
combination circuit on a breadboard:
A series-parallel combination circuit
250 Q
24V —
200 22
The recommended way to do so on a breadboard would be to
arrange the resistors in approximately the same pattern as
seen in the schematic, for ease of relation to the schematic. If
24 volts is required and we only have 6-volt batteries
available, four may be connected in series to achieve the
same effect:
oe os
©o00 00000 °©o00 0000
oooogo ogo oo 00000000 0
This is by no means the only way to connect these four
resistors together to form the circuit shown in the schematic.
Consider this alternative layout:
6 volts 6 volts 6 volts 6 volts
oooogo oo ogo o08 6006006000600 060 06
° ooo :°o oo 0
ooooo oo o0,88 80 0006000600600 0
ooooo oo 0000000000 0
If greater permanence is desired without resorting to
soldering or wire-wrapping, one could choose to construct this
circuit on a terminal strip (also called a barrier strip, or
terminal block). In this method, components and wires are
secured by mechanical tension underneath screws or heavy
clips attached to small metal bars. The metal bars, in turn, are
mounted on a nonconducting body to keep them electrically
isolated from each other.
Building a circuit with components secured to a terminal strip
isn't as easy as plugging components into a breadboard,
principally because the components cannot be physically
arranged to resemble the schematic layout. Instead, the
builder must understand how to "bend" the schematic's
representation into the real-world layout of the strip. Consider
one example of how the same four-resistor circuit could be
built on a terminal strip:
6 volts
6 volts 6 volts 6volts
Another terminal strip layout, simpler to understand and
relate to the schematic, involves anchoring parallel resistors
(R,//Rz and R3//R,) to the same two terminal points on the
strip like this:
6 volts 6 volts 6 volts
a \ |
@} |@| |S} |S] |S} |S] |S} |S! |S} |S! |S} |e |e} |e@
te
6 volts
Building more complex circuits on a terminal strip involves
the same spatial-reasoning skills, but of course requires
greater care and planning. Take for instance this complex
circuit, represented in schematic form:
The terminal strip used in the prior example barely has
enough terminals to mount all seven resistors required for this
circuit! It will be a challenge to determine all the necessary
wire connections between resistors, but with patience it can
be done. First, begin by installing and labeling all resistors on
the strip. The original schematic diagram will be shown next
to the terminal strip circuit for reference:
A, /
@| |S} |S} |S} |S} |S} |S} |S} |S} |S] |S} |S} |S] |S} |S
Next, begin connecting components together wire by wire as
shown in the schematic. Over-draw connecting lines in the
schematic to indicate completion in the real circuit. Watch
this sequence of illustrations as each individual wire is
identified in the schematic, then added to the real circuit:
Although there are minor variations possible with this
terminal strip circuit, the choice of connections shown in this
example sequence is both electrically accurate (electrically
identical to the schematic diagram) and carries the additional
benefit of not burdening any one screw terminal on the strip
with more than two wire ends, a good practice in any terminal
strip circuit.
An example of a "variant" wire connection might be the very
last wire added (step 11), which | placed between the left
terminal of Rp and the left terminal of R3. This last wire
completed the parallel connection between R> and R3 in the
circuit. However, | could have placed this wire instead
between the left terminal of Ry and the right terminal of Rj,
since the right terminal of R; is already connected to the left
terminal of R3 (having been placed there in step 9) and so is
electrically common with that one point. Doing this, though,
would have resulted in three wires secured to the right
terminal of R; instead of two, which is a faux pax in terminal
strip etiquette. Would the circuit have worked this way?
Certainly! It's just that more than two wires secured ata
single terminal makes for a "messy" connection: one that is
aesthetically unpleasing and may place undue stress on the
screw terminal.
Another variation would be to reverse the terminal
connections for resistor Rz. As shown in the last diagram, the
voltage polarity across R7 is negative on the left and positive
on the right (- , +), whereas all the other resistor polarities are
positive on the left and negative on the right (+ , -):
While this poses no electrical problem, it might cause
confusion for anyone measuring resistor voltage drops with a
voltmeter, especially an analog voltmeter which will "peg"
downscale when subjected to a voltage of the wrong polarity.
For the sake of consistency, it might be wise to arrange all
wire connections so that all resistor voltage drop polarities are
the same, like this:
Though electrons do not care about such consistency in
component layout, people do. This illustrates an important
aspect of any engineering endeavor: the human factor.
Whenever a design may be modified for easier
comprehension and/or easier maintenance -- with no sacrifice
of functional performance -- it should be done so.
e REVIEW:
e Circuits built on terminal strips can be difficult to lay out,
but when built they are robust enough to be considered
permanent, yet easy to modify.
e It is bad practice to secure more than two wire ends
and/or component leads under a single terminal screw or
clip on a terminal strip. Try to arrange connecting wires so
as to avoid this condition.
e Whenever possible, build your circuits with clarity and
ease of understanding in mind. Even though component
and wiring layout is usually of little consequence in DC
circuit function, it matters significantly for the sake of the
person who has to modify or troubleshoot it later.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Tony Armstrong (January 23, 2003): Suggested reversing
polarity on resistor R; in last terminal strip circuit.
Jason Starck (June 2000): HTML document formatting, which
led to a much better-looking second edition.
Ron LaPlante (October 1998): helped create "table" method
of series and parallel circuit analysis.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+4]l—
—| | +4/l—
Lessons In Electric Circuits
-- Volume |
Chapter 8
DC METERING CIRCUITS
What is a meter?
Voltmeter design
Voltmeter impact on measured circuit
Ammeter design
Ammeter impact on measured circuit
Ohmmeter design
High voltage ohmmeters
e Multimeters
Kelvin (4-wire) resistance measurement
Bridge circuits
e Wattmeter design
e Creating custom calibration resistances
Contributors
What is a meter?
A meteris any device built to accurately detect and display
an electrical quantity in a form readable by a human being.
Usually this "readable form" is visual: motion of a pointer on
a scale, a series of lights arranged to form a "bargraph," or
some sort of display composed of numerical figures. In the
analysis and testing of circuits, there are meters designed to
accurately measure the basic quantities of voltage, current,
and resistance. There are many other types of meters as well,
but this chapter primarily covers the design and operation of
the basic three.
Most modern meters are "digital" in design, meaning that
their readable display is in the form of numerical digits. Older
designs of meters are mechanical in nature, using some kind
of pointer device to show quantity of measurement. In either
case, the principles applied in adapting a display unit to the
measurement of (relatively) large quantities of voltage,
Current, or resistance are the same.
The display mechanism of a meter is often referred to as a
movement, borrowing from its mechanical nature to move a
pointer along a scale so that a measured value may be read.
Though modern digital meters have no moving parts, the
term "movement" may be applied to the same basic device
performing the display function.
The design of digital "Movements" is beyond the scope of
this chapter, but mechanical meter movement designs are
very understandable. Most mechanical movements are based
on the principle of electromagnetism: that electric current
through a conductor produces a magnetic field perpendicular
to the axis of electron flow. The greater the electric current,
the stronger the magnetic field produced. If the magnetic
field formed by the conductor is allowed to interact with
another magnetic field, a physical force will be generated
between the two sources of fields. If one of these sources is
free to move with respect to the other, it will do so as current
is conducted through the wire, the motion (usually against
the resistance of a spring) being proportional to strength of
current.
The first meter movements built were known as
galvanometers, and were usually designed with maximum
sensitivity in mind. A very simple galvanometer may be
made from a magnetized needle (such as the needle from a
magnetic compass) suspended from a string, and positioned
within a coil of wire. Current through the wire coil will
produce a magnetic field which will deflect the needle from
pointing in the direction of earth's magnetic field. An antique
string galvanometer is shown in the following photograph:
Such instruments were useful in their time, but have little
place in the modern world except as proof-of-concept and
elementary experimental devices. They are highly
susceptible to motion of any kind, and to any disturbances in
the natural magnetic field of the earth. Now, the term
“galvanometer" usually refers to any design of
electromagnetic meter movement built for exceptional
sensitivity, and not necessarily a crude device such as that
shown in the photograph. Practical electromagnetic meter
movements can be made now where a pivoting wire coil is
suspended in a strong magnetic field, shielded from the
majority of outside influences. Such an instrument design is
generally known as a permanent-magnet, moving coil, or
PMMC movement:
Permanent magnet, moving coil (PMMC) meter movement
current through wire coil
causes needle to deflect
meter terminal
connections
In the picture above, the meter movement "needle" is shown
pointing somewhere around 35 percent of full-scale, zero
being full to the left of the arc and full-scale being
completely to the right of the arc. An increase in measured
current will drive the needle to point further to the right and
a decrease will cause the needle to drop back down toward
its resting point on the left. The arc on the meter display is
labeled with numbers to indicate the value of the quantity
being measured, whatever that quantity is. In other words, if
it takes 50 microamps of current to drive the needle fully to
the right (making this a "50 WA full-scale movement"), the
scale would have 0 HA written at the very left end and 50 pA
at the very right, 25 WA being marked in the middle of the
scale. In all likelihood, the scale would be divided into much
smaller graduating marks, probably every 5 or 1 WA, to allow
whoever is viewing the movement to infer a more precise
reading from the needle's position.
The meter movement will have a pair of metal connection
terminals on the back for current to enter and exit. Most
meter movements are polarity-sensitive, one direction of
current driving the needle to the right and the other driving
it to the left. Some meter movements have a needle that is
spring-centered in the middle of the scale sweep instead of
to the left, thus enabling measurements of either polarity:
A "zero-center" meter movement
0
-100 100
Common polarity-sensitive movements include the
D'Arsonval and Weston designs, both PMMC-type
instruments. Current in one direction through the wire will
produce a clockwise torque on the needle mechanism, while
current the other direction will produce a counter-clockwise
torque.
Some meter movements are polarity-/nsensitive, relying on
the attraction of an unmagnetized, movable iron vane toward
a stationary, current-carrying wire to deflect the needle. Such
meters are ideally suited for the measurement of alternating
current (AC). A polarity-sensitive movement would just
vibrate back and forth uselessly if connected to a source of
AC.
While most mechanical meter movements are based on
electromagnetism (electron flow through a conductor
creating a perpendicular magnetic field), a few are based on
electrostatics: that is, the attractive or repulsive force
generated by electric charges across space. This is the same
phenomenon exhibited by certain materials (such as wax and
wool) when rubbed together. If a voltage is applied between
two conductive surfaces across an air gap, there will bea
physical force attracting the two surfaces together capable of
moving some kind of indicating mechanism. That physical
force is directly proportional to the voltage applied between
the plates, and inversely proportional to the square of the
distance between the plates. The force is also irrespective of
polarity, making this a polarity-insensitive type of meter
movement:
Electrostatic meter movement
————_
force
Me
Voltage to be measured
Unfortunately, the force generated by the electrostatic
attraction is very small for common voltages. In fact, it is so
small that such meter movement designs are impractical for
use in general test instruments. Typically, electrostatic meter
movements are used for measuring very high voltages (many
thousands of volts). One great advantage of the electrostatic
meter movement, however, is the fact that it has extremely
high resistance, whereas electromagnetic movements (which
depend on the flow of electrons through wire to generate a
magnetic field) are much lower in resistance. As we will see
in greater detail to come, greater resistance (resulting in less
current drawn from the circuit under test) makes for a better
voltmeter.
A much more common application of electrostatic voltage
measurement is seen in an device known as a Cathode Ray
Tube, or CRT. These are special glass tubes, very similar to
television viewscreen tubes. In the cathode ray tube, a beam
of electrons traveling in a vacuum are deflected from their
course by voltage between pairs of metal plates on either
side of the beam. Because electrons are negatively charged,
they tend to be repelled by the negative plate and attracted
to the positive plate. A reversal of voltage polarity across the
two plates will result in a deflection of the electron beam in
the opposite direction, making this type of meter
"movement" polarity-sensitive:
voltage to be measured
electron "gun" view-
- (vacuum) screen
» electrons
. electrons
= light
The electrons, having much less mass than metal plates, are
moved by this electrostatic force very quickly and readily.
Their deflected path can be traced as the electrons impinge
on the glass end of the tube where they strike a coating of
phosphorus chemical, emitting a glow of light seen outside of
the tube. The greater the voltage between the deflection
plates, the further the electron beam will be "bent" from its
straight path, and the further the glowing spot will be seen
from center on the end of the tube.
A photograph of a CRT is shown here:
In a real CRT, as shown in the above photograph, there are
two pairs of deflection plates rather than just one. In order to
be able to sweep the electron beam around the whole area of
the screen rather than just in a straight line, the beam must
be deflected in more than one dimension.
Although these tubes are able to accurately register small
voltages, they are bulky and require electrical power to
operate (unlike electromagnetic meter movements, which are
more compact and actuated by the power of the measured
signal current going through them). They are also much more
fragile than other types of electrical metering devices.
Usually, cathode ray tubes are used in conjunction with
precise external circuits to form a larger piece of test
equipment known as an oscilloscope, which has the ability to
display a graph of voltage over time, a tremendously useful
tool for certain types of circuits where voltage and/or current
levels are dynamically changing.
Whatever the type of meter or size of meter movement, there
will be a rated value of voltage or current necessary to give
full-scale indication. In electromagnetic movements, this will
be the "full-scale deflection current" necessary to rotate the
needle so that it points to the exact end of the indicating
scale. In electrostatic movements, the full-scale rating will be
expressed as the value of voltage resulting in the maximum
deflection of the needle actuated by the plates, or the value
of voltage in a cathode-ray tube which deflects the electron
beam to the edge of the indicating screen. In digital
“movements,” it is the amount of voltage resulting in a "full-
count" indication on the numerical display: when the digits
cannot display a larger quantity.
The task of the meter designer is to take a given meter
movement and design the necessary external circuitry for
full-scale indication at some specified amount of voltage or
current. Most meter movements (electrostatic movements
excepted) are quite sensitive, giving full-scale indication at
only a small fraction of a volt or an amp. This is impractical
for most tasks of voltage and current measurement. What the
technician often requires is a meter capable of measuring
high voltages and currents.
By making the sensitive meter movement part of a voltage or
current divider circuit, the movement's useful measurement
range may be extended to measure far greater levels than
what could be indicated by the movement alone. Precision
resistors are used to create the divider circuits necessary to
divide voltage or current appropriately. One of the lessons
you will learn in this chapter is how to design these divider
circuits.
e REVIEW:
e A"movement' is the display mechanism of a meter.
e Electromagnetic movements work on the principle of a
magnetic field being generated by electric current
through a wire. Examples of electromagnetic meter
movements include the D'Arsonval, Weston, and iron-
vane designs.
e Electrostatic movements work on the principle of
physical force generated by an electric field between two
plates.
e Cathode Ray Tubes (CRT's) use an electrostatic field to
bend the path of an electron beam, providing indication
of the beam's position by light created when the beam
strikes the end of the glass tube.
Voltmeter design
As was stated earlier, most meter movements are sensitive
devices. Some D'Arsonval movements have full-scale
deflection current ratings as little as 50 UA, with an (internal)
wire resistance of less than 1000 Q. This makes for a
voltmeter with a full-scale rating of only 50 millivolts (50 pA
X 1000 Q)! In order to build voltmeters with practical (higher
voltage) scales from such sensitive movements, we need to
find some way to reduce the measured quantity of voltage
down to a level the movement can handle.
Let's start our example problems with a D'Arsonval meter
movement having a full-scale deflection rating of 1 mA anda
coil resistance of 500 Q:
500 Q F.S=lmA
black test red test
lead lead
Using Ohm's Law (E=IR), we can determine how much
voltage will drive this meter movement directly to full scale:
E = (1 mA)(500 Q)
E = 0.5 volts
If all we wanted was a meter that could measure 1/2 of a volt,
the bare meter movement we have here would suffice. But to
measure greater levels of voltage, something more is needed.
To get an effective voltmeter meter range in excess of 1/2
volt, we'll need to design a circuit allowing only a precise
proportion of measured voltage to drop across the meter
movement. This will extend the meter movement's range to
higher voltages. Correspondingly, we will need to re-label the
scale on the meter face to indicate its new measurement
range with this proportioning circuit connected.
But how do we create the necessary proportioning circuit?
Well, if our intention is to allow this meter movement to
measure a greater vo/tage than it does now, what we need is
a voltage divider circuit to proportion the total measured
voltage into a lesser fraction across the meter movement's
connection points. Knowing that voltage divider circuits are
built from series resistances, we'll connect a resistor in series
with the meter movement (using the movement's own
internal resistance as the second resistance in the divider):
500 Q F.S.=1mA
R
+ multiplier
black test red test
lead lead
The series resistor is called a "multiplier" resistor because it
multiplies the working range of the meter movement as it
proportionately divides the measured voltage across it.
Determining the required multiplier resistance value is an
easy task if you're familiar with series circuit analysis.
For example, let's determine the necessary multiplier value
to make this 1 mA, 500 OQ movement read exactly full-scale at
an applied voltage of 10 volts. To do this, we first need to set
up an E/I/R table for the two series components:
Movement R Total
multiplier
E Volts
| Amps
R Ohms
Knowing that the movement will be at full-scale with 1 mA of
current going through it, and that we want this to happen at
an applied (total series circuit) voltage of 10 volts, we can fill
in the table as such:
Movement R Total
multiplier
There are a couple of ways to determine the resistance value
of the multiplier. One way is to determine total circuit
resistance using Ohm's Law in the "total" column (R=E/I),
then subtract the 500 Q of the movement to arrive at the
value for the multiplier:
Movement R Total
multiplier
Another way to figure the same value of resistance would be
to determine voltage drop across the movement at full-scale
deflection (E=IR), then subtract that voltage drop from the
total to arrive at the voltage across the multiplier resistor.
Finally, Ohm's Law could be used again to determine
resistance (R=E/I) for the multiplier:
Movement Rrutipier ‘Total
Either way provides the same answer (9.5 kQ), and one
method could be used as verification for the other, to check
accuracy of work.
Meter movement ranged for 10 volts full-scale
500Q F.S.=1l1mA
R
+ multiplier
red test
10 volts gives full-scale
deflection of needle
With exactly 10 volts applied between the meter test leads
(from some battery or precision power supply), there will be
exactly 1 mA of current through the meter movement, as
restricted by the "multiplier" resistor and the movement's
own internal resistance. Exactly 1/2 volt will be dropped
across the resistance of the movement's wire coil, and the
needle will be pointing precisely at full-scale. Having re-
labeled the scale to read from 0 to 10 V (instead of 0 to 1
mA), anyone viewing the scale will interpret its indication as
ten volts. Please take note that the meter user does not have
to be aware at all that the movement itself is actually
measuring just a fraction of that ten volts from the external
source. All that matters to the user is that the circuit as a
whole functions to accurately display the total, applied
voltage.
This is how practical electrical meters are designed and used:
a sensitive meter movement is built to operate with as little
voltage and current as possible for maximum sensitivity,
then it is "fooled" by some sort of divider circuit built of
precision resistors so that it indicates full-scale when a much
larger voltage or current is impressed on the circuit as a
whole. We have examined the design of a simple voltmeter
here. Ammeters follow the same general rule, except that
parallel-connected "shunt" resistors are used to create a
current divider circuit as opposed to the series-connected
voltage divider "multiplier" resistors used for voltmeter
designs.
Generally, it is useful to have multiple ranges established for
an electromechanical meter such as this, allowing it to read a
broad range of voltages with a single movement mechanism.
This is accomplished through the use of a multi-pole switch
and several multiplier resistors, each one sized for a
particular voltage range:
A multi-range voltmeter
500Q2 F.S.=lmA
range selector
switch
_———_
black test red test
lead lead
The five-position switch makes contact with only one resistor
at a time. In the bottom (full clockwise) position, it makes
contact with no resistor at all, providing an "off" setting. Each
resistor is sized to provide a particular full-scale range for the
voltmeter, all based on the particular rating of the meter
movement (1 mA, 500 Q). The end result is a voltmeter with
four different full-scale ranges of measurement. Of course, in
order to make this work sensibly, the meter movement's
scale must be equipped with labels appropriate for each
range.
With such a meter design, each resistor value is determined
by the same technique, using a known total voltage,
movement full-scale deflection rating, and movement
resistance. For a voltmeter with ranges of 1 volt, 10 volts,
100 volts, and 1000 volts, the multiplier resistances would be
as follows:
500 2 FS.=l mA
R, = 999.5 kQ
range selector 3 R,=99.5ko
R; =9.5 kQ
R,=5002
switch
black test
lead
Note the multiplier resistor values used for these ranges, and
how odd they are. It is highly unlikely that a 999.5 kQ
precision resistor will ever be found in a parts bin, so
voltmeter designers often opt for a variation of the above
design which uses more common resistor values:
500Q FS.=1mA
range selector
switch
R, = 900 kQ
R, = 90 kQ
black test R,;=9kQ
lead R, = 500 2
With each successively higher voltage range, more multiplier
resistors are pressed into service by the selector switch,
making their series resistances add for the necessary total.
For example, with the range selector switch set to the 1000
volt position, we need a total multiplier resistance value of
999.5 kQ. With this meter design, that's exactly what we'll
get:
Rtotal = Rg + R3 + Ro + Ry
Rrotal = 900 kN + 90kQN +9kQ+ 5000
Rrotal = 999.5 kQ
The advantage, of course, is that the individual multiplier
resistor values are more common (900k, 90k, 9k) than some
of the odd values in the first design (999.5k, 99.5k, 9.5k).
From the perspective of the meter user, however, there will
be no discernible difference in function.
e REVIEW:
e Extended voltmeter ranges are created for sensitive
meter movements by adding series "multiplier" resistors
to the movement circuit, providing a precise voltage
division ratio.
Voltmeter impact on measured circuit
Every meter impacts the circuit it is measuring to some
extent, just as any tire-pressure gauge changes the
measured tire pressure slightly as some air is let out to
operate the gauge. While some impact is inevitable, it can be
minimized through good meter design.
Since voltmeters are always connected in parallel with the
component or components under test, any current through
the voltmeter will contribute to the overall current in the
tested circuit, potentially affecting the voltage being
measured. A perfect voltmeter has infinite resistance, so that
it draws no current from the circuit under test. However,
perfect voltmeters only exist in the pages of textbooks, not in
real life! Take the following voltage divider circuit as an
extreme example of how a realistic voltmeter might impact
the circuit its measuring:
250 MQ
+
250 MQ c ) voltmeter
With no voltmeter connected to the circuit, there should be
exactly 12 volts across each 250 MOQ resistor in the series
circuit, the two equal-value resistors dividing the total
voltage (24 volts) exactly in half. However, if the voltmeter in
question has a lead-to-lead resistance of 10 MQ (a common
amount for a modern digital voltmeter), its resistance will
create a parallel subcircuit with the lower resistor of the
divider when connected:
250 MQ
24V —
+
voltmeter
250 MQ (Vv) (10 MQ)
This effectively reduces the lower resistance from 250 MQ to
9.615 MQ (250 MQ and 10 MQ in parallel), drastically altering
voltage drops in the circuit. The lower resistor will now have
far less voltage across it than before, and the upper resistor
far more.
23.1111 3 250 MQ
9.615 MQ
(250 MQ // 10 MQ)
24V —
A voltage divider with resistance values of 250 MQ and 9.615
MQ will divide 24 volts into portions of 23.1111 volts and
0.8889 volts, respectively. Since the voltmeter is part of that
9.615 MO resistance, that is what it will indicate: 0.8889
volts.
Now, the voltmeter can only indicate the voltage its
connected across. It has no way of "Knowing" there was a
potential of 12 volts dropped across the lower 250 MQ
resistor before it was connected across it. The very act of
connecting the voltmeter to the circuit makes it part of the
circuit, and the voltmeter's own resistance alters the
resistance ratio of the voltage divider circuit, consequently
affecting the voltage being measured.
Imagine using a tire pressure gauge that took so great a
volume of air to operate that it would deflate any tire it was
connected to. The amount of air consumed by the pressure
gauge in the act of measurement is analogous to the current
taken by the voltmeter movement to move the needle. The
less air a pressure gauge requires to operate, the less it will
deflate the tire under test. The less current drawn by a
voltmeter to actuate the needle, the less it will burden the
circuit under test.
This effect is called /oading, and it is present to some degree
in every instance of voltmeter usage. The scenario shown
here is worst-case, with a voltmeter resistance substantially
lower than the resistances of the divider resistors. But there
always will be some degree of loading, causing the meter to
indicate less than the true voltage with no meter connected.
Obviously, the higher the voltmeter resistance, the less
loading of the circuit under test, and that is why an ideal
voltmeter has infinite internal resistance.
Voltmeters with electromechanical movements are typically
given ratings in "ohms per volt" of range to designate the
amount of circuit impact created by the current draw of the
movement. Because such meters rely on different values of
multiplier resistors to give different measurement ranges,
their lead-to-lead resistances will change depending on what
range they're set to. Digital voltmeters, on the other hand,
often exhibit a constant resistance across their test leads
regardless of range setting (but not always!), and as such are
usually rated simply in ohms of input resistance, rather than
“ohms per volt" sensitivity.
What "ohms per volt" means is how many ohms of lead-to-
lead resistance for every volt of range setting on the selector
switch. Let's take our example voltmeter from the last section
as an example:
500 2 FS.=1 mA
R, = 999.5 kQ
range selector i 2 R, =99.5kQ
switch
R;=95 kQ
R, = 5002
black test
lead
On the 1000 volt scale, the total resistance is 1 MQ (999.5 kQ
+ 500Q), giving 1,000,000 © per 1000 volts of range, or
1000 ohms per volt (1 kQ/V). This ohms-per-volt "sensitivity"
rating remains constant for any range of this meter:
100 volt range 100 ko — 1000 Q/V sensitivity
100 V
10 volt range _10kQ _ 1000 2/V sensitivity
10 V
1 kQ
1 volt range 1000 Q/V sensitivity
1V
The astute observer will notice that the ohms-per-volt rating
of any meter is determined by a single factor: the full-scale
current of the movement, in this case 1 mA. "Ohms per volt"
is the mathematical reciprocal of "volts per ohm," which is
defined by Ohm's Law as current (I=E/R). Consequently, the
full-scale current of the movement dictates the Q/volt
sensitivity of the meter, regardless of what ranges the
designer equips it with through multiplier resistors. In this
case, the meter movement's full-scale current rating of 1 mA
gives it a voltmeter sensitivity of 1000 Q/V regardless of how
we range it with multiplier resistors.
To minimize the loading of a voltmeter on any circuit, the
designer must seek to minimize the current draw of its
movement. This can be accomplished by re-designing the
movement itself for maximum sensitivity (less current
required for full-scale deflection), but the tradeoff here is
typically ruggedness: a more sensitive movement tends to be
more fragile.
Another approach is to electronically boost the current sent
to the movement, so that very little current needs to be
drawn from the circuit under test. This special electronic
circuit is known as an amplifier, and the voltmeter thus
constructed is an amplified voltmeter.
Amplified voltmeter
red test
lead
Amplifier
black test
lead Battery
The internal workings of an amplifier are too complex to be
discussed at this point, but suffice it to say that the circuit
allows the measured voltage to contro/ how much battery
current is sent to the meter movement. Thus, the
movement's current needs are supplied by a battery internal
to the voltmeter and not by the circuit under test. The
amplifier still loads the circuit under test to some degree, but
generally hundreds or thousands of times less than the meter
movement would by itself.
Before the advent of semiconductors known as "field-effect
transistors," vacuum tubes were used as amplifying devices
to perform this boosting. Such vacuum-tube voltmeters, or
(VTVM's) were once very popular instruments for electronic
test and measurement. Here is a photograph of a very old
VTVM, with the vacuum tube exposed!
Now, solid-state transistor amplifier circuits accomplish the
same task in digital meter designs. While this approach (of
using an amplifier to boost the measured signal current)
works well, it vastly complicates the design of the meter,
making it nearly impossible for the beginning electronics
student to comprehend its internal workings.
A final, and ingenious, solution to the problem of voltmeter
loading is that of the potentiometric or null-balance
instrument. It requires no advanced (electronic) circuitry or
sensitive devices like transistors or vacuum tubes, but it does
require greater technician involvement and skill. In a
potentiometric instrument, a precision adjustable voltage
source iS Compared against the measured voltage, and a
sensitive device called a nu// detector is used to indicate
when the two voltages are equal. In some circuit designs, a
precision potentiometer is used to provide the adjustable
voltage, hence the label potentiometric. When the voltages
are equal, there will be zero current drawn from the circuit
under test, and thus the measured voltage should be
unaffected. It is easy to show how this works with our last
example, the high-resistance voltage divider circuit:
Potentiometric voltage measurement
250 MQ
"null" detector
adjustable
voltage
source
The "null detector" is a sensitive device capable of indicating
the presence of very small voltages. If an electromechanical
meter movement is used as the null detector, it will have a
spring-centered needle that can deflect in either direction so
as to be useful for indicating a voltage of either polarity. As
the purpose of a null detector is to accurately indicate a
condition of zero voltage, rather than to indicate any specific
(nonzero) quantity as a normal voltmeter would, the scale of
the instrument used is irrelevant. Null detectors are typically
designed to be as sensitive as possible in order to more
precisely indicate a "null" or "balance" (zero voltage)
condition.
An extremely simple type of null detector is a set of audio
headphones, the speakers within acting as a kind of meter
movement. When a DC voltage is initially applied to a
speaker, the resulting current through it will move the
Speaker cone and produce an audible "click." Another "click"
sound will be heard when the DC source is disconnected.
Building on this principle, a sensitive null detector may be
made from nothing more than headphones and a momentary
contact switch:
Headphones
Pushbutton
switch
If aset of "8 ohm" headphones are used for this purpose, its
sensitivity may be greatly increased by connecting it toa
device called a transformer. The transformer exploits
principles of electromagnetism to "transform" the voltage
and current levels of electrical energy pulses. In this case,
the type of transformer used is a step-down transformer, and
it converts low-current pulses (created by closing and
opening the pushbutton switch while connected to a small
voltage source) into higher-current pulses to more efficiently
drive the speaker cones inside the headphones. An "audio
output" transformer with an impedance ratio of 1000:8 is
ideal for this purpose. The transformer also increases
detector sensitivity by accumulating the energy of a low-
current signal in a magnetic field for sudden release into the
headphone speakers when the switch is opened. Thus, it will
produce louder "clicks" for detecting smaller signals:
Audio output
transformer Headphones
Test
leads
Connected to the potentiometric circuit as a null detector,
the switch/transformer/headphone arrangement is used as
such:
adiistable
voltage
Source
The purpose of any null detector is to act like a laboratory
balance scale, indicating when the two voltages are equal
(absence of voltage between points 1 and 2) and nothing
more. The laboratory scale balance beam doesn't actually
weigh anything; rather, it simply indicates equality between
the unknown mass and the pile of standard (calibrated)
masses.
unknown mass mass standards
Likewise, the null detector simply indicates when the voltage
between points 1 and 2 are equal, which (according to
Kirchhoff's Voltage Law) will be when the adjustable voltage
source (the battery symbol with a diagonal arrow going
through it) is precisely equal in voltage to the drop across R>.
To operate this instrument, the technician would manually
adjust the output of the precision voltage source until the
null detector indicated exactly zero (if using audio
headphones as the null detector, the technician would
repeatedly press and release the pushbutton switch, listening
for silence to indicate that the circuit was "balanced"), and
then note the source voltage as indicated by a voltmeter
connected across the precision voltage source, that
indication being representative of the voltage across the
lower 250 MO resistor:
250 M2
"null" detector
adjustable
voltage
source
Adjust voltage source until null detector registers zero.
Then, read voltmeter indication for voltage across R).
The voltmeter used to directly measure the precision source
need not have an extremely high Q/V sensitivity, because the
source will supply all the current it needs to operate. So long
as there is zero voltage across the null detector, there will be
zero current between points 1 and 2, equating to no loading
of the divider circuit under test.
It is worthy to reiterate the fact that this method, properly
executed, places a/most zero load upon the measured circuit.
Ideally, it places absolutely no load on the tested circuit, but
to achieve this ideal goal the null detector would have to
have absolutely zero voltage across it, which would require
an infinitely sensitive null meter and a perfect balance of
voltage from the adjustable voltage source. However, despite
its practical inability to achieve absolute zero loading, a
potentiometric circuit is still an excellent technique for
measuring voltage in high-resistance circuits. And unlike the
electronic amplifier solution, which solves the problem with
advanced technology, the potentiometric method achieves a
hypothetically perfect solution by exploiting a fundamental
law of electricity (KVL).
e REVIEW:
e An ideal voltmeter has infinite resistance.
e Too low of an internal resistance in a voltmeter will
adversely affect the circuit being measured.
e Vacuum tube voltmeters (VTVM's), transistor voltmeters,
and potentiometric circuits are all means of minimizing
the load placed on a measured circuit. Of these methods,
the potentiometric ("null-balance") technique is the only
one capable of placing zero load on the circuit.
e A null detector is a device built for maximum sensitivity
to small voltages or currents. It is used in potentiometric
voltmeter circuits to indicate the absence of voltage
between two points, thus indicating a condition of
balance between an adjustable voltage source and the
voltage being measured.
Ammeter design
A meter designed to measure electrical current is popularly
called an "ammeter" because the unit of measurement is
"amps."
In ammeter designs, external resistors added to extend the
usable range of the movement are connected in paral//e/ with
the movement rather than in series as is the case for
voltmeters. This is because we want to divide the measured
current, not the measured voltage, going to the movement,
and because current divider circuits are always formed by
parallel resistances.
Taking the same meter movement as the voltmeter example,
we can see that it would make a very limited instrument by
itself, full-scale deflection occurring at only 1 mA:
As is the case with extending a meter movement's voltage-
measuring ability, we would have to correspondingly re-label
the movement's scale so that it read differently for an
extended current range. For example, if we wanted to design
an ammeter to have a full-scale range of 5 amps using the
Same meter movement as before (having an intrinsic full-
scale range of only 1 mA), we would have to re-label the
movement's scale to read 0 A on the far left and 5 A on the
far right, rather than 0 mA to 1 mA as before. Whatever
extended range provided by the parallel-connected resistors,
we would have to represent graphically on the meter
movement face.
500Q FS=I1mA
black test red test
lead lead
Using 5 amps as an extended range for our sample
movement, let's determine the amount of parallel resistance
necessary to "shunt," or bypass, the majority of current so
that only 1 mA will go through the movement with a total
current of 5 A:
500Q FS.=1mA
—
+
black test red test
lead lead
Movement Raunt Total
From our given values of movement current, movement
resistance, and total circuit (measured) current, we can
determine the voltage across the meter movement (Ohm's
Law applied to the center column, E=IR):
Movement R
shunt
Knowing that the circuit formed by the movement and the
shunt is of a parallel configuration, we know that the voltage
across the movement, shunt, and test leads (total) must be
the same:
Movement R
shunt
We also know that the current through the shunt must be the
difference between the total current (5 amps) and the
current through the movement (1 mA), because branch
currents add in a parallel configuration:
Movement R
shunt
Then, using Ohm's Law (R=E/I) in the right column, we can
determine the necessary shunt resistance:
Movement = Raunt Total
Of course, we could have calculated the same value of just
over 100 milli-ohms (100 mQ) for the shunt by calculating
total resistance (R=E/I; 0.5 volts/5 amps = 100 mQ exactly),
then working the parallel resistance formula backwards, but
the arithmetic would have been more challenging:
l
l l
100m 500
Rou nt —
R = 100.02 mQ
shunt
In real life, the shunt resistor of an ammeter will usually be
encased within the protective metal housing of the meter
unit, hidden from sight. Note the construction of the
ammeter in the following photograph:
This particular ammeter is an automotive unit manufactured
by Stewart-Warner. Although the D'Arsonval meter
movement itself probably has a full scale rating in the range
of milliamps, the meter as a whole has a range of +/- 60
amps. The shunt resistor providing this high current range is
enclosed within the metal housing of the meter. Note also
with this particular meter that the needle centers at zero
amps and can indicate either a "positive" current ora
"negative" current. Connected to the battery charging circuit
of an automobile, this meter is able to indicate a charging
condition (electrons flowing from generator to battery) ora
discharging condition (electrons flowing from battery to the
rest of the car's loads).
As is the case with multiple-range voltmeters, ammeters can
be given more than one usable range by incorporating
several shunt resistors switched with a multi-pole switch:
A multirange ammeter
500 Q FS.=1mA
range selector
switch
black test
lead
Notice that the range resistors are connected through the
switch so as to be in parallel with the meter movement,
rather than in series as it was in the voltmeter design. The
five-position switch makes contact with only one resistor at a
time, of course. Each resistor is sized accordingly for a
different full-scale range, based on the particular rating of
the meter movement (1 mA, 500 Q).
With such a meter design, each resistor value is determined
by the same technique, using a known total current,
movement full-scale deflection rating, and movement
resistance. For an ammeter with ranges of 100 mA, 1A, 104A,
and 100 A, the shunt resistances would be as such:
5002 FS.=1 mA
R, = 5.00005 mQ
range selector ) R, = 50.005 mQ
emicn R, = 500.5005 m
R, = 5.05051 2
black test red test
lead lead
Notice that these shunt resistor values are very low! 5.00005
mQ is 5.00005 milli-ohms, or 0.00500005 ohms! To achieve
these low resistances, ammeter shunt resistors often have to
be custom-made from relatively large-diameter wire or solid
pieces of metal.
One thing to be aware of when sizing ammeter shunt
resistors is the factor of power dissipation. Unlike the
voltmeter, an ammeter's range resistors have to carry large
amounts of current. If those shunt resistors are not sized
accordingly, they may overheat and suffer damage, or at the
very least lose accuracy due to overheating. For the example
meter above, the power dissipations at full-scale indication
are (the double-squiggly lines represent "approximately
equal to" in mathematics):
EF @svy
R,; 5.00005 mQ
a”)
jie
|
|
Q
wn
©
=
EF. @svy
~ R, 50.005 m2
7 SY :
Po = = 0.5 W
R; 500.5 mQ
: Svy
Boe te ON). cea Sani
R, 5.05 Q
An 1/8 watt resistor would work just fine for Ry, a 1/2 watt
resistor would suffice for R3 and a 5 watt for R> (although
resistors tend to maintain their long-term accuracy better if
not operated near their rated power dissipation, so you might
want to over-rate resistors Rz and R3), but precision 50 watt
resistors are rare and expensive components indeed. A
custom resistor made from metal stock or thick wire may
have to be constructed for R; to meet both the requirements
of low resistance and high power rating.
Sometimes, shunt resistors are used in conjunction with
voltmeters of high input resistance to measure current. In
these cases, the current through the voltmeter movement is
small enough to be considered negligible, and the shunt
resistance can be sized according to how many volts or
millivolts of drop will be produced per amp of current:
current to be
measured
f
Rehunt (Vv) voltmeter
tT
current to be
measured
If, for example, the shunt resistor in the above circuit were
sized at precisely 1 Q, there would be 1 volt dropped across it
for every amp of current through it. The voltmeter indication
could then be taken as a direct indication of current through
the shunt. For measuring very small currents, higher values
of shunt resistance could be used to generate more voltage
drop per given unit of current, thus extending the usable
range of the (volt)meter down into lower amounts of current.
The use of voltmeters in conjunction with low-value shunt
resistances for the measurement of current is something
commonly seen in industrial applications.
The use of a shunt resistor along with a voltmeter to measure
current can be a useful trick for simplifying the task of
frequent current measurements in a circuit. Normally, to
measure current through a circuit with an ammeter, the
circuit would have to be broken (interrupted) and the
ammeter inserted between the separated wire ends, like this:
re
am
Load
If we have a circuit where current needs to be measured
often, or we would just like to make the process of current
measurement more convenient, a shunt resistor could be
placed between those points and left there permanently,
current readings taken with a voltmeter as needed without
interrupting continuity in the circuit:
e
shunt
ame Load
Of course, care must be taken in sizing the shunt resistor low
enough so that it doesn't adversely affect the circuit's normal
operation, but this is generally not difficult to do. This
technique might also be useful in computer circuit analysis,
where we might want to have the computer display current
through a circuit in terms of a voltage (with SPICE, this would
allow us to avoid the idiosyncrasy of reading negative
current values):
R
1 shunt 2
1Q
shunt resistor example circuit
vl 10
rshunt 12 1
rload 2 0 15k
.dc vl 12 12 1
print dec v(1,2)
.end
vl v(1,2)
1.200E+01 7.999E-04
We would interpret the voltage reading across the shunt
resistor (between circuit nodes 1 and 2 in the SPICE
simulation) directly as amps, with 7.999E-04 being 0.7999
mA, or 799.9 WA. Ideally, 12 volts applied directly across 15
kQ would give us exactly 0.8 mA, but the resistance of the
shunt lessens that current just a tiny bit (as it would in real
life). However, such a tiny error is generally well within
acceptable limits of accuracy for either a simulation or a real
circuit, and so shunt resistors can be used in all but the most
demanding applications for accurate current measurement.
e REVIEW:
e Ammeter ranges are created by adding parallel "shunt"
resistors to the movement circuit, providing a precise
Current division.
e Shunt resistors may have high power dissipations, so be
careful when choosing parts for such meters!
e Shunt resistors can be used in conjunction with high-
resistance voltmeters as well as low-resistance ammeter
movements, producing accurate voltage drops for given
amounts of current. Shunt resistors should be selected
for as low a resistance value as possible to minimize their
impact upon the circuit under test.
Ammeter impact on measured circuit
Just like voltmeters, ammeters tend to influence the amount
of current in the circuits they're connected to. However,
unlike the ideal voltmeter, the ideal ammeter has zero
internal resistance, so as to drop as little voltage as possible
as electrons flow through it. Note that this ideal resistance
value is exactly opposite as that of a voltmeter. With
voltmeters, we want as little current to be drawn as possible
from the circuit under test. With ammeters, we want as little
voltage to be dropped as possible while conducting current.
Here is an extreme example of an ammeter's effect upon a
circuit:
+
R; nternal
> O53:
With the ammeter disconnected from this circuit, the current
through the 3 Q resistor would be 666.7 mA, and the current
through the 1.5 Q resistor would be 1.33 amps. If the
ammeter had an internal resistance of 1/2 Q, and it were
inserted into one of the branches of this circuit, though, its
resistance would seriously affect the measured branch
current:
571.43 mA 3 eee
70.52
Having effectively increased the left branch resistance from 3
Q to 3.5 QO, the ammeter will read 571.43 mA instead of 666.7
mA. Placing the same ammeter in the right branch would
affect the current to an even greater extent:
+
R internal
0.5 Q
666.7 mA
Now the right branch current is 1 amp instead of 1.333 amps,
due to the increase in resistance created by the addition of
the ammeter into the current path.
When using standard ammeters that connect in series with
the circuit being measured, it might not be practical or
possible to redesign the meter for a lower input (lead-to-lead)
resistance. However, if we were selecting a value of shunt
resistor to place in the circuit for a current measurement
based on voltage drop, and we had our choice of a wide
range of resistances, it would be best to choose the lowest
practical resistance for the application. Any more resistance
than necessary and the shunt may impact the circuit
adversely by adding excessive resistance in the current path.
One ingenious way to reduce the impact that a current-
measuring device has on a circuit is to use the circuit wire as
part of the ammeter movement itself. All current-carrying
wires produce a magnetic field, the strength of which is in
direct proportion to the strength of the current. By building
an instrument that measures the strength of that magnetic
field, a no-contact ammeter can be produced. Such a meter
is able to measure the current through a conductor without
even having to make physical contact with the circuit, much
less break continuity or insert additional resistance.
magnetic field
encircling the
current-Carryin
conductor
clamp-on
QSynrn"
Vi
current to be
measured
Ammeters of this design are made, and are called "clamp-on"
meters because they have "jaws" which can be opened and
then secured around a circuit wire. Clamp-on ammeters make
for quick and safe current measurements, especially on high-
power industrial circuits. Because the circuit under test has
had no additional resistance inserted into it by a clamp-on
meter, there is no error induced in taking a current
measurement.
magnetic field
encircling the
current-carrying
conductor
clamp-on
ammeter
current to be
measured
The actual movement mechanism of a clamp-on ammeter is
much the same as for an iron-vane instrument, except that
there is no internal wire coil to generate the magnetic field.
More modern designs of clamp-on ammeters utilize a small
magnetic field detector device called a Hall-effect sensor to
accurately determine field strength. Some clamp-on meters
contain electronic amplifier circuitry to generate a small
voltage proportional to the current in the wire between the
jaws, that small voltage connected to a voltmeter for
convenient readout by a technician. Thus, a clamp-on unit
can be an accessory device to a voltmeter, for current
measurement.
A less accurate type of magnetic-field-sensing ammeter than
the clamp-on style is shown in the following photograph:
The operating principle for this ammeter is identical to the
clamp-on style of meter: the circular magnetic field
surrounding a current-carrying conductor deflects the
meter's needle, producing an indication on the scale. Note
how there are two current scales on this particular meter: +/-
75 amps and +/- 400 amps. These two measurement scales
correspond to the two sets of notches on the back of the
meter. Depending on which set of notches the current-
carrying conductor is laid in, a given strength of magnetic
field will have a different amount of effect on the needle. In
effect, the two different positions of the conductor relative to
the movement act as two different range resistors in a direct-
connection style of ammeter.
e REVIEW:
e An ideal ammeter has zero resistance.
e A'"clamp-on" ammeter measures current through a wire
by measuring the strength of the magnetic field around it
rather than by becoming part of the circuit, making it an
ideal ammeter.
e Clamp-on meters make for quick and safe current
measurements, because there is no conductive contact
between the meter and the circuit.
Ohmmeter design
Though mechanical ohmmeter (resistance meter) designs are
rarely used today, having largely been superseded by digital
instruments, their operation is nonetheless intriguing and
worthy of study.
The purpose of an ohmmeter, of course, is to measure the
resistance placed between its leads. This resistance reading
IS indicated through a mechanical meter movement which
operates on electric current. The onmmeter must then have
an internal source of voltage to create the necessary current
to operate the movement, and also have appropriate ranging
resistors to allow just the right amount of current through the
movement at any given resistance.
Starting with a simple movement and battery circuit, let's
see how it would function as an ohmmeter:
A simple ohmmeter
500Q F.S.=1mA
9V
|
: +
black test red test
lead lead
When there is infinite resistance (no continuity between test
leads), there is zero current through the meter movement,
and the needle points toward the far left of the scale. In this
regard, the ohmmeter indication is "backwards" because
maximum indication (infinity) is on the left of the scale, while
voltage and current meters have zero at the left of their
scales.
If the test leads of this ohmmeter are directly shorted
together (measuring zero Q), the meter movement will have
a maximum amount of current through it, limited only by the
battery voltage and the movement's internal resistance:
500Q F.S.=1mA
9V
|
~— |l8mA
black test red test
lead lead
With 9 volts of battery potential and only 500 Q of movement
resistance, our circuit current will be 18 mA, which is far
beyond the full-scale rating of the movement. Such an
excess of current will likely damage the meter.
Not only that, but having such a condition limits the
usefulness of the device. If full left-of-scale on the meter face
represents an infinite amount of resistance, then full right-of-
scale should represent zero. Currently, our design "pegs" the
meter movement hard to the right when Zero resistance is
attached between the leads. We need a way to make it so
that the movement just registers full-scale when the test
leads are shorted together. This is accomplished by adding a
series resistance to the meter's circuit:
500Q FS.=1mA
9V
|
black test red test
lead lead
To determine the proper value for R, we calculate the total
circuit resistance needed to limit current to 1 mA (full-scale
deflection on the movement) with 9 volts of potential from
the battery, then subtract the movement's internal
resistance from that figure:
9V
Rita = — = ——
| lLmA
Rectal =9 kQ
R = Ryyq - 500 Q = 8.5 kQ
Now that the right value for R has been calculated, we're still
left with a problem of meter range. On the left side of the
scale we have "infinity" and on the right side we have zero.
Besides being "backwards" from the scales of voltmeters and
ammeters, this scale is strange because it goes from nothing
to everything, rather than from nothing to a finite value
(such as 10 volts, 1 amp, etc.). One might pause to wonder,
“what does middle-of-scale represent? What figure lies
exactly between zero and infinity?" Infinity is more than just
a very big amount: it is an incalculable quantity, larger than
any definite number ever could be. If half-scale indication on
any other type of meter represents 1/2 of the full-scale range
value, then what is half of infinity on an ohmmeter scale?
The answer to this paradox is a nonlinear scale. Simply put,
the scale of an ohmmeter does not smoothly progress from
zero to infinity as the needle sweeps from right to left.
Rather, the scale starts out "expanded" at the right-hand
side, with the successive resistance values growing closer
and closer to each other toward the left side of the scale:
An ohmmeter’s logarithmic scale
Infinity cannot be approached in a linear (even) fashion,
because the scale would never get there! With a nonlinear
scale, the amount of resistance spanned for any given
distance on the scale increases as the scale progresses
toward infinity, making infinity an attainable goal.
We still have a question of range for our ohmmeter, though.
What value of resistance between the test leads will cause
exactly 1/2 scale deflection of the needle? If we know that
the movement has a full-scale rating of 1 mA, then 0.5 mA
(500 UWA) must be the value needed for half-scale deflection.
Following our design with the 9 volt battery as a source we
get:
E 9V
1 500A
Ryotal =
Rectal = 18kQ
With an internal movement resistance of 500 © and a series
range resistor of 8.5 kQ, this leaves 9 kQ for an external
(lead-to-lead) test resistance at 1/2 scale. In other words, the
test resistance giving 1/2 scale deflection in an ohmmeter is
equal in value to the (internal) series total resistance of the
meter circuit.
Using Ohm's Law a few more times, we can determine the
test resistance value for 1/4 and 3/4 scale deflection as well:
1/4 scale deflection (0.25 mA of meter current):
ee
rom 1 250 pA
Ry otal = 36 kQ
Reest = Rootal a R internal
Ries, = 36 kQ-9kQ
Reest a 27 kQ
3/4 scale deflection (0.75 mA of meter current):
E 9V
Real = ~ = F559 uA
Riad = 12 hee
Reest = Reotat - Rintemal
Rie = 12 kQ-9kQ
Rest = 3 kQ
So, the scale for this ohmmeter looks something like this:
9
27k 3k
One major problem with this design is its reliance upon a
stable battery voltage for accurate resistance reading. If the
battery voltage decreases (as all chemical batteries do with
age and use), the ohmmeter scale will lose accuracy. With the
series range resistor at a constant value of 8.5 kO and the
battery voltage decreasing, the meter will no longer deflect
full-scale to the right when the test leads are shorted
together (0 Q). Likewise, a test resistance of 9 kQ will fail to
deflect the needle to exactly 1/2 scale with a lesser battery
voltage.
There are design techniques used to compensate for varying
battery voltage, but they do not completely take care of the
problem and are to be considered approximations at best. For
this reason, and for the fact of the nonlinear scale, this type
of ohmmeter is never considered to be a precision
instrument.
One final caveat needs to be mentioned with regard to
ohmmeters: they only function correctly when measuring
resistance that is not being powered by a voltage or current
source. In other words, you cannot measure resistance with
an ohmmeter on a "live" circuit! The reason for this is simple:
the ohmmeter's accurate indication depends on the only
source of voltage being its internal battery. The presence of
any voltage across the component to be measured will
interfere with the ohmmeter's operation. If the voltage is
large enough, it may even damage the ohmmeter.
e REVIEW:
¢ Ohmmeters contain internal sources of voltage to supply
power in taking resistance measurements.
e An analog ohmmeter scale is "backwards" from that of a
voltmeter or ammeter, the movement needle reading
zero resistance at full-scale and infinite resistance at rest.
e Analog ohmmeters also have nonlinear scales,
"expanded" at the low end of the scale and "compressed"
at the high end to be able to span from zero to infinite
resistance.
e Analog ohmmeters are not precision instruments.
e Ohmmeters should never be connected to an energized
circuit (that is, a circuit with its own source of voltage).
Any voltage applied to the test leads of an ohmmeter will
invalidate its reading.
High voltage ohmmeters
Most ohmmeters of the design shown in the previous section
utilize a battery of relatively low voltage, usually nine volts
or less. This is perfectly adequate for measuring resistances
under several mega-ohms (MQ), but when extremely high
resistances need to be measured, a 9 volt battery is
insufficient for generating enough current to actuate an
electromechanical meter movement.
Also, as discussed in an earlier chapter, resistance is not
always a stable (linear) quantity. This is especially true of
non-metals. Recall the graph of current over voltage for a
small air gap (less than an inch):
|
(current)
A
Seo odes on
0 3% 100 150 ©6200 250 200 350 400
E |
(voltage) |
. . . | .
ionization potential
While this is an extreme example of nonlinear conduction,
other substances exhibit similar insulating/conducting
properties when exposed to high voltages. Obviously, an
ohmmeter using a low-voltage battery as a source of power
cannot measure resistance at the ionization potential of a
gas, or at the breakdown voltage of an insulator. If such
resistance values need to be measured, nothing but a high-
voltage ohmmeter will suffice.
The most direct method of high-voltage resistance
measurement involves simply substituting a higher voltage
battery in the same basic design of ohmmeter investigated
earlier:
Simple high-voltage ohmmeter
a
black test red test
lead lead
Knowing, however, that the resistance of some materials
tends to change with applied voltage, it would be
advantageous to be able to adjust the voltage of this
ohmmeter to obtain resistance measurements under different
conditions:
: +
Ii
black test red test
lead lead
Unfortunately, this would create a calibration problem for the
meter. If the meter movement deflects full-scale with a
certain amount of current through it, the full-scale range of
the meter in ohms would change as the source voltage
changed. Imagine connecting a stable resistance across the
test leads of this ohmmeter while varying the source voltage:
as the voltage is increased, there will be more current
through the meter movement, hence a greater amount of
deflection. What we really need is a meter movement that
will produce a consistent, stable deflection for any stable
resistance value measured, regardless of the applied voltage.
Accomplishing this design goal requires a special meter
movement, one that is peculiar to megohmmeters, or
meggers, as these instruments are known.
"Megger" movement
The numbered, rectangular blocks in the above illustration
are cross-sectional representations of wire coils. These three
coils all move with the needle mechanism. There is no spring
mechanism to return the needle to a set position. When the
movement is unpowered, the needle will randomly "float."
The coils are electrically connected like this:
High voltage
Black
Test leads
With infinite resistance between the test leads (open circuit),
there will be no current through coil 1, only through coils 2
and 3. When energized, these coils try to center themselves
in the gap between the two magnet poles, driving the needle
fully to the right of the scale where it points to "infinity."
Current through coils 2 and 3;
no current through coil 1
Any current through coil 1 (through a measured resistance
connected between the test leads) tends to drive the needle
to the left of scale, back to zero. The internal resistor values
of the meter movement are calibrated so that when the test
leads are shorted together, the needle deflects exactly to the
0 Q position.
Because any variations in battery voltage will affect the
torque generated by both sets of coils (coils 2 and 3, which
drive the needle to the right, and coil 1, which drives the
needle to the left), those variations will have no effect of the
calibration of the movement. In other words, the accuracy of
this ohmmeter movement is unaffected by battery voltage: a
given amount of measured resistance will produce a certain
needle deflection, no matter how much or little battery
voltage is present.
The only effect that a variation in voltage will have on meter
indication is the degree to which the measured resistance
changes with applied voltage. So, if we were to use a megger
to measure the resistance of a gas-discharge lamp, it would
read very high resistance (needle to the far right of the scale)
for low voltages and low resistance (needle moves to the left
of the scale) for high voltages. This is precisely what we
expect from a good high-voltage ohmmeter: to provide
accurate indication of subject resistance under different
circumstances.
For maximum safety, most meggers are equipped with hand-
crank generators for producing the high DC voltage (up to
1000 volts). If the operator of the meter receives a shock
from the high voltage, the condition will be self-correcting, as
he or she will naturally stop cranking the generator!
Sometimes a "slip clutch" is used to stabilize generator
speed under different cranking conditions, so as to provide a
fairly stable voltage whether it is cranked fast or slow.
Multiple voltage output levels from the generator are
available by the setting of a selector switch.
A simple hand-crank megger is shown in this photograph:
Some meggers are battery-powered to provide greater
precision in output voltage. For safety reasons these meggers
are activated by a momentary-contact pushbutton switch, so
the switch cannot be left in the "on" position and pose a
significant shock hazard to the meter operator.
Real meggers are equipped with three connection terminals,
labeled Line, Earth, and Guard. The schematic is quite similar
to the simplified version shown earlier:
High voltage
Guard Line Earth
Resistance is measured between the Line and Earth
terminals, where current will travel through coil 1. The
"Guard" terminal is provided for special testing situations
where one resistance must be isolated from another. Take for
instance this scenario where the insulation resistance is to be
tested in a two-wire cable:
cable
sheath
\
conductor
conductor
insulation
To measure insulation resistance from a conductor to the
outside of the cable, we need to connect the "Line" lead of
the megger to one of the conductors and connect the "Earth"
lead of the megger to a wire wrapped around the sheath of
the cable:
wire wrapped
In this configuration the megger should read the resistance
between one conductor and the outside sheath. Or will it? If
we draw a schematic diagram showing all insulation
resistances as resistor symbols, what we have looks like this:
Earth
Megger
Rather than just measure the resistance of the second
conductor to the sheath (R.>.,), what we'll actually measure
is that resistance in parallel with the series combination of
conductor-to-conductor resistance (R.j-<2) and the first
conductor to the sheath (R;,j.,). If we don't care about this
fact, we can proceed with the test as configured. If we desire
to measure only the resistance between the second
conductor and the sheath (R,>..), then we need to use the
megger's "Guard" terminal:
wire wrapped
aroun
; Wy sheath
Megger with "Guard"
connected
Rais
conductor,
Earth
Guard
Connecting the "Guard" terminal to the first conductor places
the two conductors at almost equal potential. With little or no
voltage between them, the insulation resistance is nearly
infinite, and thus there will be no current between the two
conductors. Consequently, the megger's resistance
indication will be based exclusively on the current through
the second conductor's insulation, through the cable sheath,
and to the wire wrapped around, not the current leaking
through the first conductor's insulation.
Meggers are field instruments: that is, they are designed to
be portable and operated by a technician on the job site with
as much ease as a regular ohmmeter. They are very useful for
checking high-resistance "short" failures between wires
caused by wet or degraded insulation. Because they utilize
such high voltages, they are not as affected by stray voltages
(voltages less than 1 volt produced by electrochemical
reactions between conductors, or "induced" by neighboring
magnetic fields) as ordinary ohmmeters.
For a more thorough test of wire insulation, another high-
voltage ohmmeter commonly called a A/-pot tester is used.
These specialized instruments produce voltages in excess of
1 kV, and may be used for testing the insulating
effectiveness of oil, ceramic insulators, and even the
integrity of other high-voltage instruments. Because they are
capable of producing such high voltages, they must be
operated with the utmost care, and only by trained
personnel.
It should be noted that hi-pot testers and even meggers (in
certain conditions) are capable of damaging wire insulation if
incorrectly used. Once an insulating material has been
subjected to breakdown by the application of an excessive
voltage, its ability to electrically insulate will be
compromised. Again, these instruments are to be used only
by trained personnel.
Multimeters
Seeing as how a common meter movement can be made to
function as a voltmeter, ammeter, or ohmmeter simply by
connecting it to different external resistor networks, it should
make sense that a multi-purpose meter ("multimeter") could
be designed in one unit with the appropriate switch(es) and
resistors.
For general purpose electronics work, the multimeter reigns
supreme as the instrument of choice. No other device is able
to do so much with so little an investment in parts and
elegant simplicity of operation. As with most things in the
world of electronics, the advent of solid-state components
like transistors has revolutionized the way things are done,
and multimeter design is no exception to this rule. However,
in keeping with this chapter's emphasis on analog ("old-
fashioned") meter technology, I'll show you a few pre-
transistor meters.
The unit shown above is typical of a handheld analog
multimeter, with ranges for voltage, current, and resistance
measurement. Note the many scales on the face of the meter
movement for the different ranges and functions selectable
by the rotary switch. The wires for connecting this instrument
to a circuit (the "test leads") are plugged into the two copper
jacks (socket holes) at the bottom-center of the meter face
marked "- TEST +", black and red.
This multimeter (Barnett brand) takes a slightly different
design approach than the previous unit. Note how the rotary
selector switch has fewer positions than the previous meter,
but also how there are many more jacks into which the test
leads may be plugged into. Each one of those jacks is labeled
with a number indicating the respective full-scale range of
the meter.
Lastly, here is a picture of a digital multimeter. Note that the
familiar meter movement has been replaced by a blank,
gray-colored display screen. When powered, numerical digits
appear in that screen area, depicting the amount of voltage,
current, or resistance being measured. This particular brand
and model of digital meter has a rotary selector switch and
four jacks into which test leads can be plugged. Two leads --
one red and one black -- are shown plugged into the meter.
A close examination of this meter will reveal one "common"
jack for the black test lead and three others for the red test
lead. The jack into which the red lead is shown inserted is
labeled for voltage and resistance measurement, while the
other two jacks are labeled for current (A, mA, and YA)
measurement. This is a wise design feature of the
multimeter, requiring the user to move a test lead plug from
one jack to another in order to switch from the voltage
measurement to the current measurement function. It would
be hazardous to have the meter set in current measurement
mode while connected across a significant source of voltage
because of the low input resistance, and making it necessary
to move a test lead plug rather than just flip the selector
switch to a different position helps ensure that the meter
doesn't get set to measure current unintentionally.
Note that the selector switch still has different positions for
voltage and current measurement, so in order for the user to
switch between these two modes of measurement they must
switch the position of the red test lead and move the selector
switch to a different position.
Also note that neither the selector switch nor the jacks are
labeled with measurement ranges. In other words, there are
no "100 volt" or "10 volt" or "1 volt" ranges (or any
equivalent range steps) on this meter. Rather, this meter is
“autoranging," meaning that it automatically picks the
appropriate range for the quantity being measured.
Autoranging is a feature only found on digital meters, but not
all digital meters.
No two models of multimeters are designed to operate
exactly the same, even if they're manufactured by the same
company. In order to fully understand the operation of any
multimeter, the owner's manual must be consulted.
Here is a schematic for a simple analog volt/ammeter:
R
multiplier!
R shunt
Rout tiplier2
R
multipliers
"Common" A V
jack
In the switch's three lower (most counter-clockwise)
positions, the meter movement is connected to the Common
and V jacks through one of three different series range
resistors (Rmuttiptiers through Rmuttiplier3), ANd So acts as a
voltmeter. In the fourth position, the meter movement is
connected in parallel with the shunt resistor, and so acts as
an ammeter for any current entering the common jack and
exiting the A jack. In the last (furthest clockwise) position,
the meter movement is disconnected from either red jack,
but short-circuited through the switch. This short-circuiting
creates a dampening effect on the needle, guarding against
mechanical shock damage when the meter is handled and
moved.
If an ohmmeter function is desired in this multimeter design,
it may be substituted for one of the three voltage ranges as
such:
Raut plier
R
shunt
Rout tiplier2
"Common"
jack
With all three fundamental functions available, this
multimeter may also be known as a vo/t-ohm-milliammeter.
Obtaining a reading from an analog multimeter when there is
a multitude of ranges and only one meter movement may
seem daunting to the new technician. On an analog
multimeter, the meter movement is marked with several
scales, each one useful for at least one range setting. Here is
a close-up photograph of the scale from the Barnett
multimeter shown earlier in this section:
~ &pe
CURRENT
Note that there are three types of scales on this meter face: a
green scale for resistance at the top, a set of black scales for
DC voltage and current in the middle, and a set of blue scales
for AC voltage and current at the bottom. Both the DC and AC
scales have three sub-scales, one ranging O to 2.5, one
ranging 0 to 5, and one ranging O to 10. The meter operator
must choose whichever scale best matches the range switch
and plug settings in order to properly interpret the meter's
indication.
This particular multimeter has several basic voltage
measurement ranges: 2.5 volts, 10 volts, 50 volts, 250 volts,
500 volts, and 1000 volts. With the use of the voltage range
extender unit at the top of the multimeter, voltages up to
5000 volts can be measured. Suppose the meter operator
chose to switch the meter into the "volt" function and plug
the red test lead into the 10 volt jack. To interpret the
needle's position, he or she would have to read the scale
ending with the number "10". If they moved the red test plug
into the 250 volt jack, however, they would read the meter
indication on the scale ending with "2.5", multiplying the
direct indication by a factor of 100 in order to find what the
measured voltage was.
If current is measured with this meter, another jack is chosen
for the red plug to be inserted into and the range Is selected
via a rotary switch. This close-up photograph shows the
switch set to the 2.5 mA position:
R100 Pyioo0
We Rx10000
Note how all current ranges are power-of-ten multiples of the
three scale ranges shown on the meter face: 2.5, 5, and 10.
In some range settings, such as the 2.5 mA for example, the
meter indication may be read directly on the O to 2.5 scale.
For other range settings (250 WA, 50 mA, 100 mA, and 500
mA), the meter indication must be read off the appropriate
scale and then multiplied by either 10 or 100 to obtain the
real figure. The highest current range available on this meter
iS Obtained with the rotary switch in the 2.5/10 amp position.
The distinction between 2.5 amps and 10 amps is made by
the red test plug position: a special "10 amp" jack next to the
regular current-measuring jack provides an alternative plug
setting to select the higher range.
Resistance in ohms, of course, is read by a nonlinear scale at
the top of the meter face. It is "backward," just like all
battery-operated analog ohmmeters, with zero at the right-
hand side of the face and infinity at the left-hand side. There
IS Only One jack provided on this particular multimeter for
"ohms," so different resistance-measuring ranges must be
selected by the rotary switch. Notice on the switch how five
different "multiplier" settings are provided for measuring
resistance: Rx1, Rx10, Rx100, Rx1000, and Rx10000. Just as
you might suspect, the meter indication is given by
multiplying whatever needle position is shown on the meter
face by the power-of-ten multiplying factor set by the rotary
switch.
Kelvin (4-wire) resistance
measurement
Suppose we wished to measure the resistance of some
component located a significant distance away from our
ohmmeter. Such a scenario would be problematic, because
an ohmmeter measures a// resistance in the circuit loop,
which includes the resistance of the wires (Ryire) connecting
the ohmmeter to the component being measured (Reupject):
R
subject
+R +R
wire subject wire
Ohmmeter indicates R
Usually, wire resistance is very small (only a few ohms per
hundreds of feet, depending primarily on the gauge (size) of
the wire), but if the connecting wires are very long, and/or
the component to be measured has a very low resistance
anyway, the measurement error introduced by wire
resistance will be substantial.
An ingenious method of measuring the subject resistance in
a situation like this involves the use of both an ammeter and
a voltmeter. We know from Ohm's Law that resistance is
equal to voltage divided by current (R = E/I). Thus, we should
be able to determine the resistance of the subject component
if we measure the current going through it and the voltage
dropped across it:
Ammeter R
Voltmeter
subject
_ Voltmeter indication
ubiect Ammeter indication
Current is the same at all points in the circuit, because it is a
series loop. Because we're only measuring voltage dropped
across the subject resistance (and not the wires' resistances),
though, the calculated resistance is indicative of the subject
component's resistance (Reupject) alone.
Our goal, though, was to measure this subject resistance
from a distance, so our voltmeter must be located
somewhere near the ammeter, connected across the subject
resistance by another pair of wires containing resistance:
Ammeter R
R
subject
aor cs Voltmeter indication
subject” Ammeter indication
At first it appears that we have lost any advantage of
measuring resistance this way, because the voltmeter now
has to measure voltage through a long pair of (resistive)
wires, introducing stray resistance back into the measuring
circuit again. However, upon closer inspection it is seen that
nothing is lost at all, because the voltmeter's wires carry
miniscule current. Thus, those long lengths of wire
connecting the voltmeter across the subject resistance will
drop insignificant amounts of voltage, resulting ina
voltmeter indication that is very nearly the same as if it were
connected directly across the subject resistance:
Ammeter
— Ryire <— —
subject
ee
Any voltage dropped across the main current-carrying wires
will not be measured by the voltmeter, and so do not factor
into the resistance calculation at all. Measurement accuracy
may be improved even further if the voltmeter's current is
kept to a minimum, either by using a high-quality (low full-
scale current) movement and/or a potentiometric (null-
balance) system.
This method of measurement which avoids errors caused by
wire resistance is called the Ke/vin, or 4-wire method. Special
connecting clips called Ke/vin clips are made to facilitate this
kind of connection across a subject resistance:
Kelvin clips
Cc clip
Pp, | 4-wire cable Da
___ R
Ce ——
clip
subject
In regular, "alligator" style clips, both halves of the jaw are
electrically common to each other, usually joined at the
hinge point. In Kelvin clips, the jaw halves are insulated from
each other at the hinge point, only contacting at the tips
where they clasp the wire or terminal of the subject being
measured. Thus, current through the "C" ("current") jaw
halves does not go through the "P" ("potential," or vo/tage)
jaw halves, and will not create any error-inducing voltage
drop along their length:
4-wire cable
C
_ Voltmeter indication
ubject “Ammeter indication
The same principle of using different contact points for
current conduction and voltage measurement is used in
precision shunt resistors for measuring large amounts of
current. As discussed previously, shunt resistors function as
current measurement devices by dropping a precise amount
of voltage for every amp of current through them, the
voltage drop being measured by a voltmeter. In this sense, a
precision shunt resistor "converts" a current value into a
proportional voltage value. Thus, current may be accurately
measured by measuring voltage dropped across the shunt:
current to be
measured
tT
current to be T
measure
4
Rehunt @ voltmeter
Current measurement using a shunt resistor and voltmeter is
particularly well-suited for applications involving particularly
large magnitudes of current. In such applications, the shunt
resistor's resistance will likely be in the order of milliohms or
microohms, so that only a modest amount of voltage will be
dropped at full current. Resistance this low is comparable to
wire connection resistance, which means voltage measured
across such a shunt must be done so in such a way as to
avoid detecting voltage dropped across the current-carrying
wire connections, lest huge measurement errors be induced.
In order that the voltmeter measure only the voltage dropped
by the shunt resistance itself, without any stray voltages
Originating from wire or connection resistance, shunts are
usually equipped with four connection terminals:
T Measured current
Voltmeter
T Measured current
In metrological (metrology = "the science of measurement")
applications, where accuracy is of paramount importance,
highly precise "standard" resistors are also equipped with
four terminals: two for carrying the measured current, and
two for conveying the resistor's voltage drop to the
voltmeter. This way, the voltmeter only measures voltage
dropped across the precision resistance itself, without any
stray voltages dropped across current-carrying wires or wire-
to-terminal connection resistances.
The following photograph shows a precision standard resistor
of 1 O value immersed in a temperature-controlled oil bath
with a few other standard resistors. Note the two large, outer
terminals for current, and the two small connection terminals
for voltage:
Here is another, older (pre-World War II) standard resistor of
German manufacture. This unit has a resistance of 0.001 Q,
and again the four terminal connection points can be seen as
black knobs (metal pads underneath each knob for direct
metal-to-metal connection with the wires), two large knobs
for securing the current-carrying wires, and two smaller
knobs for securing the voltmeter ("potential") wires:
Appreciation is extended to the Fluke Corporation in Everett,
Washington for allowing me to photograph these expensive
and somewhat rare standard resistors in their primary
standards laboratory.
It should be noted that resistance measurement using both
an ammeter and a voltmeter is subject to compound error.
Because the accuracy of both instruments factors in to the
final result, the overall measurement accuracy may be worse
than either instrument considered alone. For instance, if the
ammeter is accurate to +/- 1% and the voltmeter is also
accurate to +/- 1%, any measurement dependent on the
indications of both instruments may be inaccurate by as
much as +/- 2%.
Greater accuracy may be obtained by replacing the ammeter
with a standard resistor, used as a Current-measuring shunt.
There will still be compound error between the standard
resistor and the voltmeter used to measure voltage drop, but
this will be less than with a voltmeter + ammeter
arrangement because typical standard resistor accuracy far
exceeds typical ammeter accuracy. Using Kelvin clips to
make connection with the subject resistance, the circuit looks
something like this:
clip
>
P Reabject
S r
S >
C clip
q
All current-carrying wires in the above circuit are shown in
"bold," to easily distinguish them from wires connecting the
voltmeter across both resistances (Reupject ANd Retandard)-
Ideally, a potentiometric voltmeter is used to ensure as little
current through the "potential" wires as possible.
Ren tacts
Power supply
a oe nt Riamp
Voltmeter
current |: as
The Kelvin measurement can be a practical tool for finding
poor connections or unexpected resistance in an electrical
circuit. Connect a DC power supply to the circuit and adjust
the power supply so that it supplies a constant current to the
circuit as shown in the diagram above (within the circuit's
capabilities, of course). With a digital multimeter set to
measure DC voltage, measure the voltage drop across
various points in the circuit. If you know the wire size, you
can estimate the voltage drop you should see and compare
this to the voltage drop you measure. This can be a quick
and effective method of finding poor connections in wiring
exposed to the elements, such as in the lighting circuits of a
trailer. It can also work well for unpowered AC conductors
(make sure the AC power cannot be turned on). For example,
you can measure the voltage drop across a light switch and
determine if the wiring connections to the switch or the
Switch's contacts are suspect. To be most effective using this
technique, you should also measure the same type of circuits
after they are newly made so you have a feel for the "correct"
values. If you use this technique on new circuits and put the
results in a log book, you have valuable information for
troubleshooting in the future.
Bridge circuits
No text on electrical metering could be called complete
without a section on bridge circuits. These ingenious circuits
make use of a null-balance meter to compare two voltages,
just like the laboratory balance scale compares two weights
and indicates when they're equal. Unlike the "potentiometer"
circuit used to simply measure an unknown voltage, bridge
circuits can be used to measure all kinds of electrical values,
not the least of which being resistance.
The standard bridge circuit, often called a Wheatstone
bridge, looks something like this:
When the voltage between point 1 and the negative side of
the battery is equal to the voltage between point 2 and the
negative side of the battery, the null detector will indicate
zero and the bridge is said to be "balanced." The bridge's
state of balance is solely dependent on the ratios of R,/Rp
and R,/R>, and is quite independent of the supply voltage
(battery). To measure resistance with a Wheatstone bridge,
an unknown resistance is connected in the place of R, or Rp,
while the other three resistors are precision devices of known
value. Either of the other three resistors can be replaced or
adjusted until the bridge is balanced, and when balance has
been reached the unknown resistor value can be determined
from the ratios of the known resistances.
A requirement for this to be a measurement system is to have
a set of variable resistors available whose resistances are
precisely known, to serve as reference standards. For
example, if we connect a bridge circuit to measure an
unknown resistance R,, we will have to know the exact
values of the other three resistors at balance to determine
the value of R,:
Bridge circuit is
balanced when:
Ra Ry
R, . R,
Each of the four resistances in a bridge circuit are referred to
as arms. The resistor in series with the unknown resistance
R, (this would be R, in the above schematic) is commonly
called the rheostat of the bridge, while the other two
resistors are called the ratio arms of the bridge.
Accurate and stable resistance standards, thankfully, are not
that difficult to construct. In fact, they were some of the first
electrical "standard" devices made for scientific purposes.
Here is a photograph of an antique resistance standard unit:
This resistance standard shown here is variable in discrete
steps: the amount of resistance between the connection
terminals could be varied with the number and pattern of
removable copper plugs inserted into sockets.
Wheatstone bridges are considered a superior means of
resistance measurement to the series battery-movement-
resistor meter circuit discussed in the last section. Unlike that
circuit, with all its nonlinearities (nonlinear scale) and
associated inaccuracies, the bridge circuit is linear (the
mathematics describing its operation are based on simple
ratios and proportions) and quite accurate.
Given standard resistances of sufficient precision and a null
detector device of sufficient sensitivity, resistance
measurement accuracies of at least +/- 0.05% are attainable
with a Wheatstone bridge. It is the preferred method of
resistance measurement in calibration laboratories due to its
high accuracy.
There are many variations of the basic Wheatstone bridge
circuit. Most DC bridges are used to measure resistance,
while bridges powered by alternating current (AC) may be
used to measure different electrical quantities like
inductance, capacitance, and frequency.
An interesting variation of the Wheatstone bridge is the
Kelvin Double bridge, used for measuring very low
resistances (typically less than 1/10 of an ohm). Its schematic
diagram is as such:
Kelvin Double bridge
R. and R, are low-value resistances
The low-value resistors are represented by thick-line symbols,
and the wires connecting them to the voltage source
(carrying high current) are likewise drawn thickly in the
schematic. This oddly-configured bridge is perhaps best
understood by beginning with a standard Wheatstone bridge
set up for measuring low resistance, and evolving it step-by-
step into its final form in an effort to overcome certain
problems encountered in the standard Wheatstone
configuration.
If we were to use a standard Wheatstone bridge to measure
low resistance, it would look something like this:
When the null detector indicates zero voltage, we know that
the bridge is balanced and that the ratios R,/R, and Ry/Ry
are mathematically equal to each other. Knowing the values
of Rz, Ry, and Ry therefore provides us with the necessary
data to solve for R, . . . almost.
We have a problem, in that the connections and connecting
wires between R, and R, possess resistance as well, and this
stray resistance may be substantial compared to the low
resistances of R, and R,. These stray resistances will drop
substantial voltage, given the high current through them,
and thus will affect the null detector's indication and thus
the balance of the bridge:
Stray Evwire dere ig will corrupt
x
the accuracy of R,’s measurement
Since we don't want to measure these stray wire and
connection resistances, but only measure R,, we must find
some way to connect the null detector so that it won't be
influenced by voltage dropped across them. If we connect
the null detector and Ry/Ry ratio arms directly across the
ends of R, and R,, this gets us closer to a practical solution:
Now, only the two E,,;,. voltages
are part of the null detector loop
Now the top two E,,;,2 voltage drops are of no effect to the
null detector, and do not influence the accuracy of R,'s
resistance measurement. However, the two remaining Eyjre
voltage drops will cause problems, as the wire connecting the
lower end of R, with the top end of R, is now shunting across
those two voltage drops, and will conduct substantial
current, introducing stray voltage drops along its own length
as well.
Knowing that the left side of the null detector must connect
to the two near ends of R, and R, in order to avoid
introducing those Ey;,2 voltage drops into the null detector's
loop, and that any direct wire connecting those ends of R,
and R, will itself carry substantial current and create more
stray voltage drops, the only way out of this predicament is
to make the connecting path between the lower end of R,
and the upper end of R, substantially resistive:
We can manage the stray voltage drops between R, and R,
by sizing the two new resistors so that their ratio from upper
to lower is the same ratio as the two ratio arms on the other
side of the null detector. This is why these resistors were
labeled R,, and R, in the original Kelvin Double bridge
schematic: to signify their proportionality with Ry and Ry:
Kelvin Double bridge
R, and R, are low-value resistances
With ratio R,,/R, set equal to ratio Ry/Ry, rheostat arm
resistor R, is adjusted until the null detector indicates
balance, and then we can say that R,/R, is equal to Ry/Ry, or
simply find R, by the following equation:
Ry
R,=R,
M
The actual balance equation of the Kelvin Double bridge is as
follows (Rwire is the resistance of the thick, connecting wire
between the low-resistance standard R, and the test
resistance R,):
R, Ry R
x
4 wire ( rs )( Ry R, )
4s | a, head
R, Ry R, Ry, + R, + ae Ryu Ry
So long as the ratio between Ry and Ry is equal to the ratio
between R,, and R,, the balance equation is no more
complex than that of a regular Wheatstone bridge, with R,/R,
equal to Rj/Ry, because the last term in the equation will be
zero, canceling the effects of all resistances except R,, Rz,
Ry, and Ry.
In many Kelvin Double bridge circuits, Ry=R,, and Ry=R,.-
However, the lower the resistances of R,, and R,, the more
sensitive the null detector will be, because there is less
resistance in series with it. Increased detector sensitivity is
good, because it allows smaller imbalances to be detected,
and thus a finer degree of bridge balance to be attained.
Therefore, some high-precision Kelvin Double bridges use R,,
and R,, values as low as 1/100 of their ratio arm counterparts
(Ry and Ry, respectively). Unfortunately, though, the lower
the values of R,, and R,, the more current they will carry,
which will increase the effect of any junction resistances
present where R,, and R,, connect to the ends of R, and R,.
As you can see, high instrument accuracy demands that a//
error-producing factors be taken into account, and often the
best that can be achieved is a compromise minimizing two or
more different kinds of errors.
e REVIEW:
e Bridge circuits rely on sensitive null-voltage meters to
compare two voltages for equality.
e A Wheatstone bridge can be used to measure resistance
by comparing the unknown resistor against precision
resistors of known value, much like a laboratory scale
measures an unknown weight by comparing it against
known standard weights.
¢ A Kelvin Double bridge is a variant of the Wheatstone
bridge used for measuring very low resistances. Its
additional complexity over the basic Wheatstone design
Is necessary for avoiding errors otherwise incurred by
stray resistances along the current path between the low-
resistance standard and the resistance being measured.
Wattmeter design
Power in an electric circuit is the product (multiplication) of
voltage and current, so any meter designed to measure
power must account for both of these variables.
A special meter movement designed especially for power
measurement is called the dynamometer movement, and is
similar to a D'Arsonval or Weston movement in that a
lightweight coil of wire is attached to the pointer mechanism.
However, unlike the D'Arsonval or Weston movement,
another (stationary) coil is used instead of a permanent
magnet to provide the magnetic field for the moving coil to
react against. The moving coil is generally energized by the
voltage in the circuit, while the stationary coil is generally
energized by the current in the circuit. A dynamometer
movement connected in a circuit looks something like this:
Electrodynamometer movement
— a Load
The top (horizontal) coil of wire measures load current while
the bottom (vertical) coil measures load voltage. Just like the
lightweight moving coils of voltmeter movements, the
(moving) voltage coil of a dynamometer is typically
connected in series with a range resistor so that full load
voltage is not applied to it. Likewise, the (stationary) current
coil of adynamometer may have precision shunt resistors to
divide the load current around it. With custom-built
dynamometer movements, shunt resistors are less likely to
be needed because the stationary coil can be constructed
with as heavy of wire as needed without impacting meter
response, unlike the moving coil which must be constructed
of lightweight wire for minimum inertia.
Electrodynamometer movement
Rit
voltage
coll (moving)
current
_ Col
(stationary) 7
multiplier
e REVIEW:
e Wattmeters are often designed around dynamometer
meter movements, which employ both voltage and
current coils to move a needle.
Creating custom calibration
resistances
Often in the course of designing and building electrical meter
circuits, it is necessary to have precise resistances to obtain
the desired range(s). More often than not, the resistance
values required cannot be found in any manufactured
resistor unit and therefore must be built by you.
One solution to this dilemma is to make your own resistor out
of a length of special high-resistance wire. Usually, a small
"bobbin" is used as a form for the resulting wire coil, and the
coil is wound in such a way as to eliminate any
electromagnetic effects: the desired wire length is folded in
half, and the looped wire wound around the bobbin so that
current through the wire winds clockwise around the bobbin
for half the wire's length, then counter-clockwise for the
other half. This is known as a bifilar winding. Any magnetic
fields generated by the current are thus canceled, and
external magnetic fields cannot induce any voltage in the
resistance wire coil:
Before winding coil Completed resistor
Bobbin
Special
resistance
wire
As you might imagine, this can be a labor-intensive process,
especially if more than one resistor must be built! Another,
easier solution to the dilemma of a custom resistance is to
connect multiple fixed-value resistors together in series-
parallel fashion to obtain the desired value of resistance. This
solution, although potentially time-intensive in choosing the
best resistor values for making the first resistance, can be
duplicated much faster for creating multiple custom
resistances of the same value:
Ry
total
R
A disadvantage of either technique, though, is the fact that
both result in a fixed resistance value. In a perfect world
where meter movements never lose magnetic strength of
their permanent magnets, where temperature and time have
no effect on component resistances, and where wire
connections maintain zero resistance forever, fixed-value
resistors work quite well for establishing the ranges of
precision instruments. However, in the real world, it is
advantageous to have the ability to calibrate, or adjust, the
instrument in the future.
It makes sense, then, to use potentiometers (connected as
rheostats, usually) as variable resistances for range resistors.
The potentiometer may be mounted inside the instrument
case so that only a service technician has access to change
its value, and the shaft may be locked in place with thread-
fastening compound (ordinary nail polish works well for this!)
so that it will not move if subjected to vibration.
However, most potentiometers provide too large a resistance
Span over their mechanically-short movement range to allow
for precise adjustment. Suppose you desired a resistance of
8.335 kQ +/- 1 QO, and wanted to use a 10 kQ potentiometer
(rheostat) to obtain it. A precision of 1 QO out of a span of 10
kQ is 1 part in 10,000, or 1/100 of a percent! Even with a 10-
turn potentiometer, it will be very difficult to adjust it to any
value this finely. Such a feat would be nearly impossible
using a standard 3/4 turn potentiometer. So how can we get
the resistance value we need and still have room for
adjustment?
The solution to this problem is to use a potentiometer as part
of a larger resistance network which will create a limited
adjustment range. Observe the following example:
8kQ 1LkQ
Rota
8kQ to9 kQ
adjustable range
Here, the 1 kO potentiometer, connected as a rheostat,
provides by itself a 1 kQ span (a range of O Oto 1 kQ).
Connected in series with an 8 kQ resistor, this offsets the
total resistance by 8,000 Q, giving an adjustable range of 8
kQ to 9 kQ. Now, a precision of +/- 1 QO represents 1 part in
1000, or 1/10 of a percent of potentiometer shaft motion.
This is ten times better, in terms of adjustment sensitivity,
than what we had using a 10 kQ potentiometer.
If we desire to make our adjustment capability even more
precise -- SO we can Set the resistance at 8.335 kO with even
greater precision -- we may reduce the span of the
potentiometer by connecting a fixed-value resistor in parallel
with it:
8kQ LkQ
R otal
8kQ to 8.5 kQ
adjustable range
Now, the calibration span of the resistor network is only 500
Q, from 8 kQ to 8.5 kQ. This makes a precision of +/- 1 QO
equal to 1 part in 500, or 0.2 percent. The adjustment is now
half as sensitive as it was before the addition of the parallel
resistor, facilitating much easier calibration to the target
value. The adjustment will not be linear, unfortunately
(halfway on the potentiometer's shaft position will not result
in 8.25 kQ total resistance, but rather 8.333 kQ). Still, it is an
improvement in terms of sensitivity, and it is a practical
solution to our problem of building an adjustable resistance
for a precision instrument!
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—|| +4]
—/ | 4]
Lessons In Electric Circuits
-- Volume |
Chapter 9
ELECTRICAL
INSTRUMENTATION
SIGNALS
Analog_and digital signals
Voltage signal systems
Current signal systems
Tachogenerators
Thermocouples
pDH measurement
Strain gauges
Contributors
Analog and digital signals
Instrumentation is a field of study and work centering on
measurement and control of physical processes. These
physical processes include pressure, temperature, flow rate,
and chemical consistency. An instrument is a device that
measures and/or acts to control any kind of physical process.
Due to the fact that electrical quantities of voltage and
Current are easy to measure, manipulate, and transmit over
long distances, they are widely used to represent such
physical variables and transmit the information to remote
locations.
A signal is any kind of physical quantity that conveys
information. Audible speech is certainly a kind of signal, as it
conveys the thoughts (information) of one person to another
through the physical medium of sound. Hand gestures are
signals, too, conveying information by means of light. This
text is another kind of signal, interpreted by your English-
trained mind as information about electric circuits. In this
chapter, the word signa! will be used primarily in reference
to an electrical quantity of voltage or current that is used to
represent or signify some other physical quantity.
An analog signal is a kind of signal that is continuously
variable, as opposed to having a limited number of steps
along its range (called digita/). A well-known example of
analog vs. digital is that of clocks: analog being the type
with pointers that slowly rotate around a circular scale, and
digital being the type with decimal number displays or a
"second-hand" that jerks rather than smoothly rotates. The
analog clock has no physical limit to how finely it can
display the time, as its "hands" move in a smooth, pauseless
fashion. The digital clock, on the other hand, cannot convey
any unit of time smaller than what its display will allow for.
The type of clock with a "second-hand" that jerks in 1-
second intervals is a digital device with a minimum
resolution of one second.
Both analog and digital signals find application in modern
electronics, and the distinctions between these two basic
forms of information is something to be covered in much
greater detail later in this book. For now, | will limit the
scope of this discussion to analog signals, since the systems
using them tend to be of simpler design.
With many physical quantities, especially electrical, analog
variability is easy to come by. If such a physical quantity is
used as a Signal medium, it will be able to represent
variations of information with almost unlimited resolution.
In the early days of industrial instrumentation, compressed
air was used as a Signaling medium to convey information
from measuring instruments to indicating and controlling
devices located remotely. The amount of air pressure
corresponded to the magnitude of whatever variable was
being measured. Clean, dry air at approximately 20 pounds
per square inch (PSI) was supplied from an air compressor
through tubing to the measuring instrument and was then
regulated by that instrument according to the quantity
being measured to produce a corresponding output signal.
For example, a pneumatic (air signal) level "transmitter"
device set up to measure height of water (the "process
variable") in a storage tank would output a low air pressure
when the tank was empty, a medium pressure when the
tank was partially full, and a high pressure when the tank
was completely full.
Storage tank
pipe or tube
—+— air flow
20 PSI pret lala
air supply
analog, air posers
Signa
f
pipe or tube
water "level transmitter"
water "level indicator"
(LT)
(Ll)
The "water level indicator" (LI) is nothing more than a
pressure gauge measuring the air pressure in the pneumatic
signal line. This air pressure, being a signal, is inturna
representation of the water level in the tank. Any variation
of level in the tank can be represented by an appropriate
variation in the pressure of the pneumatic signal. Aside from
certain practical limits imposed by the mechanics of air
pressure devices, this pneumatic signal is infinitely variable,
able to represent any degree of change in the water's level,
and is therefore analog in the truest sense of the word.
Crude as it may appear, this kind of pneumatic signaling
system formed the backbone of many industrial
measurement and control systems around the world, and
still sees use today due to its simplicity, safety, and
reliability. Air pressure signals are easily transmitted through
inexpensive tubes, easily measured (with mechanical
pressure gauges), and are easily manipulated by mechanical
devices using bellows, diaphragms, valves, and other
pneumatic devices. Air pressure signals are not only useful
for measuring physical processes, but for controlling them as
well. With a large enough piston or diaphragm, a small air
pressure signal can be used to generate a large mechanical
force, which can be used to move a valve or other
controlling device. Complete automatic control systems
have been made using air pressure as the signal medium.
They are simple, reliable, and relatively easy to understand.
However, the practical limits for air pressure signal accuracy
can be too limiting in some cases, especially when the
compressed air is not clean and dry, and when the
possibility for tubing leaks exist.
With the advent of solid-state electronic amplifiers and other
technological advances, electrical quantities of voltage and
current became practical for use as analog instrument
signaling media. Instead of using pneumatic pressure
signals to relay information about the fullness of a water
storage tank, electrical signals could relay that same
information over thin wires (instead of tubing) and not
require the support of such expensive equipment as air
compressors to operate:
Storage tank
water "level transmitter"
(LT)
analog electric
current signal
————
water "level indicator"
(Ll)
—_——
Analog electronic signals are still the primary kinds of
signals used in the instrumentation world today (January of
2001), but it is giving way to digital modes of
communication in many applications (more on that subject
later). Despite changes in technology, it is always good to
have a thorough understanding of fundamental principles,
so the following information will never really become
obsolete.
One important concept applied in many analog
instrumentation signal systems is that of "live zero," a
standard way of scaling a signal so that an indication of 0
percent can be discriminated from the status of a "dead"
system. Take the pneumatic signal system as an example: if
the signal pressure range for transmitter and indicator was
designed to be 0 to 12 PSI, with 0 PSI representing 0 percent
of process measurement and 12 PSI representing 100
percent, a received signal of 0 percent could be a legitimate
reading of 0 percent measurement or it could mean that the
system was malfunctioning (air compressor stopped, tubing
broken, transmitter malfunctioning, etc.). With the 0 percent
point represented by O PSI, there would be no easy way to
distinguish one from the other.
If, however, we were to scale the instruments (transmitter
and indicator) to use a scale of 3 to 15 PSI, with 3 PSI
representing O percent and 15 PSI representing 100 percent,
any kind of a malfunction resulting in zero air pressure at
the indicator would generate a reading of -25 percent (0
PSI), which is clearly a faulty value. The person looking at
the indicator would then be able to immediately tell that
something was wrong.
Not all signal standards have been set up with live zero
baselines, but the more robust signals standards (3-15 PSI,
4-20 mA) have, and for good reason.
e REVIEW:
e A signal is any kind of detectable quantity used to
communicate information.
e An analog signal is a signal that can be continuously, or
infinitely, varied to represent any small amount of
change.
e Pneumatic, or air pressure, signals used to be used
predominately in industrial instrumentation signal
systems. This has been largely superseded by analog
electrical signals such as voltage and current.
e A live zero refers to an analog signal scale using a non-
zero quantity to represent 0 percent of real-world
measurement, so that any system malfunction resulting
in a natural "rest" state of zero signal pressure, voltage,
or current can be immediately recognized.
Voltage signal systems
The use of variable voltage for instrumentation signals
seems a rather obvious option to explore. Let's see how a
voltage signal instrument might be used to measure and
relay information about water tank level:
Level transmitter
Level indicator
potentiometer
moved by float
two-conductor cable
float
The "transmitter" in this diagram contains its own precision
regulated source of voltage, and the potentiometer setting is
varied by the motion of a float inside the water tank
following the water level. The "indicator" is nothing more
than a voltmeter with a scale calibrated to read in some unit
height of water (inches, feet, meters) instead of volts.
As the water tank level changes, the float will move. As the
float moves, the potentiometer wiper will correspondingly be
moved, dividing a different proportion of the battery voltage
to go across the two-conductor cable and on to the level
indicator. As a result, the voltage received by the indicator
will be representative of the level of water in the storage
tank.
This elementary transmitter/indicator system is reliable and
easy to understand, but it has its limitations. Perhaps
greatest is the fact that the system accuracy can be
influenced by excessive cable resistance. Remember that
real voltmeters draw small amounts of current, even though
it is ideal for a voltmeter not to draw any current at all. This
being the case, especially for the kind of heavy, rugged
analog meter movement likely used for an industrial-quality
system, there will be a small amount of current through the
2-conductor cable wires. The cable, having a small amount
of resistance along its length, will consequently drop a small
amount of voltage, leaving less voltage across the
indicator's leads than what is across the leads of the
transmitter. This loss of voltage, however small, constitutes
an error in measurement:
Level transmitter
Level indicator
potentiometer
moved by float
voltage drop
4
+
+
C}—- Voltage drop
float Due to voltage drops along
cable conductors, there will be
slightly less voltage across the
indicator (meter) than there is
at the output of the transmitter.
Resistor symbols have been added to the wires of the cable
to show what is happening in a real system. Bear in mind
that these resistances can be minimized with heavy-gauge
wire (at additional expense) and/or their effects mitigated
through the use of a high-resistance (null-balance?)
voltmeter for an indicator (at additional complexity).
Despite this inherent disadvantage, voltage signals are still
used in many applications because of their extreme design
simplicity. One common signal standard is 0-10 volts,
meaning that a signal of 0 volts represents O percent of
measurement, 10 volts represents 100 percent of
measurement, 5 volts represents 50 percent of
measurement, and so on. Instruments designed to output
and/or accept this standard signal range are available for
purchase from major manufacturers. A more common
voltage range is 1-5 volts, which makes use of the "live zero"
concept for circuit fault indication.
e REVIEW:
e DC voltage can be used as an analog signal to relay
information from one location to another.
e A major disadvantage of voltage signaling is the
possibility that the voltage at the indicator (voltmeter)
will be less than the voltage at the signal source, due to
line resistance and indicator current draw. This drop in
voltage along the conductor length constitutes a
measurement error from transmitter to indicator.
Current signal systems
It is possible through the use of electronic amplifiers to
design a circuit outputting a constant amount of current
rather than a constant amount of voltage. This collection of
components is collectively Known as a current source, and
its symbol looks like this:
o current source
+
A current source generates as much or as little voltage as
needed across its leads to produce a constant amount of
current through it. This is just the opposite of a voltage
source (an ideal battery), which will output as much or as
little current as demanded by the external circuit in
maintaining its output voltage constant. Following the
"conventional flow" symbology typical of electronic devices,
the arrow points against the direction of electron motion.
Apologies for this confusing notation: another legacy of
Benjamin Franklin's false assumption of electron flow!
electron flow
——.
(+) current source
—<_—
electron flow
Current in this circuit remains
constant, regardless of circuit
resistance. Only voltage will
change!
Current sources can be built as variable devices, just like
voltage sources, and they can be designed to produce very
precise amounts of current. If a transmitter device were to
be constructed with a variable current source instead of a
variable voltage source, we could design an instrumentation
signal system based on current instead of voltage:
Level transmitter
Level indicator
voltage drop
+
fy
float position changes [voltage drop] ;
voltage drop} Being a simple series
output of current source circuit, current is equal
at all points, regardless
C}-—- of any voltage drops!
float
The internal workings of the transmitter's current source
need not be a concern at this point, only the fact that its
output varies in response to changes in the float position,
just like the potentiometer setup in the voltage signal
system varied voltage output according to float position.
Notice now how the indicator is an ammeter rather than a
voltmeter (the scale calibrated in inches, feet, or meters of
water in the tank, as always). Because the circuit is a series
configuration (accounting for the cable resistances), current
will be precisely equa! through all components. With or
without cable resistance, the current at the indicator is
exactly the same as the current at the transmitter, and
therefore there is no error incurred as there might be with a
voltage signal system. This assurance of zero signal
degradation is a decided advantage of current signal
systems over voltage signal systems.
The most common current signal standard in modern use is
the 4 to 20 milliamp (4-20 mA) loop, with 4 milliamps
representing 0 percent of measurement, 20 milliamps
representing 100 percent, 12 milliamps representing 50
percent, and so on. A convenient feature of the 4-20 mA
standard is its ease of signal conversion to 1-5 volt
indicating instruments. A simple 250 ohm precision resistor
connected in series with the circuit will produce 1 volt of
drop at 4 milliamps, 5 volts of drop at 20 milliamps, etc:
Indicator (1-5 V instrument)
4 -20 mA current signal
—_—>- —_—> —_—
Transmitter Indicator
(4-20 mA instrument)
| Percent of | 4-20 mA_ | 1-5 V |
| measurement | Signal | Signal |
, 4 a | 4.0m | lov |
| 1 | 56m | 1.4Vv |
[- 20°. - f° Bam aveey. 3]
| 2 | 80m | 2.0V |
| 30 | 88m | 2.2Vv |
| 40 | 10.4mA | 2.6V |
| 50 | 120m | 3.0V |
| 68 | 136m | 3.4V |
| 70 | 152m | 3.8V |
The current loop scale of 4-20 milliamps has not always
been the standard for current instruments: for a while there
was also a 10-50 milliamp standard, but that standard has
since been obsoleted. One reason for the eventual
supremacy of the 4-20 milliamp loop was safety: with lower
circuit voltages and lower current levels than in 10-50 mA
system designs, there was less chance for personal shock
injury and/or the generation of sparks capable of igniting
flammable atmospheres in certain industrial environments.
e REVIEW:
e A current source is a device (usually constructed of
several electronic components) that outputs a constant
amount of current through a circuit, much like a voltage
source (ideal battery) outputting a constant amount of
voltage to a circuit.
A current "loop" instrumentation circuit relies on the
series circuit principle of current being equal through all
components to insure no signal error due to wiring
resistance.
The most common analog current signal standard in
modern use is the "4 to 20 milliamp current loop."
Tachogenerators
An electromechanical generator is a device capable of
producing electrical power from mechanical energy, usually
the turning of a shaft. When not connected to a load
resistance, generators will generate voltage roughly
proportional to shaft speed. With precise construction and
design, generators can be built to produce very precise
voltages for certain ranges of shaft speeds, thus making
them well-suited as measurement devices for shaft speed in
mechanical equipment. A generator specially designed and
constructed for this use is called a tachometer or
tachogenerator. Often, the word "tach" (pronounced "tack")
is used rather than the whole word.
Tachogenerator
voltmeter with shaft
scale calibrated
in RPM (Revolution
Per Minute)
By measuring the voltage produced by a tachogenerator,
you can easily determine the rotational soeed of whatever
its mechanically attached to. One of the more common
voltage signal ranges used with tachogenerators is 0 to 10
volts. Obviously, since a tachogenerator cannot produce
voltage when its not turning, the zero cannot be "live" in
this signal standard. Tachogenerators can be purchased with
different "full-scale" (10 volt) speeds for different
applications. Although a voltage divider could theoretically
be used with a tachogenerator to extend the measurable
speed range in the 0-10 volt scale, it is not advisable to
significantly overspeed a precision instrument like this, or its
life will be shortened.
Tachogenerators can also indicate the direction of rotation
by the polarity of the output voltage. When a permanent-
magnet style DC generator's rotational direction is reversed,
the polarity of its output voltage will switch. In measurement
and control systems where directional indication is needed,
tachogenerators provide an easy way to determine that.
Tachogenerators are frequently used to measure the speeds
of electric motors, engines, and the equipment they power:
conveyor belts, machine tools, mixers, fans, etc.
Thermocouples
An interesting phenomenon applied in the field of
instrumentation is the Seebeck effect, which is the
production of a small voltage across the length of a wire due
to a difference in temperature along that wire. This effect is
most easily observed and applied with a junction of two
dissimilar metals in contact, each metal producing a
different Seebeck voltage along its length, which translates
to a voltage between the two (unjoined) wire ends. Most any
pair of dissimilar metals will produce a measurable voltage
when their junction is heated, some combinations of metals
producing more voltage per degree of temperature than
others:
Seebeck voltage
unction
theated)
iron wire —
4 +—| small voltage between wires:
more voltage produced as
——- ~«—| junction temperature increases.
copper wire
The Seebeck effect is fairly linear; that is, the voltage
produced by a heated junction of two wires is directly
proportional to the temperature. This means that the
temperature of the metal wire junction can be determined
by measuring the voltage produced. Thus, the Seebeck
effect provides for us an electric method of temperature
measurement.
Seebeck voltage
When a pair of dissimilar metals are joined together for the
purpose of measuring temperature, the device formed is
called a thermocouple. Thermocouples made for
instrumentation use metals of high purity for an accurate
temperature/voltage relationship (as linear and as
predictable as possible).
Seebeck voltages are quite small, in the tens of millivolts for
most temperature ranges. This makes them somewhat
difficult to measure accurately. Also, the fact that any
junction between dissimilar metals will produce
temperature-dependent voltage creates a problem when we
try to connect the thermocouple to a voltmeter, completing
a circuit:
a second iron/copper
junction is formed!
‘ iron wire + | - copper wire 4
junction
The second iron/copper junction formed by the connection
between the thermocouple and the meter on the top wire
will produce a temperature-dependent voltage opposed in
polarity to the voltage produced at the measurement
junction. This means that the voltage between the
voltmeter's copper leads will be a function of the difference
in temperature between the two junctions, and not the
temperature at the measurement junction alone. Even for
thermocouple types where copper is not one of the
dissimilar metals, the combination of the two metals joining
the copper leads of the measuring instrument forms a
junction equivalent to the measurement junction:
These two junctions in series form
the equivalent of a single iron/constantan
junction in opposition to the measurement
junction on the lett.
iron/copper
4 iron wire copper wire ¥
measurement
junction
constantan wire copper wire -
constantan/copper
This second junction is called the reference or co/d junction,
to distinguish it from the junction at the measuring end, and
there is no way to avoid having one in a thermocouple
circuit. In some applications, a differential temperature
measurement between two points is required, and this
inherent property of thermocouples can be exploited to
make a very simple measurement system.
iron wire iron wire
junction + + junction
es
copper wire copper wire
However, in most applications the intent is to measure
temperature at a single point only, and in these cases the
second junction becomes a liability to function.
Compensation for the voltage generated by the reference
junction is typically performed by a special circuit designed
to measure temperature there and produce a corresponding
voltage to counter the reference junction's effects. At this
point you may wonder, "If we have to resort to some other
form of temperature measurement just to overcome an
idiosyncrasy with thermocouples, then why bother using
thermocouples to measure temperature at all? Why not just
use this other form of temperature measurement, whatever
it may be, to do the job?" The answer is this: because the
other forms of temperature measurement used for reference
junction compensation are not as robust or versatile as a
thermocouple junction, but do the job of measuring room
temperature at the reference junction site quite well. For
example, the thermocouple measurement junction may be
inserted into the 1800 degree (F) flue of a foundry holding
furnace, while the reference junction sits a hundred feet
away in a metal cabinet at ambient temperature, having its
temperature measured by a device that could never survive
the heat or corrosive atmosphere of the furnace.
The voltage produced by thermocouple junctions is strictly
dependent upon temperature. Any current in a
thermocouple circuit is a function of circuit resistance in
opposition to this voltage (I=E/R). In other words, the
relationship between temperature and Seebeck voltage is
fixed, while the relationship between temperature and
current is variable, depending on the total resistance of the
circuit. With heavy enough thermocouple conductors,
currents upwards of hundreds of amps can be generated
from a single pair of thermocouple junctions! (I've actually
seen this in a laboratory experiment, using heavy bars of
copper and copper/nickel alloy to form the junctions and the
circuit conductors.)
For measurement purposes, the voltmeter used ina
thermocouple circuit is designed to have a very high
resistance so as to avoid any error-inducing voltage drops
along the thermocouple wire. The problem of voltage drop
along the conductor length is even more severe here than
with the DC voltage signals discussed earlier, because here
we only have a few millivolts of voltage produced by the
junction. We simply cannot afford to have even a single
millivolt of drop along the conductor lengths without
incurring serious temperature measurement errors.
Ideally, then, current in a thermocouple circuit is zero. Early
thermocouple indicating instruments made use of null-
balance potentiometric voltage measurement circuitry to
measure the junction voltage. The early Leeds & Northrup
"Speedomax" line of temperature indicator/recorders were a
good example of this technology. More modern instruments
use semiconductor amplifier circuits to allow the
thermocouple's voltage signal to drive an indication device
with little or no current drawn in the circuit.
Thermocouples, however, can be built from heavy-gauge
wire for low resistance, and connected in such a way so as to
generate very high currents for purposes other than
temperature measurement. One such purpose is electric
power generation. By connecting many thermocouples in
series, alternating hot/cold temperatures with each junction,
a device called a thermopile can be constructed to produce
substantial amounts of voltage and current:
~«<—_—_ output voltage _____,
copper wire
iron wire
copper wire
iron wire
copper wire "Thermopile”
iron wire
copper wire
iron wire
copper wire
iron wire
copper wire
With the left and right sets of junctions at the same
temperature, the voltage at each junction will be equal and
the opposing polarities would cancel to a final voltage of
zero. However, if the left set of junctions were heated and
the right set cooled, the voltage at each left junction would
be greater than each right junction, resulting in a total
output voltage equal to the sum of all junction pair
differentials. In a thermopile, this is exactly how things are
set up. A source of heat (combustion, strong radioactive
substance, solar heat, etc.) is applied to one set of junctions,
while the other set is bonded to a heat sink of some sort (air-
or water-cooled). Interestingly enough, as electrons flow
through an external load circuit connected to the
thermopile, heat energy is transferred from the hot junctions
to the cold junctions, demonstrating another thermo-electric
phenomenon: the so-called Pe/tier Effect (electric current
transferring heat energy).
Another application for thermocouples is in the
measurement of average temperature between several
locations. The easiest way to do this is to connect several
thermocouples in parallel with each other. The millivolt
signal produced by each thermocouple will average out at
the parallel junction point. The voltage differences between
the junctions drop along the resistances of the thermocouple
wires:
iron wire copper wire
junction
#1 constantan wire copper wire
iron wire
junction
#2 constantan wire : ;
~— reference junctions
iron wire
junction
#3 constantan wire
iron wire
junction
#4 7 constantan wire
Unfortunately, though, the accurate averaging of these
Seebeck voltage potentials relies on each thermocouple's
wire resistances being equal. If the thermocouples are
located at different places and their wires join in parallel ata
single location, equal wire length will be unlikely. The
thermocouple having the greatest wire length from point of
measurement to parallel connection point will tend to have
the greatest resistance, and will therefore have the least
effect on the average voltage produced.
To help compensate for this, additional resistance can be
added to each of the parallel thermocouple circuit branches
to make their respective resistances more equal. Without
custom-sizing resistors for each branch (to make resistances
precisely equal between all the thermocouples), it is
acceptable to simply install resistors with equal values,
significantly higher than the thermocouple wires'
resistances so that those wire resistances will have a much
smaller impact on the total branch resistance. These
resistors are called swamping resistors, because their
relatively high values overshadow or "swamp" the
resistances of the thermocouple wires themselves:
iron wire Revamp copper wire
+
junction
#1 constantan wire copper wire
iron wire
junction
#9 . The meter will register
constantan wire a more realistic average
; of all junction temperatures
Iron WIre with the 'swamping"
junction resistors in place.
#3 constantan wire
iron wire
junction
#4
constantan wire
Because thermocouple junctions produce such low voltages,
it is imperative that wire connections be very clean and tight
for accurate and reliable operation. Also, the location of the
reference junction (the place where the dissimilar-metal
thermocouple wires join to standard copper) must be kept
close to the measuring instrument, to ensure that the
instrument can accurately compensate for reference
junction temperature. Despite these seemingly restrictive
requirements, thermocouples remain one of the most robust
and popular methods of industrial temperature
measurement in modern use.
e REVIEW:
e The Seebeck Effect is the production of a voltage
between two dissimilar, joined metals that is
proportional to the temperature of that junction.
e In any thermocouple circuit, there are two equivalent
junctions formed between dissimilar metals. The
junction placed at the site of intended measurement is
called the measurement junction, while the other (single
or equivalent) junction is called the reference junction.
Two thermocouple junctions can be connected in
opposition to each other to generate a voltage signal
proportional to differential temperature between the two
junctions. A collection of junctions so connected for the
purpose of generating electricity is called a thermopile.
When electrons flow through the junctions of a
thermopile, heat energy is transferred from one set of
junctions to the other. This is known as the Peltier Effect.
Multiple thermocouple junctions can be connected in
parallel with each other to generate a voltage signal
representing the average temperature between the
junctions. "Swamping" resistors may be connected in
series with each thermocouple to help maintain equality
between the junctions, so the resultant voltage will be
more representative of a true average temperature.
It is imperative that current in a thermocouple circuit be
kept as low as possible for good measurement accuracy.
Also, all related wire connections should be clean and
tight. Mere millivolts of drop at any place in the circuit
will cause substantial measurement errors.
pH measurement
A very important measurement in many liquid chemical
processes (industrial, pharmaceutical, manufacturing, food
production, etc.) is that of pH: the measurement of hydrogen
ion concentration in a liquid solution. A solution with a low
DH value is called an "acid," while one with a high pH is
called a "caustic." The common pH scale extends from 0
(strong acid) to 14 (strong caustic), with 7 in the middle
representing pure water (neutral):
The pH scale
012 3 4 5 67 8 9 10 1112 13 14
1 111-14
Acid ~— —~ Caustic
Neutral
DH is defined as follows: the lower-case letter "p" in pH
stands for the negative common (base ten) logarithm, while
the upper-case letter "H" stands for the element hydrogen.
Thus, pH is a logarithmic measurement of the number of
moles of hydrogen ions (H*) per liter of solution.
Incidentally, the "p" prefix is also used with other types of
chemical measurements where a logarithmic scale is
desired, pCO2 (Carbon Dioxide) and pO2 (Oxygen) being
two such examples.
The logarithmic pH scale works like this: a solution with 10°
12 moles of H* ions per liter has a pH of 12; a solution with
10-3 moles of Ht ions per liter has a pH of 3. While very
uncommon, there is such a thing as an acid with a pH
measurement below O and a caustic with a pH above 14.
Such solutions, understandably, are quite concentrated and
extremely reactive.
While pH can be measured by color changes in certain
chemical powders (the "litmus strip" being a familiar
example from high school chemistry classes), continuous
process monitoring and control of pH requires a more
sophisticated approach. The most common approach is the
use of a specially-prepared electrode designed to allow
hydrogen ions in the solution to migrate through a selective
barrier, producing a measurable potential (voltage)
difference proportional to the solution's pH:
Voltage produced between
electrodes is proportional
to the pH of the solution
f
The design and operational theory of pH electrodes is a very
complex subject, explored only briefly here. What is
important to understand is that these two electrodes
generate a voltage directly proportional to the pH of the
solution. At a pH of 7 (neutral), the electrodes will produce 0
volts between them. At a low pH (acid) a voltage will be
developed of one polarity, and at a high pH (caustic) a
voltage will be developed of the opposite polarity.
An unfortunate design constraint of pH electrodes is that
one of them (called the measurement electrode) must be
constructed of special glass to create the ion-selective
barrier needed to screen out hydrogen ions from all the
other ions floating around in the solution. This glass is
chemically doped with lithium ions, which is what makes it
react electrochemically to hydrogen ions. Of course, glass is
not exactly what you would call a "conductor;" rather, it is
an extremely good insulator. This presents a major problem
if our intent is to measure voltage between the two
electrodes. The circuit path from one electrode contact,
through the glass barrier, through the solution, to the other
electrode, and back through the other electrode's contact, is
one of extremely high resistance.
The other electrode (called the reference electrode) is made
from a chemical solution of neutral (7) pH buffer solution
(usually potassium chloride) allowed to exchange ions with
the process solution through a porous separator, forming a
relatively low resistance connection to the test liquid. At
first, one might be inclined to ask: why not just dip a metal
wire into the solution to get an electrical connection to the
liquid? The reason this will not work is because metals tend
to be highly reactive in ionic solutions and can produce a
significant voltage across the interface of metal-to-liquid
contact. The use of a wet chemical interface with the
measured solution is necessary to avoid creating such a
voltage, which of course would be falsely interpreted by any
measuring device as being indicative of pH.
Here is an illustration of the measurement electrode's
construction. Note the thin, lithium-doped glass membrane
across which the pH voltage is generated:
wire connection point
MEASUREMENT
ELECTRODE glass body
bulb filled with
potassium chloride :
buffer" solution +)
very thin glass bulb.
chemically “doped" with
lithium ions so as to react
with hydrogen ions outside
the bulb.
voltage produced
across thickness of
glass membrane
Here is an illustration of the reference electrode's
construction. The porous junction shown at the bottom of
the electrode is where the potassium chloride buffer and
process liquid interface with each other:
wire connection point
REFERENCE
ELECTRODE «glass or plastic body
filled with = —
potassium chloride
"puffer" solution
silver Chloride
tip
porous junction
The measurement electrode's purpose is to generate the
voltage used to measure the solution's pH. This voltage
appears across the thickness of the glass, placing the silver
wire on one side of the voltage and the liquid solution on the
other. The reference electrode's purpose is to provide the
stable, zero-voltage connection to the liquid solution so that
a complete circuit can be made to measure the glass
electrode's voltage. While the reference electrode's
connection to the test liquid may only be a few kilo-ohms,
the glass electrode's resistance may range from ten to nine
hundred mega-ohms, depending on electrode design! Being
that any current in this circuit must travel through both
electrodes' resistances (and the resistance presented by the
test liquid itself), these resistances are in series with each
other and therefore add to make an even greater total.
An ordinary analog or even digital voltmeter has much too
low of an internal resistance to measure voltage in sucha
high-resistance circuit. The equivalent circuit diagram of a
typical pH probe circuit illustrates the problem:
Hh gauirenint electrode
voltage
produced by —
electrodes © ———
+ precision voltmeter
Tatwtanee electrode
=3kOQ
Even avery small circuit current traveling through the high
resistances of each component in the circuit (especially the
measurement electrode's glass membrane), will produce
relatively substantial voltage drops across those resistances,
seriously reducing the voltage seen by the meter. Making
matters worse is the fact that the voltage differential
generated by the measurement electrode is very small, in
the millivolt range (ideally 59.16 millivolts per pH unit at
room temperature). The meter used for this task must be
very sensitive and have an extremely high input resistance.
The most common solution to this measurement problem is
to use an amplified meter with an extremely high internal
resistance to measure the electrode voltage, so as to draw as
little current through the circuit as possible. With modern
semiconductor components, a voltmeter with an input
resistance of up to 10!” QO can be built with little difficulty.
Another approach, seldom seen in contemporary use, is to
use a potentiometric "null-balance" voltage measurement
setup to measure this voltage without drawing any current
from the circuit under test. If a technician desired to check
the voltage output between a pair of pH electrodes, this
would probably be the most practical means of doing so
using only standard benchtop metering equipment:
eas ment electrode
=400 MQ
oA by pees )
electrodes voltage
R source
reference electrode
=3kQ
As uSual, the precision voltage supply would be adjusted by
the technician until the null detector registered zero, then
the voltmeter connected in parallel with the supply would
be viewed to obtain a voltage reading. With the detector
"nulled" (registering exactly zero), there should be zero
Current in the pH electrode circuit, and therefore no voltage
dropped across the resistances of either electrode, giving
the real electrode voltage at the voltmeter terminals.
Wiring requirements for pH electrodes tend to be even more
severe than thermocouple wiring, demanding very clean
connections and short distances of wire (10 yards or less,
even with gold-plated contacts and shielded cable) for
accurate and reliable measurement. As with thermocouples,
however, the disadvantages of electrode pH measurement
are offset by the advantages: good accuracy and relative
technical simplicity.
Few instrumentation technologies inspire the awe and
mystique commanded by pH measurement, because it is so
widely misunderstood and difficult to troubleshoot. Without
elaborating on the exact chemistry of pH measurement, a
few words of wisdom can be given here about pH
measurement systems:
All pH electrodes have a finite life, and that lifespan
depends greatly on the type and severity of service. In
some applications, a pH electrode life of one month may
be considered long, and in other applications the same
electrode(s) may be expected to last for over a year.
Because the glass (measurement) electrode is
responsible for generating the pH-proportional voltage,
it is the one to be considered suspect if the
measurement system fails to generate sufficient voltage
change for a given change in PH (approximately 59
millivolts per pH unit), or fails to respond quickly enough
to a fast change in test liquid pH.
If a pH measurement system "drifts," creating offset
errors, the problem likely lies with the reference
electrode, which is supposed to provide a zero-voltage
connection with the measured solution.
Because pH measurement is a logarithmic
representation of ion concentration, there is an
incredible range of process conditions represented in the
seemingly simple 0-14 pH scale. Also, due to the
nonlinear nature of the logarithmic scale, a change of 1
pH at the top end (say, from 12 to 13 pH) does not
represent the same quantity of chemical activity change
as a change of 1 pH at the bottom end (say, from 2 to 3
pH). Control system engineers and technicians must be
aware of this dynamic if there is to be any hope of
controlling process pH at a stable value.
e The following conditions are hazardous to measurement
(glass) electrodes: high temperatures, extreme pH levels
(either acidic or alkaline), high ionic concentration in the
liquid, abrasion, hydrofluoric acid in the liquid (HF acid
dissolves glass!), and any kind of material coating on
the surface of the glass.
e Temperature changes in the measured liquid affect both
the response of the measurement electrode to a given
DH level (ideally at 59 mV per pH unit), and the actual
DH of the liquid. Temperature measurement devices can
be inserted into the liquid, and the signals from those
devices used to compensate for the effect of
temperature on pH measurement, but this will only
compensate for the measurement electrode's mV/pH
response, not the actual pH change of the process
liquid!
Advances are still being made in the field of pH
measurement, some of which hold great promise for
overcoming traditional limitations of pH electrodes. One
such technology uses a device called a field-effect transistor
to electrostatically measure the voltage produced by an ion-
permeable membrane rather than measure the voltage with
an actual voltmeter circuit. While this technology harbors
limitations of its own, it is at least a pioneering concept, and
may prove more practical at a later date.
REVIEW:
pH is a representation of hydrogen ion activity ina
liquid. It is the negative logarithm of the amount of
hydrogen ions (in moles) per liter of liquid. Thus: 10-14
moles of hydrogen ions in 1 liter of liquid = 11 pH. 10°>-3
moles of hydrogen ions in 1 liter of liquid = 5.3 pH.
The basic pH scale extends from O (strong acid) to 7
(neutral, pure water) to 14 (strong caustic). Chemical
solutions with pH levels below zero and above 14 are
possible, but rare.
pH can be measured by measuring the voltage produced
between two special electrodes immersed in the liquid
solution.
One electrode, made of a special glass, is called the
measurement electrode. It's job it to generate a small
voltage proportional to pH (ideally 59.16 mV per pH
unit).
e The other electrode (called the reference electrode) uses
a porous junction between the measured liquid and a
stable, neutral pH buffer solution (usually potassium
chloride) to create a zero-voltage electrical connection
to the liquid. This provides a point of continuity fora
complete circuit so that the voltage produced across the
thickness of the glass in the measurement electrode can
be measured by an external voltmeter.
e The extremely high resistance of the measurement
electrode's glass membrane mandates the use of a
voltmeter with extremely high internal resistance, ora
null-balance voltmeter, to measure the voltage.
Strain gauges
If a strip of conductive metal is stretched, it will become
Sskinnier and longer, both changes resulting in an increase of
electrical resistance end-to-end. Conversely, if a strip of
conductive metal is placed under compressive force (without
buckling), it will broaden and shorten. If these stresses are
kept within the elastic limit of the metal strip (so that the
strip does not permanently deform), the strip can be used as
a measuring element for physical force, the amount of
applied force inferred from measuring its resistance.
Such a device is called a strain gauge. Strain gauges are
frequently used in mechanical engineering research and
development to measure the stresses generated by
machinery. Aircraft component testing is one area of
application, tiny strain-gauge strips glued to structural
members, linkages, and any other critical component of an
airframe to measure stress. Most strain gauges are smaller
than a postage stamp, and they look something like this:
Tension causes
resistance increase Bonded strain gauge
Gauge insensitive “~— Resistance measured
to lateral forces between these points
Pel
Compression causes
resistance decrease
A strain gauge's conductors are very thin: if made of round
wire, about 1/1000 inch in diameter. Alternatively, strain
gauge conductors may be thin strips of metallic film
deposited on a nonconducting substrate material called the
carrier. The latter form of strain gauge is represented in the
previous illustration. The name "bonded gauge" is given to
strain gauges that are glued to a larger structure under
stress (called the test soecimen). The task of bonding strain
gauges to test specimens may appear to be very simple, but
it is not. "Gauging" is a craft in its own right, absolutely
essential for obtaining accurate, stable strain
measurements. It is also possible to use an unmounted
gauge wire stretched between two mechanical points to
measure tension, but this technique has its limitations.
Typical strain gauge resistances range from 30 Q to 3 kO
(unstressed). This resistance may change only a fraction of a
percent for the full force range of the gauge, given the
limitations imposed by the elastic limits of the gauge
material and of the test specimen. Forces great enough to
induce greater resistance changes would permanently
deform the test specimen and/or the gauge conductors
themselves, thus ruining the gauge as a measurement
device. Thus, in order to use the strain gauge as a practical
instrument, we must measure extremely small changes in
resistance with high accuracy.
Such demanding precision calls for a bridge measurement
circuit. Unlike the Wheatstone bridge shown in the last
chapter using a null-balance detector and a human operator
to maintain a state of balance, a strain gauge bridge circuit
indicates measured strain by the degree of imbalance, and
uses a precision voltmeter in the center of the bridge to
provide an accurate measurement of that imbalance:
Quarter-bridge strain gauge circuit
strain gauge
Typically, the rheostat arm of the bridge (R> in the diagram)
is set at a value equal to the strain gauge resistance with no
force applied. The two ratio arms of the bridge (R, and R3)
are set equal to each other. Thus, with no force applied to
the strain gauge, the bridge will be symmetrically balanced
and the voltmeter will indicate zero volts, representing zero
force on the strain gauge. As the strain gauge is either
compressed or tensed, its resistance will decrease or
increase, respectively, thus unbalancing the bridge and
producing an indication at the voltmeter. This arrangement,
with a single element of the bridge changing resistance in
response to the measured variable (mechanical force), is
known as a guarter-bridge circuit.
As the distance between the strain gauge and the three
other resistances in the bridge circuit may be substantial,
wire resistance has a significant impact on the operation of
the circuit. To illustrate the effects of wire resistance, I'll
show the same schematic diagram, but add two resistor
symbols in series with the strain gauge to represent the
wires:
The strain gauge's resistance (Rgauge) is not the only
resistance being measured: the wire resistances Ryj-e; and
Rwire2, being in series with Rgauge, also contribute to the
resistance of the lower half of the rheostat arm of the bridge,
and consequently contribute to the voltmeter's indication.
This, of course, will be falsely interpreted by the meter as
physical strain on the gauge.
While this effect cannot be completely eliminated in this
configuration, it can be minimized with the addition of a
third wire, connecting the right side of the voltmeter directly
to the upper wire of the strain gauge:
Three-wire, quarter-bridge
strain gauge circuit
Ry rel
Because the third wire carries practically no current (due to
the voltmeter's extremely high internal resistance), its
resistance will not drop any substantial amount of voltage.
Notice how the resistance of the top wire (Ryj,e,) has been
"bypassed" now that the voltmeter connects directly to the
top terminal of the strain gauge, leaving only the lower
wire's resistance (Rwire2) to contribute any stray resistance in
series with the gauge. Not a perfect solution, of course, but
twice as good as the last circuit!
There is a way, however, to reduce wire resistance error far
beyond the method just described, and also help mitigate
another kind of measurement error due to temperature. An
unfortunate characteristic of strain gauges is that of
resistance change with changes in temperature. This is a
property common to all conductors, some more than others.
Thus, our quarter-bridge circuit as shown (either with two or
with three wires connecting the gauge to the bridge) works
as a thermometer just as well as it does a strain indicator. If
all we want to do is measure strain, this is not good. We can
transcend this problem, however, by using a "dummy" strain
gauge in place of R5, so that both elements of the rheostat
arm will change resistance in the same proportion when
temperature changes, thus canceling the effects of
temperature change:
Quarter-bridge strain gauge circuit
with temperature compensation
strain gauge
(unstressed)
strain gauge
(stressed)
Resistors R; and R3 are of equal resistance value, and the
strain gauges are identical to one another. With no applied
force, the bridge should be in a perfectly balanced condition
and the voltmeter should register 0 volts. Both gauges are
bonded to the same test specimen, but only one is placed in
a position and orientation so as to be exposed to physical
strain (the active gauge). The other gauge is isolated from
all mechanical stress, and acts merely as a temperature
compensation device (the "dummy" gauge). If the
temperature changes, both gauge resistances will change by
the same percentage, and the bridge's state of balance will
remain unaffected. Only a differential resistance (difference
of resistance between the two strain gauges) produced by
physical force on the test specimen can alter the balance of
the bridge.
Wire resistance doesn't impact the accuracy of the circuit as
much as before, because the wires connecting both strain
gauges to the bridge are approximately equal length.
Therefore, the upper and lower sections of the bridge's
rheostat arm contain approximately the same amount of
stray resistance, and their effects tend to cancel:
strain gauge
(unstressed)
R
wire 1
strain gauge
(stressed)
Even though there are now two strain gauges in the bridge
circuit, only one is responsive to mechanical strain, and thus
we would still refer to this arrangement as a gquarter-bridge.
However, if we were to take the upper strain gauge and
position it so that it is exposed to the opposite force as the
lower gauge (i.e. when the upper gauge is compressed, the
lower gauge will be stretched, and vice versa), we will have
both gauges responding to strain, and the bridge will be
more responsive to applied force. This utilization is known as
a half-bridge. Since both strain gauges will either increase or
decrease resistance by the same proportion in response to
changes in temperature, the effects of temperature change
remain canceled and the circuit will suffer minimal
temperature-induced measurement error:
Half-bridge strain gauge circuit
strain gauge
(stressed)
strain gauge
(stressed)
An example of how a pair of strain gauges may be bonded to
a test specimen so as to yield this effect is illustrated here:
Strain gauge #1
Strain gauge #2
(-)
Bridge balanced
With no force applied to the test specimen, both strain
gauges have equal resistance and the bridge circuit is
balanced. However, when a downward force is applied to the
free end of the specimen, it will bend downward, stretching
gauge #1 and compressing gauge #2 at the same time:
Strain gauge #1
Strain gauge #2
(-)
Bridge unbalanced
In applications where such complementary pairs of strain
gauges can be bonded to the test specimen, it may be
advantageous to make all four elements of the bridge
"active" for even greater sensitivity. This is called a full-
bridge circuit:
Full-bridge strain gauge circuit
strain gauge strain gauge
(stressed) (
2
=
D
W
ie?)
=
I
strain gauge strain gauge
(stressed) (stressed
—
Both half-bridge and full-bridge configurations grant greater
sensitivity over the quarter-bridge circuit, but often it is not
possible to bond complementary pairs of strain gauges to
the test specimen. Thus, the quarter-bridge circuit is
frequently used in strain measurement systems.
When possible, the full-bridge configuration is the best to
use. This is true not only because it is more sensitive than
the others, but because it is /inear while the others are not.
Quarter-bridge and half-bridge circuits provide an output
(imbalance) signal that is only approximately proportional to
applied strain gauge force. Linearity, or proportionality, of
these bridge circuits is best when the amount of resistance
change due to applied force is very small compared to the
nominal resistance of the gauge(s). With a full-bridge,
however, the output voltage is directly proportional to
applied force, with no approximation (provided that the
change in resistance caused by the applied force is equal for
all four strain gauges!).
Unlike the Wheatstone and Kelvin bridges, which provide
measurement at a condition of perfect balance and therefore
function irrespective of source voltage, the amount of source
(or "excitation") voltage matters in an unbalanced bridge
like this. Therefore, strain gauge bridges are rated in
millivolts of imbalance produced per volt of excitation, per
unit measure of force. A typical example for a strain gauge
of the type used for measuring force in industrial
environments is 15 mV/V at 1000 pounds. That is, at exactly
1000 pounds applied force (either compressive or tensile),
the bridge will be unbalanced by 15 millivolts for every volt
of excitation voltage. Again, such a figure is precise if the
bridge circuit is full-active (four active strain gauges, one in
each arm of the bridge), but only approximate for half-bridge
and quarter-bridge arrangements.
Strain gauges may be purchased as complete units, with
both strain gauge elements and bridge resistors in one
housing, sealed and encapsulated for protection from the
elements, and equipped with mechanical fastening points
for attachment to a machine or structure. Such a package is
typically called a /oad cell.
Like many of the other topics addressed in this chapter,
strain gauge systems can become quite complex, and a full
dissertation on strain gauges would be beyond the scope of
this book.
¢ REVIEW:
e A strain gauge is a thin strip of metal designed to
measure mechanical load by changing resistance when
stressed (stretched or compressed within its elastic
limit).
e Strain gauge resistance changes are typically measured
in a bridge circuit, to allow for precise measurement of
the small resistance changes, and to provide
compensation for resistance variations due to
temperature.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] +4] l—
—/ | 4]
Lessons In Electric Circuits
-- Volume |
Chapter 10
DC NETWORK ANALYSIS
What is network analysis?
Branch current method
Mesh current method
o Mesh Current, conventional method
o Mesh current by inspection
Node voltage method
Introduction to network theorems
Millman's Theorem
Superposition Theorem
Thevenin's Theorem
Norton's Theorem
Thevenin-Norton equivalencies
Millman's Theorem revisited
Maximum Power Transfer Theorem
A-Y and Y-A conversions
Contributors
Bibliography
What is network analysis?
Generally speaking, network analysis is any structured
technique used to mathematically analyze a circuit (a
“network” of interconnected components). Quite often the
technician or engineer will encounter circuits containing
multiple sources of power or component configurations
which defy simplification by series/parallel analysis
techniques. In those cases, he or she will be forced to use
other means. This chapter presents a few techniques useful
in analyzing such complex circuits.
To illustrate how even a simple circuit can defy analysis by
breakdown into series and parallel portions, take start with
this series-parallel circuit:
To analyze the above circuit, one would first find the
equivalent of R> and R3 in parallel, then add R, in series to
arrive at a total resistance. Then, taking the voltage of
battery B, with that total circuit resistance, the total current
could be calculated through the use of Ohm's Law (I=E/R),
then that current figure used to calculate voltage drops in
the circuit. All in all, a fairly simple procedure.
However, the addition of just one more battery could change
all of that:
Resistors R> and R3 are no longer in parallel with each other,
because B, has been inserted into R3's branch of the circuit.
Upon closer inspection, it appears there are no two resistors
in this circuit directly in series or parallel with each other.
This is the crux of our problem: in series-parallel analysis, we
started off by identifying sets of resistors that were directly
in series or parallel with each other, reducing them to single
equivalent resistances. If there are no resistors in a simple
series or parallel configuration with each other, then what
can we do?
It should be clear that this seemingly simple circuit, with
only three resistors, is impossible to reduce as a combination
of simple series and simple parallel sections: it is something
different altogether. However, this is not the only type of
circuit defying series/parallel analysis:
Here we have a bridge circuit, and for the sake of example
we will suppose that it is not balanced (ratio Rj/R, not equal
to ratio R>/R5). If it were balanced, there would be zero
current through R3, and it could be approached as a
series/parallel combination circuit (Rj--Ry // Ro--Rs).
However, any current through R3 makes a series/parallel
analysis impossible. R, is not in series with R, because
there's another path for electrons to flow through R3. Neither
is R> in series with Rs for the same reason. Likewise, Rj is
not in parallel with Rz because R3 is separating their bottom
leads. Neither is Ry in parallel with Rs. Aaarrggghhhh!
Although it might not be apparent at this point, the heart of
the problem is the existence of multiple unknown quantities.
At least in a series/parallel combination circuit, there was a
way to find total resistance and total voltage, leaving total
current as a single unknown value to calculate (and then
that current was used to satisfy previously unknown
variables in the reduction process until the entire circuit
could be analyzed). With these problems, more than one
parameter (variable) is unknown at the most basic level of
circuit simplification.
With the two-battery circuit, there is no way to arrive ata
value for “total resistance,” because there are two sources of
power to provide voltage and current (we would need two
“total” resistances in order to proceed with any Ohm's Law
calculations). With the unbalanced bridge circuit, there is
such a thing as total resistance across the one battery
(paving the way for a calculation of total current), but that
total current immediately splits up into unknown proportions
at each end of the bridge, so no further Ohm's Law
calculations for voltage (E=IR) can be carried out.
So what can we do when we're faced with multiple
unknowns in a circuit? The answer is initially found ina
mathematical process known as simultaneous equations or
systems of equations, whereby multiple unknown variables
are solved by relating them to each other in multiple
equations. In a scenario with only one unknown (such as
every Ohm's Law equation we've dealt with thus far), there
only needs to be a single equation to solve for the single
unknown:
E=1R (Eis unknown; 1 andR are known )
HOF 52
I -—. ( I is unknown; E andR are known )
OF 6s
R=— (R is unknown; E and 1 are known )
However, when we're solving for multiple unknown values,
we need to have the same number of equations as we have
unknowns in order to reach a solution. There are several
methods of solving simultaneous equations, all rather
intimidating and all too complex for explanation in this
chapter. However, many scientific and programmable
calculators are able to solve for simultaneous unknowns, so
it is recommended to use such a calculator when first
learning how to analyze these circuits.
This is not as scary as it may seem at first. Trust me!
Later on we'll see that some clever people have found tricks
to avoid having to use simultaneous equations on these
types of circuits. We call these tricks network theorems, and
we will explore a few later in this chapter.
e REVIEW:
e Some circuit configurations (“networks”) cannot be
solved by reduction according to series/parallel circuit
rules, due to multiple unknown values.
e Mathematical techniques to solve for multiple unknowns
(called “simultaneous equations” or “systems”) can be
applied to basic Laws of circuits to solve networks.
Branch current method
The first and most straightforward network analysis
technique is called the Branch Current Method. |n this
method, we assume directions of currents in a network, then
write equations describing their relationships to each other
through Kirchhoff's and Ohm's Laws. Once we have one
equation for every unknown current, we can solve the
simultaneous equations and determine all currents, and
therefore all voltage drops in the network.
Let's use this circuit to illustrate the method:
The first step is to choose a node (junction of wires) in the
circuit to use as a point of reference for our unknown
currents. I'll choose the node joining the right of R,, the top
of R>, and the left of R3.
chosen node
At this node, guess which directions the three wires' currents
take, labeling the three currents as lj, Ip, and I3, respectively.
Bear in mind that these directions of current are speculative
at this point. Fortunately, if it turns out that any of our
guesses were wrong, we will know when we mathematically
solve for the currents (any “wrong” current directions will
show up as negative numbers in our solution).
Kirchhoff's Current Law (KCL) tells us that the algebraic sum
of currents entering and exiting a node must equal zero, so
we can relate these three currents (1;, lp, and I3) to each
other in a single equation. For the sake of convention, I'll
denote any current entering the node as positive in sign,
and any current exiting the node as negative in sign:
Kirchhoff's Current Law (KCL)
applied to currents at node
-1,+1,-1,=0
The next step is to label all voltage drop polarities across
resistors according to the assumed directions of the
currents. Remember that the “upstream” end of a resistor
will always be negative, and the “downstream” end of a
resistor positive with respect to each other, since electrons
are negatively charged:
The battery polarities, of course, remain as they were
according to their symbology (short end negative, long end
positive). It is OK if the polarity of a resistor's voltage drop
doesn't match with the polarity of the nearest battery, so
long as the resistor voltage polarity is correctly based on the
assumed direction of current through it. In some cases we
may discover that current will be forced backwards through
a battery, causing this very effect. The important thing to
remember here is to base all your resistor polarities and
subsequent calculations on the directions of current(s)
initially assumed. As stated earlier, if your assumption
happens to be incorrect, it will be apparent once the
equations have been solved (by means of a negative
solution). The magnitude of the solution, however, will still
be correct.
Kirchhoff's Voltage Law (KVL) tells us that the algebraic sum
of all voltages in a loop must equal zero, so we can create
more equations with current terms (Ij, l5, and I3) for our
simultaneous equations. To obtain a KVL equation, we must
tally voltage drops in a loop of the circuit, as though we were
measuring with a real voltmeter. I'll choose to trace the left
loop of this circuit first, starting from the upper-left corner
and moving counter-clockwise (the choice of starting points
and directions is arbitrary). The result will look like this:
Voltmeter indicates: -28V
Voltmeter indicates: OV
Voltmeter indicates: a positive voltage
+ Ep>
Voltmeter indicates: a positive voltage
+ Ep>
Having completed our trace of the left loop, we add these
voltage indications together for a sum of zero:
Kirchhoff’s Voltage Law (KVL)
applied to voltage drops in left loop
-28+0+E,,+E,,=0
Of course, we don't yet know what the voltage is across R
or R>, SO we can't insert those values into the equation as
numerical figures at this point. However, we do know that all
three voltages must algebraically add to zero, so the
equation is true. We can go a step further and express the
unknown voltages as the product of the corresponding
unknown currents (Il, and I,) and their respective resistors,
following Ohm's Law (E=IR), as well as eliminate the 0 term:
-28+E,,+E,,=0
Ohm's Law: E=1R
... Substituting IR for E in the KVL equation .. .
-28+1LR,+1,R,=0
Since we know what the values of all the resistors are in
ohms, we can just substitute those figures into the equation
to simplify things a bit:
- 28 +21, +41, =0
You might be wondering why we went through all the trouble
of manipulating this equation from its initial form (-28 + Eps
+ Ep,). After all, the last two terms are still unknown, so
what advantage is there to expressing them in terms of
unknown voltages or as unknown currents (multiplied by
resistances)? The purpose in doing this is to get the KVL
equation expressed using the same unknown variables as
the KCL equation, for this is a necessary requirement for any
simultaneous equation solution method. To solve for three
unknown currents (Ij, lp, and l3), we must have three
equations relating these three currents (not vo/tages!)
together.
Applying the same steps to the right loop of the circuit
(starting at the chosen node and moving counter-clockwise),
we get another KVL equation:
Voltmeter indicates: a negative voltage
=E..
Voltmeter indicates: OV
Voltmeter indicates: +7V
Voltmeter indicates: a negative voltage
Kirchhoff's Voltage Law (KVL)
applied to voltage drops in right loop
-E,, +0+7-E,,=0
Knowing now that the voltage across each resistor can be
and should be expressed as the product of the
corresponding current and the (Known) resistance of each
resistor, we can re-write the equation as such:
-21,+7-11,=0
Now we have a mathematical system of three equations (one
KCL equation and two KVL equations) and three unknowns:
-1,+1-1,=0 Kirchhoff's Current Law
- 28+ 21, + 41,=0 Kirchhoff's Voltage Law
-21,+7- 11,=0 Kirchhoff's Voltage Law
For some methods of solution (especially any method
involving a calculator), it is helpful to express each unknown
term in each equation, with any constant value to the right
of the equal sign, and with any “unity” terms expressed with
an explicit coefficient of 1. Re-writing the equations again,
we have:
- ll,+ 11,- 0,=0 Kirchhoff's Current Law
41, + 21, + O1; = 28 Kirchhoff's Voltage Law
Ol, - 21, - 11, =-7 Kirchhoff's Voltage Law
All three variables represented
in all three equations
Using whatever solution techniques are available to us, we
should arrive at a solution for the three unknown current
values:
Solutions:
1=5A
1L=4A
L=-1A
So, |; is 5 amps, ly is 4 amps, and l3 is a negative 1 amp. But
what does “negative” current mean? In this case, it means
that our assumed direction for lz; was opposite of its rea/
direction. Going back to our original circuit, we can re-draw
the current arrow for l3 (and re-draw the polarity of R3's
voltage drop to match):
Notice how current is being pushed backwards through
battery 2 (electrons flowing “up”) due to the higher voltage
of battery 1 (whose current is pointed “down” as it normally
would)! Despite the fact that battery B's polarity is trying to
push electrons down in that branch of the circuit, electrons
are being forced backwards through it due to the superior
voltage of battery B,. Does this mean that the stronger
battery will always “win” and the weaker battery always get
current forced through it backwards? No! It actually depends
on both the batteries’ relative voltages and the resistor
values in the circuit. The only sure way to determine what's
going on is to take the time to mathematically analyze the
network.
Now that we know the magnitude of all currents in this
circuit, we can calculate voltage drops across all resistors
with Ohm's Law (E=IR):
Ex; =1,R;= (1 A) Q)=1V
Let us now analyze this network using SPICE to verify our
voltage figures.[spi] We could analyze current as well with
SPICE, but since that requires the insertion of extra
components into the circuit, and because we know that if
the voltages are all the same and all the resistances are the
same, the currents must all be the same, I'll opt for the less
complex analysis. Here's a re-drawing of our circuit,
complete with node numbers for SPICE to reference:
network analysis example
vl 1 0
v2 3 0 dc 7
rl 124
r2 20 2
m3. 2.3.1
.dc vl 28 28 1
.print dc v(1,2) v(2,0) v(2,3)
.end
vl v(1,2) v(2) v(2,3)
2.800E+01 2.000E+01 8.000E+00 1.000E+00
Sure enough, the voltage figures all turn out to be the same:
20 volts across R; (nodes 1 and 2), 8 volts across R» (nodes
2 and 0), and 1 volt across R3 (nodes 2 and 3). Take note of
the signs of all these voltage figures: they're all positive
values! SPICE bases its polarities on the order in which
nodes are listed, the first node being positive and the
second node negative. For example, a figure of positive (+)
20 volts between nodes 1 and 2 means that node 1 is
positive with respect to node 2. If the figure had come out
negative in the SPICE analysis, we would have known that
our actual polarity was “backwards” (node 1 negative with
respect to node 2). Checking the node orders in the SPICE
listing, we can see that the polarities all match what we
determined through the Branch Current method of analysis.
e REVIEW:
Steps to follow for the “Branch Current” method of
analysis:
(1) Choose a node and assume directions of currents.
(2) Write a KCL equation relating currents at the node.
(3) Label resistor voltage drop polarities based on
assumed currents.
(4) Write KVL equations for each loop of the circuit,
substituting the product IR for E in each resistor term of
the equations.
(5) Solve for unknown branch currents (simultaneous
equations).
e (6) If any solution is negative, then the assumed
direction of current for that solution is wrong!
e (7) Solve for voltage drops across all resistors (E=IR).
Mesh current method
The Mesh Current Method, also Known as the Loop Current
Method, is quite similar to the Branch Current method in
that it uses simultaneous equations, Kirchhoff's Voltage Law,
and Ohm's Law to determine unknown currents in a network.
It differs from the Branch Current method in that it does not
use Kirchhoff's Current Law, and it is usually able to solve a
circuit with less unknown variables and less simultaneous
equations, which is especially nice if you're forced to solve
without a calculator.
Mesh Current, conventional method
Let's see how this method works on the same example
problem:
The first step in the Mesh Current method is to identify
“loops” within the circuit encompassing all components. In
our example circuit, the loop formed by Bj, Rj, and R> will
be the first while the loop formed by B3, R>, and R3 will be
the second. The strangest part of the Mesh Current method
iS envisioning circulating currents in each of the loops. In
fact, this method gets its name from the idea of these
currents meshing together between loops like sets of
spinning gears:
The choice of each current's direction is entirely arbitrary,
just as in the Branch Current method, but the resulting
equations are easier to solve if the currents are going the
Same direction through intersecting components (note how
currents |, and I5 are both going “up” through resistor R>,
where they “mesh,” or intersect). If the assumed direction of
a mesh current is wrong, the answer for that current will
have a negative value.
The next step is to label all voltage drop polarities across
resistors according to the assumed directions of the mesh
currents. Remember that the “upstream” end of a resistor
will always be negative, and the “downstream” end of a
resistor positive with respect to each other, since electrons
are negatively charged. The battery polarities, of course, are
dictated by their symbol orientations in the diagram, and
may or may not “agree” with the resistor polarities (assumed
current directions):
Using Kirchhoff's Voltage Law, we can now step around each
of these loops, generating equations representative of the
component voltage drops and polarities. As with the Branch
Current method, we will denote a resistor's voltage drop as
the product of the resistance (in ohms) and its respective
mesh current (that quantity being unknown at this point).
Where two currents mesh together, we will write that term in
the equation with resistor current being the sum of the two
meshing currents.
Tracing the left loop of the circuit, starting from the upper-
left corner and moving counter-clockwise (the choice of
starting points and directions is ultimately irrelevant),
counting polarity as if we had a voltmeter in hand, red lead
on the point ahead and black lead on the point behind, we
get this equation:
- 28+ 2(1,+1L)+41,=0
Notice that the middle term of the equation uses the sum of
mesh currents |, and I> as the current through resistor R3>.
This is because mesh currents I, and I, are going the same
direction through R>, and thus complement each other.
Distributing the coefficient of 2 to the |, and I, terms, and
then combining I, terms in the equation, we can simplify as
such:
- 28+ 2(1,+1)+41,=0 Original form of equation
. .. distributing to terms within parentheses .. .
- 28+ 21, +21, + 41,=0
... combining like terms .. .
- 28+ 61, + 21,=0 Simplified form of equation
At this time we have one equation with two unknowns. To be
able to solve for two unknown mesh currents, we must have
two equations. If we trace the other loop of the circuit, we
can obtain another KVL equation and have enough data to
solve for the two currents. Creature of habit that | am, I'll
start at the upper-left hand corner of the right loop and trace
counter-clockwise:
- 2(1,+1)+7- 11,=0
Simplifying the equation as before, we end up with:
- 21, - 31,+7=0
Now, with two equations, we can use one of several methods
to mathematically solve for the unknown currents |, and I>:
- 28 + 61, + 21, =0
- 21, - 31,+7=0
. .. rearranging equations for easier solution. . .
61, + 21, =28
7 Pee | mes
Solutions:
1=5A
L=-1A
Knowing that these solutions are values for mesh currents,
not branch currents, we must go back to our diagram to see
how they fit together to give currents through all
components:
The solution of -1 amp for I, means that our initially
assumed direction of current was incorrect. In actuality, I> is
flowing in a counter-clockwise direction at a value of
(positive) 1 amp:
This change of current direction from what was first assumed
will alter the polarity of the voltage drops across R> and R3
due to current I5. From here, we can say that the current
through R, is 5 amps, with the voltage drop across R, being
the product of current and resistance (E=IR), 20 volts
(positive on the left and negative on the right). Also, we can
safely say that the current through R3 is 1 amp, with a
voltage drop of 1 volt (E=IR), positive on the left and
negative on the right. But what is happening at R>?
Mesh current |, is going “up” through R>, while mesh current
Il, is going “down” through R>. To determine the actual
current through R>, we must see how mesh currents |, and I>
interact (in this case they're in opposition), and algebraically
add them to arrive at a final value. Since I, is going “up” at
5 amps, and I, is going “down” at 1 amp, the rea/ current
through R> must be a value of 4 amps, going “up:”
A current of 4 amps through R,'s resistance of 2 O gives usa
voltage drop of 8 volts (E=IR), positive on the top and
negative on the bottom.
The primary advantage of Mesh Current analysis is that it
generally allows for the solution of a large network with
fewer unknown values and fewer simultaneous equations.
Our example problem took three equations to solve the
Branch Current method and only two equations using the
Mesh Current method. This advantage is much greater as
networks increase in complexity:
To solve this network using Branch Currents, we'd have to
establish five variables to account for each and every
unique current in the circuit (Il; through I;). This would
require five equations for solution, in the form of two KCL
equations and three KVL equations (two equations for KCL at
the nodes, and three equations for KVL in each loop):
-1,+1,+1,=0 Kirchhoff's Current Law at node 1
-1,+1,-1,=0 Kirchhoff's Current Law at node 2
-E3,+1LR,+1,R,=0 Kirchhoff's Voltage Law in left loop
-LR,+1,R,+1,R;=0 Kirchhoff's Voltage Law in middle loop
-1,R,+E,,-1,R;=90 Kirchhoff's Voltage Law in right loop
| suppose if you have nothing better to do with your time
than to solve for five unknown variables with five equations,
you might not mind using the Branch Current method of
analysis for this circuit. For those of us who have better
things to do with our time, the Mesh Current method is a
whole lot easier, requiring only three unknowns and three
equations to solve:
-E,, + R01, +1)+1,R,=0 Kirchhoff's Voltage Law
in left loop
- R,(1, +1,)- R,0, + 1,)- LR; =0 Kirchhoff's Voltage Law
: : in middle loop
(1, +1,)+ E,,+1,R,=0 Kirchhoff's Voltage Law
5 +1) + Feo + LRs in right t loop”
Less equations to work with is a decided advantage,
especially when performing simultaneous equation solution
by hand (without a calculator).
Another type of circuit that lends itself well to Mesh Current
is the unbalanced Wheatstone Bridge. Take this circuit, for
example:
Since the ratios of R;/R4 and R>/Rs are unequal, we know
that there will be voltage across resistor R3, and some
amount of current through it. As discussed at the beginning
of this chapter, this type of circuit is irreducible by normal
series-parallel analysis, and may only be analyzed by some
other method.
We could apply the Branch Current method to this circuit,
but it would require six currents (Il, through I,), leading toa
very large set of simultaneous equations to solve. Using the
Mesh Current method, though, we may solve for all currents
and voltages with much fewer variables.
The first step in the Mesh Current method is to draw just
enough mesh currents to account for all components in the
circuit. Looking at our bridge circuit, it should be obvious
where to place two of these currents:
The directions of these mesh currents, of course, is arbitrary.
However, two mesh currents is not enough in this circuit,
because neither I, nor |, goes through the battery. So, we
must add a third mesh current, I3:
Here, | have chosen I3 to loop from the bottom side of the
battery, through Rg, through Rj, and back to the top side of
the battery. This is not the only path | could have chosen for
Iz, but it seems the simplest.
Now, we must label the resistor voltage drop polarities,
following each of the assumed currents’ directions:
Notice something very important here: at resistor Ry, the
polarities for the respective mesh currents do not agree. This
iS because those mesh currents (I5 and I3) are going through
R, in different directions. This does not preclude the use of
the Mesh Current method of analysis, but it does complicate
it a bit. Though later, we will show how to avoid the Ry
current clash. (See Example below)
Generating a KVL equation for the top loop of the bridge,
starting from the top node and tracing in a clockwise
direction:
501, + LOO(1, + 1,) + 150(1, +1,)=0 Original form of equation
... distributing to terms within parentheses. . . .
501, + 1001, + 1001, + 1501, + 1501, =0
... combining like terms .. .
3001, + LOOL, + 1501, = 0 Simplified form of equation
In this equation, we represent the common directions of
currents by their sums through common resistors. For
example, resistor R3, with a value of 100 Q, has its voltage
drop represented in the above KVL equation by the
expression 100(I; + I5), since both currents |, and Iz go
through R3 from right to left. The same may be said for
resistor R,, with its voltage drop expression shown as 150(I,
+ l3), since both I, and l3 go from bottom to top through that
resistor, and thus work together to generate its voltage drop.
Generating a KVL equation for the bottom loop of the bridge
will not be so easy, since we have two currents going against
each other through resistor Ry. Here is how | do it (starting at
the right-hand node, and tracing counter-clockwise):
100(1, + 1,) + 300(1, - 1,) + 2501, = 0 Original form of equation
... distributing to terms within parentheses . . .
1001, + 1001, + 3001, - 3001, + 2501, = 0
... combining like terms .. .
LOOI, + 6501, - 3001, = 0 Simplified form of equation
Note how the second term in the equation's original form
has resistor R,'s value of 300 Q multiplied by the difference
between I> and I3 (I> - Iz). This is how we represent the
combined effect of two mesh currents going in opposite
directions through the same component. Choosing the
appropriate mathematical signs is very important here:
300(I, - lz) does not mean the same thing as 300(I3 - Ip). |
chose to write 300(I> - 13) because | was thinking first of I's
effect (creating a positive voltage drop, measuring with an
imaginary voltmeter across Ry, red lead on the bottom and
black lead on the top), and secondarily of I3's effect
(creating a negative voltage drop, red lead on the bottom
and black lead on the top). If | had thought in terms of I3's
effect first and I,'s effect secondarily, holding my imaginary
voltmeter leads in the same positions (red on bottom and
black on top), the expression would have been -300(I3 - Ip).
Note that this expression /s mathematically equivalent to
the first one: +300(Ip - I3).
Well, that takes care of two equations, but | still need a third
equation to complete my simultaneous equation set of three
variables, three equations. This third equation must also
include the battery's voltage, which up to this point does
not appear in either two of the previous KVL equations. To
generate this equation, | will trace a loop again with my
imaginary voltmeter starting from the battery's bottom
(negative) terminal, stepping clockwise (again, the direction
in which | step is arbitrary, and does not need to be the
same as the direction of the mesh current in that loop):
24 - 150(1; +1,) - 300(1; - 1,)=0 Original form of equation
.. . distributing to terms within parentheses . . .
24 - 15OL, - 1501, - 3001, + 3001, = 0
... combining like terms .. .
-1501, + 3001, - 4501, = -24 Simplified form of equation
Solving for I, I>, and lz using whatever simultaneous
equation method we prefer:
3001, + 1001, + 1501, =0
L001, + 6501, - 3001, = 0
-1501, + 3001, - 4501, = -24
Solutions:
1, = -93.793 mA
1, = 77.241 mA
1, = 136.092 mA
Example:
Use Octave to find the solution for l,, Iz, and l3 from the
above simplified form of equations. [octav]
Solution:
In Octave, an open source Matlab® clone, enter the
coefficients into the A matrix between square brackets with
column elements comma separated, and rows semicolon
separated.[octav] Enter the voltages into the column vector:
b. The unknown currents: Ij, lz, and l3 are calculated by the
command: x=A\b. These are contained within the x column
vector.
octave:1>A = [ 300,100,150; 100,650, -300; -150,300, -450]
A =
300 100 150
100 650 -300
-150 300 -450
octave:2> b = [ 0; 0; -24]
b=
0
0
-24
octave:3> x = A\b
xX =
-0.093793
0.077241
0.136092
The negative value arrived at for |, tells us that the assumed
direction for that mesh current was incorrect. Thus, the
actual current values through each resistor is as such:
1>1>L
le, = 15-1, = 136.092 mA - 93.793 mA = 42.299 mA
Igo = 1; = 93.793 mA
le3 = 1, -1, = 93.793 mA - 77.241 mA = 16.552 mA
ly = 1; -1, = 136.092 mA - 77.241 mA = 58.851 mA
les = 1, = 77.241 mA
Calculating voltage drops across each resistor:
Ep; = 1gR, = (42.299 mA)(150 Q) = 6.3448 V
Eo = IgoR> = (93.793 mA)(50 Q) = 4.6897 V
Ep; = lp3R; = (16.552 mA)(100 Q) = 1.6552 V
Eps = IpgRy = (58.851 mA)(300 Q) = 17.6552 V
Eps = IpsR; = (77.241 mA)(250 Q) = 19.3103 V
A SPICE simulation confirms the accuracy of our voltage
calculations:[spi]
1 1
unbalanced wheatstone bridge
vl 10
rl 1 2 150
r2 1 3 50
r3 2 3 100
r4 2 0 300
r5 3 0 250
.dc vl 24 24 1
.print de v(1,2) v(1,3) v(3,2) v(2,0) v(3,0)
end
vl v(1,2) v(1,3) v(3,2) v(2)
v(3)
2.400E+01 6.345E+00 4.690E+00 1.655E+00 1.766E+01
1.931E+01
Example:
(a) Find a new path for current I3 that does not produce a
conflicting polarity on any resistor compared to |, or I>. Ra
was the offending component. (b) Find values for I,, lz, and
Iz. (c) Find the five resistor currents and compare to the
previous values.
Solution: [dvn]
(a) Route I3 through Rs, R3 and R, as shown:
Original form of equations
501, + 100(1, +1, +1,) + 150(, +1,) =0
3001, + 250(1, + 1,) + L001, +1,+1,)=0
24-250(1,+1,) - 100(1,+1,+1,)- 150(1,+1,)=0
Simplified form of equations
3001, + 1001, + 2501, =0
1001, + 6501, + 3501, =0
-2501, - 3501, - 5001, =-24
Note that the conflicting polarity on Ry has been removed.
Moreover, none of the other resistors have conflicting
polarities.
(bo) Octave, an open source (free) matlab clone, yields a
mesh current vector at “x”:[octav]
octave:l> A =
[ 300,100,250; 100,650,350; -250, -350, -500]
A=
300 100 250
100 650 350
-250 -350 -500
octave:2> b = [ 0; 0; -24]
b =
0
0
-24
octave:3> x = A\b
xX =
-0.093793
-0.058851
0.136092
Not all currents I,, lp, and Iz are the same (I>) as the previous
bridge because of different loop paths However, the resistor
currents compare to the previous values:
Ip, = I, + I3 = -93.793 ma + 136.092 ma = 42.299 ma
Ip = I, = -93.793 ma
Ip3 = I, + Ip + 13 = -93.793 ma -58.851 ma + 136.092
ma = -16.552 ma
Ipqg = Ip = -58.851 ma
Ip, = In + I3 = -58.851 ma + 136.092 ma = 77.241 ma
Since the resistor currents are the same as the previous
values, the resistor voltages will be identical and need not
be calculated again.
e REVIEW:
Steps to follow for the “Mesh Current” method of
analysis:
e (1) Draw mesh currents in loops of circuit, enough to
account for all components.
e (2) Label resistor voltage drop polarities based on
assumed directions of mesh currents.
(3) Write KVL equations for each loop of the circuit,
substituting the product IR for E in each resistor term of
the equation. Where two mesh currents intersect
through a component, express the current as the
algebraic sum of those two mesh currents (i.e. 1, + I>) if
the currents go in the same direction through that
component. If not, express the current as the difference
(i.e. ly = 5).
(4) Solve for unknown mesh currents (simultaneous
equations).
e (5) If any solution is negative, then the assumed current
direction is wrong!
e (6) Algebraically add mesh currents to find current in
components sharing multiple mesh currents.
e (7) Solve for voltage drops across all resistors (E=IR).
Mesh current by inspection
We take a second look at the “mesh current method” with all
the currents running counterclockwise (ccw). The motivation
is to simplify the writing of mesh equations by ignoring the
resistor voltage drop polarity. Though, we must pay
attention to the polarity of voltage sources with respect to
assumed current direction. The sign of the resistor voltage
drops will follow a fixed pattern.
If we write a set of conventional mesh current equations for
the circuit below, where we do pay attention to the signs of
the voltage drop across the resistors, we may rearrange the
coefficients into a fixed pattern:
R, R, Mesh equations
- + - + (I, - L)R,+1,R, -B, =O
| 7 LR, - (1, -L)R, -B, =0
B, Gx Sf (=) — B, Simplified
+ +] - - (RL +R, -Rib =B,
-R,L, +(R,,RyL, =B,
Once rearranged, we may write equations by inspection. The
signs of the coefficients follow a fixed pattern in the pair
above, or the set of three in the rules below.
e Mesh current rules:
e This method assumes electron flow (not conventional
current flow) voltage sources. Replace any current
source in parallel with a resistor with an equivalent
voltage source in series with an equivalent resistance.
Ignoring current direction or voltage polarity on
resistors, draw counterclockwise current loops traversing
all components. Avoid nested loops.
Write voltage-law equations in terms of unknown
currents currents: lj, Iz, and I3. Equation 1 coefficient 1,
equation 2, coefficient 2, and equation 3 coefficient 3
are the positive sums of resistors around the respective
loops.
All other coefficients are negative, representative of the
resistance common to a pair of loops. Equation 1
coefficient 2 is the resistor common to loops 1 and 2,
coefficient 3 the resistor common to loops 1 an 3. Repeat
for other equations and coefficients.
+(sum of R's loop 1)I, - (common R loop 1-2)I> -
(common R loop 1-3)I3 = Ej
-(common R loop 1-2)I, + (sum of R's loop 2)I>5 -
(common R loop 2-3)I3 = E>
-(common R loop 1-3)I, - (common R loop 2-3)I> + (sum
of R's loop 3)I3 = E3
The right hand side of the equations is equal to any
electron current flow voltage source. A voltage rise with
respect to the counterclockwise assumed current is
positive, and 0 for no voltage source.
Solve equations for mesh currents:1,, lz, and l3 . Solve for
currents through individual resistors with KCL. Solve for
voltages with Ohms Law and KVL.
While the above rules are specific for a three mesh circuit,
the rules may be extended to smaller or larger meshes. The
figure below illustrates the application of the rules. The
three currents are all drawn in the same direction,
counterclockwise. One KVL equation is written for each of
the three loops. Note that there is no polarity drawn on the
resistors. We do not need it to determine the signs of the
coefficients. Though we do need to pay attention to the
polarity of the voltage source with respect to current
direction. The lz;counterclockwise current traverses the 24V
source from (+) to (-). This is a voltage rise for electron
current flow. Therefore, the third equation right hand side is
+24V.
+/R)+R+R,iL, -(R, iL; -(RyjL, = 0
ve -R,jL, +(/R,+R,+R, iL, -(R,iL, = 0
a he -(Ry iL, -(R IL, +(Ry+R,jL, =24
T +(150+50+100)1, -(100)1, -(150)1,= 0
-(100)L, +(100+300+250)1, - (300)1,= 0
(150), - (300), +(150+300)1, =24
+(300)l, -(100}; -(150)I;= 0
-(100)1, + (650)1, -(300)1,= 0
-(150)1, -(300)1, +(450)1, =24
In Octave, enter the coefficients into the A matrix with
column elements comma separated, and rows semicolon
separated. Enter the voltages into the column vector b.
Solve for the unknown currents: Ij, Ip, and lz with the
command: x=A\b. These currents are contained within the x
column vector. The positive values indicate that the three
mesh currents all flow in the assumed counterclockwise
direction.
octave:2> A=
[ 300, -100,-150; -100,650, -300; -150, -300, 450]
A =
300 -100 -150
-100 650 -300
-150 -300 450
octave:3> b= 0; 0; 24]
b=
0
0
24
octave:4> x=A\b
xX =
0.093793
0.077241
0.136092
The mesh currents match the previous solution by a
different mesh current method.. The calculation of resistor
voltages and currents will be identical to the previous
solution. No need to repeat here.
Note that electrical engineering texts are based on
conventional current flow. The loop-current, mesh-current
method in those text will run the assumed mesh currents
clockwise.[aef] The conventional current flows out the (+)
terminal of the battery through the circuit, returning to the
(-) terminal. A conventional current voltage rise corresponds
to tracing the assumed current from (-) to (+) through any
voltage sources.
One more example of a previous circuit follows. The
resistance around loop 1 is 6 Q, around loop 2: 3 Q. The
resistance common to both loops is 2 Q. Note the coefficients
of I, and I, in the pair of equations. Tracing the assumed
counterclockwise loop 1 current through B, from (+) to (-)
corresponds to an electron current flow voltage rise. Thus,
the sign of the 28 V is positive. The loop 2 counter clockwise
assumed current traces (-) to (+) through B5, a voltage drop.
Thus, the sign of B> is negative, -7 in the 2nd mesh
equation. Once again, there are no polarity markings on the
resistors. Nor do they figure into the equations.
6I, - 21, = 28
} Mesh equations
-21, + 31, =-7
6l, - 21, = 28 61, - 21, = 28
2 61,491,=-21 61,- 2(1) =28
71, =7 61, =30
L=1 I, =5
The currents |]; = 5 A, and lp = 1A are both positive. They
both flow in the direction of the counterclockwise loops. This
compares with previous results.
¢ Summary:
e The modified mesh-current method avoids having to
determine the signs of the equation coefficients by
drawing all mesh currents counterclockwise for electron
current flow.
e However, we do need to determine the sign of any
voltage sources in the loop. The voltage source is
positive if the assumed ccw current flows with the
battery (source). The sign is negative if the assumed ccw
current flows against the battery.
e See rules above for details.
Node voltage method
The node voltage method of analysis solves for unknown
voltages at circuit nodes in terms of a system of KCL
equations. This analysis looks strange because it involves
replacing voltage sources with equivalent current sources.
Also, resistor values in ohms are replaced by equivalent
conductances in siemens, G = 1/R. The siemens (S) is the
unit of conductance, having replaced the mho unit. In any
event S = Q!. And S = mho (obsolete).
We start with a circuit having conventional voltage sources.
A common node Eg is chosen as a reference point. The node
voltages E, and E, are calculated with respect to this point.
A voltage source in series with a resistance must be replaced
by an equivalent current source in parallel with the
resistance. We will write KCL equations for each node. The
right hand side of the equation is the value of the current
source feeding the node.
1,=B,/R,=10/2=5A
(a) (b)
Replacing voltage sources and associated series resistors
with equivalent current sources and parallel resistors yields
the modified circuit. Substitute resistor conductances in
siemens for resistance in ohms.
I, = E,/R, = 10/2 =5A
I, = E,/Rs = 4/1 =4A
G, = 1/R,= 1/20 =90.5S
G) = 1/R,=1/40 = 06.255
G3; = 1/R3 = 1/2.59 = 0.45
G, = 1/R, = 1/5 0 0.25
Gj = 1/R,= 1/10 =1.0S
The Parallel conductances (resistors) may be combined by
addition of the conductances. Though, we will not redraw
the circuit. The circuit is ready for application of the node
voltage method.
Ga
Gp
G, + G
G, + Gs
25 S$ = 0.75 S$
S-= 1.2 5
0.5S5 +0.
0.2 S +1
Deriving a general node voltage method, we write a pair of
KCL equations in terms of unknown node voltages V,; and V>
this one time. We do this to illustrate a pattern for writing
equations by inspection.
GyEy + G3(E, - E>) = I, (1)
GpE> = G3(E, = E>) = I, (2)
(Gy + G3 )E, -G3E> I, (1)
-G3E, + (Gg + G3)E> = I> (2)
The coefficients of the last pair of equations above have
been rearranged to show a pattern. The sum of
conductances connected to the first node is the positive
coefficient of the first voltage in equation (1). The sum of
conductances connected to the second node is the positive
coefficient of the second voltage in equation (2). The other
coefficients are negative, representing conductances
between nodes. For both equations, the right hand side is
equal to the respective current source connected to the
node. This pattern allows us to quickly write the equations
by inspection. This leads to a set of rules for the node
voltage method of analysis.
Node voltage rules:
Convert voltage sources in series with a resistor to an
equivalent current source with the resistor in parallel.
Change resistor values to conductances.
Select a reference node(Eo)
Assign unknown voltages (E,)(E>) ... (Ey)to remaining
nodes.
Write a KCL equation for each node 1,2, ... N. The
positive coefficient of the first voltage in the first
equation is the sum of conductances connected to the
node. The coefficient for the second voltage in the
second equation is the sum of conductances connected
to that node. Repeat for coefficient of third voltage, third
equation, and other equations. These coefficients fall on
a diagonal.
All other coefficients for all equations are negative,
representing conductances between nodes. The first
equation, second coefficient is the conductance from
node 1 to node 2, the third coefficient is the
conductance from node 1 to node 3. Fill in negative
coefficients for other equations.
The right hand side of the equations is the current
source connected to the respective nodes.
Solve system of equations for unknown node voltages.
Example: Set up the equations and solve for the node
voltages using the numerical values in the above figure.
Solution:
0.540.25+0.4)E, -(0.4)E5= 5
0.4)E, +(0.44+0.2+1.0)E, = -4
1.15)E, -(0.4)E5= 5
0
> .4)E, +(1.6)E, = -4
1 = 3.8095
E, = -1.5476
The solution of two equations can be performed with a
calculator, or with octave (not shown).[octav] The solution is
verified with SPICE based on the original schematic diagram
with voltage sources. [spi] Though, the circuit with the
Current sources could have been simulated.
V1 11 0 DC 10
V2 22 0 DC -4
rl 11 12
a,
ON ©
ON s
1
1
r4 2
222 1
.DC V1 10 10 1 V2 -4 -4 1
.print DC V(1) V(2)
.end
v(1) v(2)
3.809524e+00 -1.547619e+00
One more example. This one has three nodes. We do not list
the conductances on the schematic diagram. However, G, =
1/Rj, etc.
There are three nodes to write equations for by inspection.
Note that the coefficients are positive for equation (1) Ej,
equation (2) E>, and equation (3) E3. These are the sums of
all conductances connected to the nodes. All other
coefficients are negative, representing a conductance
between nodes. The right hand side of the equations is the
associated current source, 0.136092 A for the only current
source at node 1. The other equations are zero on the right
hand side for lack of current sources. We are too lazy to
calculate the conductances for the resistors on the diagram.
Thus, the subscripted G's are the coefficients.
(Gy + G,)E, -G,E> -G5E3
= 0.136092
Gey 200. GG, ce,
= 0
-G5E, -G3E> +(G> + G3 + Gs) E3
= 0
We are so lazy that we enter reciprocal resistances and sums
of reciprocal resistances into the octave “A” matrix, letting
octave compute the matrix of conductances after “A=”.
[octav] The initial entry line was so long that it was split into
three rows. This is different than previous examples. The
entered “A” matrix is delineated by starting and ending
square brackets. Column elements are space separated.
Rows are “new line” separated. Commas and semicolons are
not need as separators. Though, the current vector at “b” is
semicolon separated to yield a column vector of currents.
octave:12> A = [1/150+1/50 -1/150 -1/50
> -1/150 1/1504+1/100+1/300 -1/100
> -1/50 -1/100 1/50+1/100+1/250]
A =
0.0266667 -0.0066667 -0.0200000
-Q0.0066667 0.0200000 -0.0100000
-0.0200000 -0.0100000 0.0340000
octave:13> b = [ 0.136092; 0; 0]
b =
0.13609
0.00000
0.00000
octave:14> x=A\b
xX =
24.000
17.655
19.310
Note that the “A” matrix diagonal coefficients are positive,
That all other coefficients are negative.
The solution as a voltage vector is at “x”. E,; = 24.000 V, E,
= 17.655 V, E3 = 19.310 V. These three voltages compare to
the previous mesh current and SPICE solutions to the
unbalanced bridge problem. This is no coincidence, for the
0.13609 A current source was purposely chosen to yield the
24 V used as a voltage source in that problem.
e Summary
e Given a network of conductances and current sources,
the node voltage method of circuit analysis solves for
unknown node voltages from KCL equations.
e See rules above for details in writing the equations by
inspection.
e The unit of conductance G is the siemens S.
Conductance is the reciprocal of resistance: G = 1/R
Introduction to network theorems
Anyone who's studied geometry should be familiar with the
concept of a theorem: a relatively simple rule used to solve
a problem, derived from a more intensive analysis using
fundamental rules of mathematics. At least hypothetically,
any problem in math can be solved just by using the simple
rules of arithmetic (in fact, this is how modern digital
computers carry out the most complex mathematical
calculations: by repeating many cycles of additions and
subtractions!), but human beings aren't as consistent or as
fast as a digital computer. We need “shortcut” methods in
order to avoid procedural errors.
In electric network analysis, the fundamental rules are
Ohm's Law and Kirchhoff's Laws. While these humble laws
may be applied to analyze just about any circuit
configuration (even if we have to resort to complex algebra
to handle multiple unknowns), there are some “shortcut”
methods of analysis to make the math easier for the average
human.
As with any theorem of geometry or algebra, these network
theorems are derived from fundamental rules. In this
chapter, I'm not going to delve into the formal proofs of any
of these theorems. If you doubt their validity, you can
always empirically test them by setting up example circuits
and calculating values using the “old” (simultaneous
equation) methods versus the “new” theorems, to see if the
answers coincide. They always should!
Millman's Theorem
In Millman's Theorem, the circuit is re-drawn as a parallel
network of branches, each branch containing a resistor or
series battery/resistor combination. Millman's Theorem is
applicable only to those circuits which can be re-drawn
accordingly. Here again is our example circuit used for the
last two analysis methods:
And here is that same circuit, re-drawn for the sake of
applying Millman's Theorem:
By considering the supply voltage within each branch and
the resistance within each branch, Millman's Theorem will
tell us the voltage across all branches. Please note that I've
labeled the battery in the rightmost branch as “B3” to
clearly denote it as being in the third branch, even though
there is no “B>” in the circuit!
Millman's Theorem is nothing more than a long equation,
applied to any circuit drawn as a set of parallel-connected
branches, each branch with its own voltage source and
series resistance:
Millman’s Theorem Equation
E E; Es;
Bl re B2 4 B3
RK.
l l
— + +
R, RR, R;
3
a
3
= Voltage across all branches
—
Substituting actual voltage and resistance figures from our
example circuit for the variable terms of this equation, we
get the following expression:
28 V OV 7¥
+ +
4Q 2Q 1Q
=8V
1 I I
ee ee
4Q 22 1Q
The final answer of 8 volts is the voltage seen across all
parallel branches, like this:
The polarity of all voltages in Millman's Theorem are
referenced to the same point. In the example circuit above, |
used the bottom wire of the parallel circuit as my reference
point, and so the voltages within each branch (28 for the Rj
branch, 0 for the R» branch, and 7 for the R3 branch) were
inserted into the equation as positive numbers. Likewise,
when the answer came out to 8 volts (positive), this meant
that the top wire of the circuit was positive with respect to
the bottom wire (the original point of reference). If both
batteries had been connected backwards (negative ends up
and positive ends down), the voltage for branch 1 would
have been entered into the equation as a -28 volts, the
voltage for branch 3 as -7 volts, and the resulting answer of
-8 volts would have told us that the top wire was negative
with respect to the bottom wire (our initial point of
reference).
To solve for resistor voltage drops, the Millman voltage
(across the parallel network) must be compared against the
voltage source within each branch, using the principle of
voltages adding in series to determine the magnitude and
polarity of voltage across each resistor:
E,, = 8 V - 28 V =-20 V (negative on top)
E,, = 8 V-0 V=8 V (positive on top)
E,; = 8V-7V=1V (positive on top)
To solve for branch currents, each resistor voltage drop can
be divided by its respective resistance (I=E/R):
ley = =5A
RI 19
7 2Q
LV
= =1A
R3 12
The direction of current through each resistor is determined
by the polarity across each resistor, not by the polarity
across each battery, as current can be forced backwards
through a battery, as is the case with B3 in the example
circuit. This is important to keep in mind, since Millman's
Theorem doesn't provide as direct an indication of “wrong”
current direction as does the Branch Current or Mesh Current
methods. You must pay close attention to the polarities of
resistor voltage drops as given by Kirchhoff's Voltage Law,
determining direction of currents from that.
lpi Ips
SA | fia
1V
Millman's Theorem is very convenient for determining the
voltage across a set of parallel branches, where there are
enough voltage sources present to preclude solution via
regular series-parallel reduction method. It also is easy in
the sense that it doesn't require the use of simultaneous
equations. However, it is limited in that it only applied to
circuits which can be re-drawn to fit this form. It cannot be
used, for example, to solve an unbalanced bridge circuit.
And, even in cases where Millman's Theorem can be applied,
the solution of individual resistor voltage drops can be a bit
daunting to some, the Millman's Theorem equation only
providing a single figure for branch voltage.
As you will see, each network analysis method has its own
advantages and disadvantages. Each method is a tool, and
there is no tool that is perfect for all jobs. The skilled
technician, however, carries these methods in his or her
mind like a mechanic carries a set of tools in his or her tool
box. The more tools you have equipped yourself with, the
better prepared you will be for any eventuality.
e REVIEW:
e Millman's Theorem treats circuits as a parallel set of
series-component branches.
e All voltages entered and solved for in Millman's Theorem
are polarity-referenced at the same point in the circuit
(typically the bottom wire of the parallel network).
Superposition Theorem
Superposition theorem is one of those strokes of genius that
takes a complex subject and simplifies it in a way that
makes perfect sense. A theorem like Millman's certainly
works well, but it is not quite obvious why it works so well.
Superposition, on the other hand, is obvious.
The strategy used in the Superposition Theorem is to
eliminate all but one source of power within a network at a
time, using series/parallel analysis to determine voltage
drops (and/or currents) within the modified network for each
power source separately. Then, once voltage drops and/or
currents have been determined for each power source
working separately, the values are all “Superimposed” on
top of each other (added algebraically) to find the actual
voltage drops/currents with all sources active. Let's look at
our example circuit again and apply Superposition Theorem
to it:
Ry R
Since we have two sources of power in this circuit, we will
have to calculate two sets of values for voltage drops and/or
currents, one for the circuit with only the 28 volt battery in
effect...
...and one for the circuit with only the 7 volt battery in
effect:
When re-drawing the circuit for series/parallel analysis with
one source, all other voltage sources are replaced by wires
(shorts), and all current sources with open circuits (breaks).
Since we only have voltage sources (batteries) in our
example circuit, we will replace every inactive source during
analysis with a wire.
Analyzing the circuit with only the 28 volt battery, we obtain
the following values for voltage and current:
R, + R,//R,
R, R, R; R,//R; Total
Analyzing the circuit with only the 7 volt battery, we obtain
another set of values for voltage and current:
R, +R /R,
R, R, R; RJ//R, Total
When superimposing these values of voltage and current,
we have to be very careful to consider polarity (voltage
drop) and direction (electron flow), as the values have to be
added algebraically.
With 28 V With 7 V
battery battery With both batteries
4V 20 V
24V
+
Ex, —W—
24V-4V=20V
+
Epo Ea
4V+4V=8V
hve
ER; -W-
4V-3V=IV
Applying these superimposed voltage figures to the circuit,
the end result looks something like this:
Currents add up algebraically as well, and can either be
superimposed as done with the resistor voltage drops, or
simply calculated from the final voltage drops and
respective resistances (I=E/R). Either way, the answers will
be the same. Here | will show the superposition method
applied to current:
With 28 V With 7 V
battery battery With both batteries
Once again applying these superimposed figures to our
circuit:
Quite simple and elegant, don't you think? It must be noted,
though, that the Superposition Theorem works only for
circuits that are reducible to series/parallel combinations for
each of the power sources at a time (thus, this theorem is
useless for analyzing an unbalanced bridge circuit), and it
only works where the underlying equations are linear (no
mathematical powers or roots). The requisite of linearity
means that Superposition Theorem is only applicable for
determining voltage and current, not power!!! Power
dissipations, being nonlinear functions, do not algebraically
add to an accurate total when only one source is considered
at a time. The need for linearity also means this Theorem
cannot be applied in circuits where the resistance of a
component changes with voltage or current. Hence,
networks containing components like lamps (incandescent
or gas-discharge) or varistors could not be analyzed.
Another prerequisite for Superposition Theorem is that all
components must be “bilateral,” meaning that they behave
the same with electrons flowing either direction through
them. Resistors have no polarity-specific behavior, and so
the circuits we've been studying so far all meet this
criterion.
The Superposition Theorem finds use in the study of
alternating current (AC) circuits, and semiconductor
(amplifier) circuits, where sometimes AC is often mixed
(Superimposed) with DC. Because AC voltage and current
equations (Ohm's Law) are linear just like DC, we can use
Superposition to analyze the circuit with just the DC power
source, then just the AC power source, combining the results
to tell what will happen with both AC and DC sources in
effect. For now, though, Superposition will suffice as a break
from having to do simultaneous equations to analyze a
Circuit.
e REVIEW:
e The Superposition Theorem states that a circuit can be
analyzed with only one source of power at a time, the
corresponding component voltages and currents
algebraically added to find out what they'll do with all
power sources in effect.
e To negate all but one power source for analysis, replace
any source of voltage (batteries) with a wire; replace any
Current source with an open (break).
Thevenin's Theorem
Thevenin's Theorem states that it is possible to simplify any
linear circuit, no matter how complex, to an equivalent
circuit with just a single voltage source and series resistance
connected to a load. The qualification of “linear” is identical
to that found in the Superposition Theorem, where all the
underlying equations must be linear (no exponents or roots).
If we're dealing with passive components (such as resistors,
and later, inductors and capacitors), this is true. However,
there are some components (especially certain gas-
discharge and semiconductor components) which are
nonlinear: that is, their opposition to current changes with
voltage and/or current. As such, we would call circuits
containing these types of components, nonlinear circuits.
Thevenin's Theorem is especially useful in analyzing power
systems and other circuits where one particular resistor in
the circuit (called the “load” resistor) is subject to change,
and re-calculation of the circuit is necessary with each trial
value of load resistance, to determine voltage across it and
current through it. Let's take another look at our example
circuit:
Let's suppose that we decide to designate R> as the “load”
resistor in this circuit. We already have four methods of
analysis at our disposal (Branch Current, Mesh Current,
Millman's Theorem, and Superposition Theorem) to use in
determining voltage across R> and current through R>, but
each of these methods are time-consuming. Imagine
repeating any of these methods over and over again to find
what would happen if the load resistance changed
(changing load resistance is very common in power systems,
as multiple loads get switched on and off as needed. the
total resistance of their parallel connections changing
depending on how many are connected at a time). This
could potentially involve a /ot of work!
Thevenin's Theorem makes this easy by temporarily
removing the load resistance from the original circuit and
reducing what's left to an equivalent circuit composed of a
single voltage source and series resistance. The load
resistance can then be re-connected to this “Thevenin
equivalent circuit” and calculations carried out as if the
whole network were nothing but a simple series circuit:
... after Thevenin conversion...
Thevenin Equivalent Circuit
R
Thevenin
E
Thevenin —
The “Thevenin Equivalent Circuit” is the electrical
equivalent of B,, R;, R3, and B> as seen from the two points
where our load resistor (R>) connects.
The Thevenin equivalent circuit, if correctly derived, will
behave exactly the same as the original circuit formed by
B,, Ry, R3, and B>. In other words, the load resistor (R>)
voltage and current should be exactly the same for the same
value of load resistance in the two circuits. The load resistor
R> cannot “tell the difference” between the original network
of B;, Rz, R3, and Bs, and the Thevenin equivalent circuit of
and Rtnevenin have been calculated correctly.
The advantage in performing the “Thevenin conversion” to
the simpler circuit, of course, is that it makes load voltage
and load current so much easier to solve than in the original
network. Calculating the equivalent Thevenin source voltage
and series resistance is actually quite easy. First, the chosen
load resistor is removed from the original circuit, replaced
with a break (open circuit):
R, R;
4Q [ 1Q
Bev moved BTV
a |
Next, the voltage between the two points where the load
resistor used to be attached is determined. Use whatever
analysis methods are at your disposal to do this. In this case,
the original circuit with the load resistor removed is nothing
more than a simple series circuit with opposing batteries,
and so we can determine the voltage across the open load
terminals by applying the rules of series circuits, Ohm's Law,
and Kirchhoff's Voltage Law:
R, 4Q , 1
1 : 1 -
16.8 V 4.2V
4 + +
By. 28 11.2V B= FV
4.24 —> 4.24 —
The voltage between the two load connection points can be
figured from the one of the battery's voltage and one of the
resistor's voltage drops, and comes out to 11.2 volts. This is
our “Thevenin voltage” (Etpevenin) in the equivalent circuit:
Thevenin Equivalent Circuit
R
Thevenin
Exnevenin — 11.2V (Load)
To find the Thevenin series resistance for our equivalent
circuit, we need to take the original circuit (with the load
resistor still removed), remove the power sources (in the
Same style as we did with the Superposition Theorem:
voltage sources replaced with wires and current sources
replaced with breaks), and figure the resistance from one
load terminal to the other:
With the removal of the two batteries, the total resistance
measured at this location is equal to R; and R3 in parallel:
0.8 Q. This is our “Thevenin resistance” (Rtpevenin) for the
equivalent circuit:
Thevenin Equivalent Circuit
Rinevenin
Enhevenin zs | th Es’ (Load)
With the load resistor (2 Q) attached between the
connection points, we can determine voltage across it and
current through it as though the whole network were
nothing more than a simple series circuit:
Rrhevenin R Load Total
Notice that the voltage and current figures for R> (8 volts, 4
amps) are identical to those found using other methods of
analysis. Also notice that the voltage and current figures for
the Thevenin series resistance and the Thevenin source
(total) do not apply to any component in the original,
complex circuit. Thevenin's Theorem is only useful for
determining what happens to a sing/e resistor in a network:
the load.
The advantage, of course, is that you can quickly determine
what would happen to that single resistor if it were of a
value other than 2 Q without having to go through a lot of
analysis again. Just plug in that other value for the load
resistor into the Thevenin equivalent circuit and a little bit of
series circuit calculation will give you the result.
e REVIEW:
e Thevenin's Theorem is a way to reduce a network to an
equivalent circuit composed of a single voltage source,
series resistance, and series load.
Steps to follow for Thevenin's Theorem:
(1) Find the Thevenin source voltage by removing the
load resistor from the original circuit and calculating
voltage across the open connection points where the
load resistor used to be.
(2) Find the Thevenin resistance by removing all power
sources in the original circuit (voltage sources shorted
and current sources open) and calculating total
resistance between the open connection points.
e (3) Draw the Thevenin equivalent circuit, with the
Thevenin voltage source in series with the Thevenin
resistance. The load resistor re-attaches between the two
open points of the equivalent circuit.
e (4) Analyze voltage and current for the load resistor
following the rules for series circuits.
Norton's Theorem
Norton's Theorem states that it is possible to simplify any
linear circuit, no matter how complex, to an equivalent
circuit with just a single current source and parallel
resistance connected to a load. Just as with Thevenin's
Theorem, the qualification of “linear” is identical to that
found in the Superposition Theorem: all underlying
equations must be linear (no exponents or roots).
Contrasting our original example circuit against the Norton
equivalent: it looks something like this:
... after Norton conversion...
Norton Equivalent Circuit
INorto n (+) nee 2 (Load }
Remember that a current source is a component whose job
is to provide a constant amount of current, outputting as
much or as little voltage necessary to maintain that constant
Current.
As with Thevenin's Theorem, everything in the original
circuit except the load resistance has been reduced to an
equivalent circuit that is simpler to analyze. Also similar to
Thevenin's Theorem are the steps used in Norton's Theorem
to calculate the Norton source current (Inorton) and Norton
resistance (Ryorton)-
As before, the first step is to identify the load resistance and
remove it from the original circuit:
R, R;
4Q [ 1Q
B, — 28V praia E — 7¥
Lo,
Then, to find the Norton current (for the current source in
the Norton equivalent circuit), place a direct wire (short)
connection between the load points and determine the
resultant current. Note that this step is exactly opposite the
respective step in Thevenin's Theorem, where we replaced
the load resistor with a break (open circuit):
R, R;
3
tA.
H14A
I hort = [py + Ip2
With zero voltage dropped between the load resistor
connection points, the current through R, is strictly a
function of B,'s voltage and R,'s resistance: 7 amps (I=E/R).
Likewise, the current through R3 is now strictly a function of
B>'s voltage and R3's resistance: 7 amps (I=E/R). The total
current through the short between the load connection
points is the sum of these two currents: 7 amps + 7 amps =
14 amps. This figure of 14 amps becomes the Norton source
current (Inorton) iN Our equivalent circuit:
Norton Equivalent Circuit
Roxon 2 (Load )
Remember, the arrow notation for a current source points in
the direction opposite that of electron flow. Again, apologies
for the confusion. For better or for worse, this is standard
electronic symbol notation. Blame Mr. Franklin again!
To calculate the Norton resistance (Ryorton), We do the exact
same thing as we did for calculating Thevenin resistance
(Rthevenin): take the original circuit (with the load resistor
still removed), remove the power sources (in the same style
as we did with the Superposition Theorem: voltage sources
replaced with wires and current sources replaced with
breaks), and figure total resistance from one load connection
point to the other:
Now our Norton equivalent circuit looks like this:
Norton Equivalent Circuit
(Load )
INorton (+)
4A
If we re-connect our original load resistance of 2 QO, we can
analyze the Norton circuit as a simple parallel arrangement:
Ryor ton R Load Total
As with the Thevenin equivalent circuit, the only useful
information from this analysis is the voltage and current
values for R>; the rest of the information is irrelevant to the
Original circuit. However, the same advantages seen with
Thevenin's Theorem apply to Norton's as well: if we wish to
analyze load resistor voltage and current over several
different values of load resistance, we can use the Norton
equivalent circuit again and again, applying nothing more
complex than simple parallel circuit analysis to determine
what's happening with each trial load.
e REVIEW:
e Norton's Theorem is a way to reduce a network to an
equivalent circuit composed of a single current source,
parallel resistance, and parallel load.
Steps to follow for Norton's Theorem:
(1) Find the Norton source current by removing the load
resistor from the original circuit and calculating current
through a short (wire) jumping across the open
connection points where the load resistor used to be.
(2) Find the Norton resistance by removing all power
sources in the original circuit (voltage sources shorted
and current sources open) and calculating total
resistance between the open connection points.
(3) Draw the Norton equivalent circuit, with the Norton
current source in parallel with the Norton resistance. The
load resistor re-attaches between the two open points of
the equivalent circuit.
e (4) Analyze voltage and current for the load resistor
following the rules for parallel circuits.
Thevenin-Norton equivalencies
Since Thevenin's and Norton's Theorems are two equally
valid methods of reducing a complex network down to
something simpler to analyze, there must be some way to
convert a Thevenin equivalent circuit to a Norton equivalent
circuit, and vice versa (just what you were dying to know,
right?). Well, the procedure is very simple.
You may have noticed that the procedure for calculating
Thevenin resistance is identical to the procedure for
calculating Norton resistance: remove all power sources and
determine resistance between the open load connection
points. As such, Thevenin and Norton resistances for the
Same original network must be equal. Using the example
circuits from the last two sections, we can see that the two
resistances are indeed equal:
Thevenin Equivalent Circuit
Ronevenin
0.8 Q
Ennevenin => Liha (Load)
Norton Equivalent Circuit
INorton gs (Load )
14a
Rohevenin = Ryorton
Considering the fact that both Thevenin and Norton
equivalent circuits are intended to behave the same as the
Original network in supplying voltage and current to the load
resistor (as seen from the perspective of the load connection
points), these two equivalent circuits, having been derived
from the same original network should behave identically.
This means that both Thevenin and Norton equivalent
circuits should produce the same voltage across the load
terminals with no load resistor attached. With the Thevenin
equivalent, the open-circuited voltage would be equal to the
Thevenin source voltage (no circuit current present to drop
voltage across the series resistor), which is 11.2 volts in this
case. With the Norton equivalent circuit, all 14 amps from
the Norton current source would have to flow through the
0.8 QO Norton resistance, producing the exact same voltage,
11.2 volts (E=IR). Thus, we can say that the Thevenin
voltage is equal to the Norton current times the Norton
resistance:
Erhevenin = INorton® Norton
So, if we wanted to convert a Norton equivalent circuit to a
Thevenin equivalent circuit, we could use the same
resistance and calculate the Thevenin voltage with Ohm's
Law.
Conversely, both Thevenin and Norton equivalent circuits
should generate the same amount of current through a short
circuit across the load terminals. With the Norton equivalent,
the short-circuit current would be exactly equal to the
Norton source current, which is 14 amps in this case. With
the Thevenin equivalent, all 11.2 volts would be applied
across the 0.8 QO Thevenin resistance, producing the exact
Same current through the short, 14 amps (I=E/R). Thus, we
can say that the Norton current is equal to the Thevenin
voltage divided by the Thevenin resistance:
Etheve nin
R
l =
Norton
Thevenin
This equivalence between Thevenin and Norton circuits can
be a useful tool in itself, as we shall see in the next section.
e REVIEW:
e Thevenin and Norton resistances are equal.
e Thevenin voltage is equal to Norton current times
Norton resistance.
e Norton current is equal to Thevenin voltage divided by
Thevenin resistance.
Millman's Theorem revisited
You may have wondered where we got that strange equation
for the determination of “Millman Voltage” across parallel
branches of a circuit where each branch contains a series
resistance and voltage source:
Millman’s Theorem Equation
Es, Es) Es;
.o
R, R, R
3
l l l
—— + ——- +
RR; ROR
2 3
= Voltage across all branches
Parts of this equation seem familiar to equations we've seen
before. For instance, the denominator of the large fraction
looks conspicuously like the denominator of our parallel
resistance equation. And, of course, the E/R terms in the
numerator of the large fraction should give figures for
current, Ohm's Law being what it is (I=E/R).
Now that we've covered Thevenin and Norton source
equivalencies, we have the tools necessary to understand
Millman's equation. What Millman's equation is actually
doing is treating each branch (with its series voltage source
and resistance) as a Thevenin equivalent circuit and then
converting each one into equivalent Norton circuits.
Thus, in the circuit above, battery B, and resistor R, are
seen as a Thevenin source to be converted into a Norton
source of 7 amps (28 volts / 4 Q) in parallel with a 4 QO
resistor. The rightmost branch will be converted into a7 amp
current source (7 volts / 1 Q) and 1 Q resistor in parallel. The
center branch, containing no voltage source at all, will be
converted into a Norton source of 0 amps in parallel with a 2
Q resistor:
raQ® o (S10
Since current sources directly add their respective currents
in parallel, the total circuit current willbe 7 +0+7,o0r14
amps. This addition of Norton source currents is what's
being represented in the numerator of the Millman equation:
Millman’s Theorem Equation
Ba. . ta... Be Fs, Eg: — Eps
+ + —» + +
i ime. ae R, R, R,
I I I
—— + +
Ri. Re oR
Listal =
i.
=
3
All the Norton resistances are in parallel with each other as
well in the equivalent circuit, so they diminish to create a
total resistance. This diminishing of source resistances is
what's being represented in the denominator of the
Millman's equation:
Millman’s Theorem Equation
Es Es» E33
+ +
I Re
Ricsat = oe
I 1 I 1 1 1
ee eee ee —> + +
R, R, R, R R R,
In this case, the resistance total will be equal to 571.43
millionms (571.43 mQ). We can re-draw our equivalent
circuit now as one with a single Norton current source and
Norton resistance:
l4A G 571.43 mQ
Ohm's Law can tell us the voltage across these two
components now (E=IR):
Evora = (14 A)(57 1.43 mQ)
Bal =8V
oN
+
14 (4) 571.43 mQ
ce
Let's summarize what we know about the circuit thus far. We
know that the total current in this circuit is given by the sum
of all the branch voltages divided by their respective
resistances. We also know that the total resistance is found
by taking the reciprocal of all the branch resistance
reciprocals. Furthermore, we should be well aware of the fact
that total voltage across all the branches can be found by
multiplying total current by total resistance (E=IR). All we
need to do is put together the two equations we had earlier
for total circuit current and total resistance, multiplying
them to find total voltage:
Ohm's Law: IXR=E
(total current) x (total resistance) = (total voltage)
E EA. s
a ee Fs = (total voltage)
l
l l
R, R, R; l
— + +
ER, 3 R,
Or.
Ex, Ex, Ex;
+ +
R, R, R,
= = (total voltage)
1 1 1
— + +
R, R, R
wo
The Millman's equation is nothing more than a Thevenin-to-
Norton conversion matched together with the parallel
resistance formula to find total voltage across all the
branches of the circuit. So, hopefully some of the mystery is
gone now!
Maximum Power Transfer Theorem
The Maximum Power Transfer Theorem is not so much a
means of analysis as it is an aid to system design. Simply
stated, the maximum amount of power will be dissipated by
a load resistance when that load resistance is equal to the
Thevenin/Norton resistance of the network supplying the
power. If the load resistance is lower or higher than the
Thevenin/Norton resistance of the source network, its
dissipated power will be less than maximum.
This is essentially what is aimed for in radio transmitter
design , where the antenna or transmission line
“impedance” is matched to final power amplifier
“impedance” for maximum radio frequency power output.
Impedance, the overall opposition to AC and DC current, is
very similar to resistance, and must be equal between
source and load for the greatest amount of power to be
transferred to the load. A load impedance that is too high
will result in low power output. A load impedance that is too
low will not only result in low power output, but possibly
overheating of the amplifier due to the power dissipated in
its internal (Thevenin or Norton) impedance.
Taking our Thevenin equivalent example circuit, the
Maximum Power Transfer Theorem tells us that the load
resistance resulting in greatest power dissipation is equal in
value to the Thevenin resistance (in this case, 0.8 Q):
Rtheven in
Ate es 11.2 V
With this value of load resistance, the dissipated power will
be 39.2 watts:
If we were to try a lower value for the load resistance (0.5 Q
instead of 0.8 O, for example), our power dissipated by the
load resistance would decrease:
Rthevenin R Load Total
Volts
Amps
08 | os | 13 | Ohms
59.38 Watts
Power dissipation increased for both the Thevenin resistance
and the total circuit, but it decreased for the load resistor.
Likewise, if we increase the load resistance (1.1 Q instead of
0.8 Q, for example), power dissipation will also be less than
it was at 0.8 O exactly:
vDUVaD —- Mm
Rthevenin Ri oad Total
| os |
38.22
If you were designing a circuit for maximum power
dissipation at the load resistance, this theorem would be
very useful. Having reduced a network down to a Thevenin
voltage and resistance (or Norton current and resistance),
you simply set the load resistance equal to that Thevenin or
Norton equivalent (or vice versa) to ensure maximum power
dissipation at the load. Practical applications of this might
include radio transmitter final amplifier stage design
(seeking to maximize power delivered to the antenna or
transmission line), a grid tied inverter loading a solar array,
or electric vehicle design (Seeking to maximize power
delivered to drive motor).
The Maximum Power Transfer Theorem is not:
Maximum power transfer does not coincide with maximum
efficiency. Application of The Maximum Power Transfer
theorem to AC power distribution will not result in maximum
or even high efficiency. The goal of high efficiency is more
important for AC power distribution, which dictates a
relatively low generator impedance compared to load
impedance.
Similar to AC power distribution, high fidelity audio
amplifiers are designed for a relatively low output
impedance and a relatively high speaker load impedance. As
a ratio, "output impdance" : "load impedance" is known as
damping factor, typically in the range of 100 to 1000. [rar]
[dfd]
Maximum power transfer does not coincide with the goal of
lowest noise. For example, the low-level radio frequency
amplifier between the antenna and a radio receiver is often
designed for lowest possible noise. This often requires a
mismatch of the amplifier input impedance to the antenna
as compared with that dictated by the maximum power
transfer theorem.
e REVIEW:
e The Maximum Power Transfer Theorem states that the
maximum amount of power will be dissipated by a load
resistance if it is equal to the Thevenin or Norton
resistance of the network supplying power.
e The Maximum Power Transfer Theorem does not satisfy
the goal of maximum efficiency.
A-Y and Y-A conversions
In many circuit applications, we encounter components
connected together in one of two ways to form a three-
terminal network: the “Delta,” or A (also Known as the “Pi,”
or tt) configuration, and the “Y” (also known as the “T”)
configuration.
Delta (A) network Wye (Y) network
A Rac C A
Rag Rec
B
Pi (x) network Tee (T) network
A Rac Cc A Ry Re S
Raz Rac Rs
B B
It is possible to calculate the proper values of resistors
necessary to form one kind of network (A or Y) that behaves
identically to the other kind, as analyzed from the terminal
connections alone. That is, if we had two separate resistor
networks, one A and one Y, each with its resistors hidden
from view, with nothing but the three terminals (A, B, and C)
exposed for testing, the resistors could be sized for the two
networks so that there would be no way to electrically
determine one network apart from the other. In other words,
equivalent A and Y networks behave identically.
There are several equations used to convert one network to
the other:
To convert a Delta (A) to a Wye (Y) To convert a Wye (Y) to a Delta (A)
Rap Rac RaRg + RaRco+RpRe
R, =———__—__ Rag = ———___—_-
Rap + Rac + Rec Re
Rag + Rac + Rac Rs
Rac Rec RaRp + RaRo+RpRe
Re = ——$——_—— a i A=
Rag + Rac + Rec Rp
A and Y networks are seen frequently in 3-phase AC power
systems (a topic covered in volume II of this book series),
but even then they're usually balanced networks (all
resistors equal in value) and conversion from one to the
other need not involve such complex calculations. When
would the average technician ever need to use these
equations?
A prime application for A-Y conversion is in the solution of
unbalanced bridge circuits, such as the one below:
Solution of this circuit with Branch Current or Mesh Current
analysis is fairly involved, and neither the Millman nor
Superposition Theorems are of any help, since there's only
one source of power. We could use Thevenin's or Norton's
Theorem, treating R3 as our load, but what fun would that
be?
If we were to treat resistors Rj, Ro, and R3 as being
connected in a A configuration (R3p, Rac, and Rye,
respectively) and generate an equivalent Y network to
replace them, we could turn this bridge circuit into a
(simpler) series/parallel combination circuit:
Selecting Delta (A) network to convert:
After the A-Y conversion...
A converted toa Y
If we perform our calculations correctly, the voltages
between points A, B, and C will be the same in the converted
circuit as in the original circuit, and we can transfer those
values back to the original bridge configuration.
R, = Co 7 216 -60
(12 Q)+ (18 Q)4+ (6 Q) 36
— (12 QY6 2) | = 7 -20
(12 Q) + (18 2) 4 (6 Q) 36
(18 Q)(6 Q) _ 108 _ 36
© (12. Q) + (18. Q) + (6 Q) 36
Resistors Ry, and Rs, of course, remain the same at 18 QO and
12 Q, respectively. Analyzing the circuit now asa
series/parallel combination, we arrive at the following
figures:
Ry Rp Ro R, R;
E Volts
| Amps
R| 6 | 2 | 3 { 18 | 2 | Ohms
Rg +R,
R3s+R, Ret+R; RetR; Total
E Volts
| Amps
R L4.571_| Ohms
We must use the voltage drops figures from the table above
to determine the voltages between points A, B, and C,
seeing how the add up (or subtract, as is the case with
voltage between points B and C):
Es = 4.706 V
E, ¢= 5.294V
Es.c¢ = 588.24 mV
Now that we know these voltages, we can transfer them to
the same points A, B, and C in the original bridge circuit:
Voltage drops across Ry and Rs, of course, are exactly the
same as they were in the converted circuit.
At this point, we could take these voltages and determine
resistor currents through the repeated use of Ohm's Law
(I=E/R):
Lys GA
122
ley = OY — = 204.12 mA
18Q
i. OR inl
62
es et
18Q
oe a ek
122
A quick simulation with SPICE will serve to verify our work:
[spi]
unbalanced bridge circuit
v1 10
rl 12 12
r2 13 18
r3 23 6
r4 2 0 18
r5 3 0 12
.dc v1 10 10 1
.print dc v(1,2) v(1,3) v(2,3) v(2,0) v(3,0)
end
vl v(1,2) v(1,3) v(2,3) v(2)
v(3)
1.000E+01 4.706E+00 5.294E+00 5.882E-01 5.294E+00
4.706E+00
The voltage figures, as read from left to right, represent
voltage drops across the five respective resistors, R; through
Rs. | could have shown currents as well, but since that would
have required insertion of “dummy” voltage sources in the
SPICE netlist, and since we're primarily interested in
validating the A-Y conversion equations and not Ohm's Law,
this will suffice.
e REVIEW:
e “Delta” (A) networks are also known as “Pi” (tt) networks.
e “Y” networks are also Known as “T” networks.
e A and Y networks can be converted to their equivalent
counterparts with the proper resistance equations. By
“equivalent,” | mean that the two networks will be
electrically identical as measured from the three
terminals (A, B, and C).
e A bridge circuit can be simplified to a series/parallel
circuit by converting half of it from a A to a Y network.
After voltage drops between the original three
connection points (A, B, and C) have been solved for,
those voltages can be transferred back to the original
bridge circuit, across those same equivalent points.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Dejan Budimir (January 2003): Suggested clarifications for
explaining the Mesh Current method of circuit analysis.
Bill Heath (December 2002): Pointed out several
typographical errors.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Davy Van Nieuwenborgh (April 2004): Pointed out error in
Mesh current section, supplied editorial material, end of
section.
Bibliography
1. [aef] A.E. Fitzergerald, David E. Higginbotham, Arvin
Grabel, Basic Electrical Engineering, (McGraw-Hill,
1975).
2. [spi] Tony Kuphaldt, Using the Spice Circuit Simulation
Program, in“Lessons in Electricity, Reference”, Volume 5,
Chapter 7, at
http://www. ibiblio.org/obp/electricCircuits/Ref/
3. [dvn] Davy Van Nieuwenborgh, private communications,
Theoretical Computer Science laboratory, Department of
Computer Science, Vrije Universiteit Brussel (4/7/2004).
4.[octav] Octave, Matrix calculator open source program
for Linux or MS Windows, at
http://www.gnu.org/software/octave/
5. [rar]Ray A. Rayburn , private communications, Senior
Consultant K2 Audio, LLC; Fellow of the Audio
Engineering Society, (6/29/2009).
6. [dfd]Damping Factor De-Mystified , at
http://www.sweetwater.com/shop/live-sound/power-
amplifiers/buying-guide.php# 2
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+]l—
—/ | 4]
Lessons In Electric Circuits
-- Volume |
Chapter 11
BATTERIES AND POWER
SYSTEMS
Electron activity in chemical reactions
Battery construction
Battery ratings
Special-purpose batteries
Practical considerations
Contributors
Bibliography
Electron activity in chemical reactions
So far in our discussions on electricity and electric circuits,
we have not discussed in any detail how batteries function.
Rather, we have simply assumed that they produce constant
voltage through some sort of mysterious process. Here, we
will explore that process to some degree and cover some of
the practical considerations involved with real batteries and
their use in power systems.
In the first chapter of this book, the concept of an atom was
discussed, as being the basic building-block of all material
objects. Atoms, in turn, are composed of even smaller pieces
of matter called particles. Electrons, protons, and neutrons
are the basic types of particles found in atoms. Each of these
particle types plays a distinct role in the behavior of an
atom. While electrical activity involves the motion of
electrons, the chemical identity of an atom (which largely
determines how conductive the material will be) is
determined by the number of protons in the nucleus
(center).
© © = electron
= proton
() = neutron
The protons in an atom's nucleus are extremely difficult to
dislodge, and so the chemical identity of any atom is very
stable. One of the goals of the ancient alchemists (to turn
lead into gold) was foiled by this sub-atomic stability. All
efforts to alter this property of an atom by means of heat,
light, or friction were met with failure. The electrons of an
atom, however, are much more easily dislodged. As we have
already seen, friction is one way in which electrons can be
transferred from one atom to another (glass and silk, wax
and wool), and so is heat (generating voltage by heating a
junction of dissimilar metals, as in the case of
thermocouples).
Electrons can do much more than just move around and
between atoms: they can also serve to link different atoms
together. This linking of atoms by electrons is called a
chemical bond. A crude (and simplified) representation of
such a bond between two atoms might look like this:
There are several types of chemical bonds, the one shown
above being representative of a cova/ent bond, where
electrons are shared between atoms. Because chemical
bonds are based on links formed by electrons, these bonds
are only as strong as the immobility of the electrons forming
them. That is to say, chemical bonds can be created or
broken by the same forces that force electrons to move:
heat, light, friction, etc.
When atoms are joined by chemical bonds, they form
materials with unique properties known as molecules. The
dual-atom picture shown above is an example of a simple
molecule formed by two atoms of the same type. Most
molecules are unions of different types of atoms. Even
molecules formed by atoms of the same type can have
radically different physical properties. Take the element
carbon, for instance: in one form, graphite, carbon atoms
link together to form flat "plates" which slide against one
another very easily, giving graphite its natural lubricating
properties. In another form, diamond, the same carbon
atoms link together in a different configuration, this time in
the shapes of interlocking pyramids, forming a material of
exceeding hardness. In yet another form, Fu//erene, dozens
of carbon atoms form each molecule, which looks something
like a soccer ball. Fullerene molecules are very fragile and
lightweight. The airy soot formed by excessively rich
combustion of acetylene gas (as in the initial ignition of an
oxy-acetylene welding/cutting torch) contains many
Fullerene molecules.
When alchemists succeeded in changing the properties of a
substance by heat, light, friction, or mixture with other
substances, they were really observing changes in the types
of molecules formed by atoms breaking and forming bonds
with other atoms. Chemistry is the modern counterpart to
alchemy, and concerns itself primarily with the properties of
these chemical bonds and the reactions associated with
them.
A type of chemical bond of particular interest to our study of
batteries is the so-called /onic bond, and it differs from the
covalent bond in that one atom of the molecule possesses
an excess of electrons while another atom lacks electrons,
the bonds between them being a result of the electrostatic
attraction between the two unlike charges. When ionic
bonds are formed from neutral atoms, there is a transfer of
electrons between the positively and negatively charged
atoms. An atom that gains an excess of electrons is said to
be reduced; an atom with a deficiency of electrons is said to
be oxidized. A mnemonic to help remember the definitions is
OIL RIG (oxidized is less; reduced is gained). It is important
to note that molecules will often contain both ionic and
covalent bonds. Sodium hydroxide (lye, NaOH) has an ionic
bond between the sodium atom (positive) and the hydroxyl
ion (negative). The hydroxyl ion has a covalent bond (shown
as a bar) between the hydrogen and oxygen atoms:
Nat O—H-
Sodium only loses one electron, so its charge is +1 in the
above example. If an atom loses more than one electron, the
resulting charge can be indicated as +2, +3, +4, etc. or bya
Roman numeral in parentheses showing the oxidation state,
such as (1), (Il), (IV), etc. Some atoms can have multiple
oxidation states, and it is sometimes important to include
the oxidation state in the molecular formula to avoid
ambiguity.
The formation of ions and ionic bonds from neutral atoms or
molecules (or vice versa) involves the transfer of electrons.
That transfer of electrons can be harnessed to generate an
electric current.A device constructed to do just this is called
a voltaic cell, or cell for short, usually consisting of two metal
electrodes immersed in a chemical mixture (called an
electrolyte) designed to facilitate such an electrochemical
(oxidation/reduction) reaction:
Voltaic cell
electrodes
The two electrodes are made of different materials,
both of which chemically react with the electrolyte
in some form of ionic bonding.
In the common "lead-acid" cell (the kind commonly used in
automobiles), the negative electrode is made of lead (Pb)
and the positive is made of lead (IV) dioxide (Pb0>), both
metallic substances. It is important to note that lead dioxide
is metallic and is an electrical conductor, unlike other metal
oxides that are usually insulators. (note: Table below) The
electrolyte solution is a dilute sulfuric acid (H3SO, + H>0). If
the electrodes of the cell are connected to an external
circuit, such that electrons have a place to flow from one to
the other, lead(IV) atoms in the positive electrode (PbO>)
will gain two electrons each to produce Pb(II)O. The oxygen
atoms which are “left over” combine with positively charged
hydrogen ions (H)tto form water (H,O). This flow of
electrons into into the lead dioxide (PbO) electrode, gives it
a positive electrical charge. Consequently, lead atoms in the
negative electrode give up two electrons each to produce
lead Pb(II), which combines with sulfate ions (SO,)
produced from the disassociation of the hydrogen ions (Ht)
from the sulfuric acid (H3SO,) to form lead sulfate (PbSO,).
The flow of electrons out of the lead electrode gives ita
negative electrical charge. These reactions are shown
diagrammitically below:[DOE]
Lead-acid cell discharging
+ -
PbO, electrode Pb electrode
At (+) electrode: Pb(IV)O, + 3H* + HSO, + 2e° —* Pb(II)SO, + 2H,O
At (-) electrode: Pb + HSO, * Pb(Il)SO,4 + H* + 2e
Overall cell: PbO, + Pb + 2H,SO, * 2PbSO, + 2H2O
Note on lead oxide nomenclature
The nomenclature for lead oxides can be confusing. The
term, lead oxide can refer to either Pb(II)O or Po(IV)O5, and
the correct compound can be determined usually from
context. Other synonyms for Pb(IV)O, are: lead dioxide,
lead peroxide, plumbic oxide, lead oxide brown, and lead
superoxide. The term, lead peroxide is particularly
confusing, as it implies a compound of lead (II) with two
oxygen atoms, Pb(Il)O2, which apparently does not exist.
Unfortunately, the term lead peroxide has persisted in
industrial literature. In this section, lead dioxide will be
used to refer to Po(IV)O2, and lead oxide will refer to
Pb(Il)O. The oxidation states will not be shown usually.
This process of the cell providing electrical energy to supply
a load is called discharging, since it is depleting its internal
chemical reserves. Theoretically, after all of the sulfuric acid
has been exhausted, the result will be two electrodes of lead
sulfate (PbSO,) and an electrolyte solution of pure water
(HO), leaving no more capacity for additional ionic bonding.
In this state, the cell is said to be fully discharged. |n a lead-
acid cell, the state of charge can be determined by an
analysis of acid strength. This is easily accomplished with a
device called a hydrometer, which measures the specific
gravity (density) of the electrolyte. Sulfuric acid is denser
than water, so the greater the charge of a cell, the greater
the acid concentration, and thus a denser electrolyte
solution.
There is no single chemical reaction representative of all
voltaic cells, so any detailed discussion of chemistry is
bound to have limited application. The important thing to
understand is that electrons are motivated to and/or from
the cell's electrodes via ionic reactions between the
electrode molecules and the electrolyte molecules. The
reaction is enabled when there is an external path for
electric current, and ceases when that path is broken.
Being that the motivation for electrons to move through a
cell is chemical in nature, the amount of voltage
(electromotive force) generated by any cell will be specific
to the particular chemical reaction for that cell type. For
instance, the lead-acid cell just described has a nominal
voltage of 2.04 volts per cell, based on a fully "charged" cell
(acid concentration strong) in good physical condition.
There are other types of cells with different specific voltage
outputs. The Edison cell, for example, with a positive
electrode made of nickel oxide, a negative electrode made
of iron, and an electrolyte solution of potassium hydroxide (a
caustic, not acid, substance) generates a nominal voltage of
only 1.2 volts, due to the specific differences in chemical
reaction with those electrode and electrolyte substances.
The chemical reactions of some types of cells can be
reversed by forcing electric current backwards through the
cell (in the negative electrode and out the positive
electrode). This process is called charging. Any such
(rechargeable) cell is called a secondary cell. A cell whose
chemistry cannot be reversed by a reverse current is called a
primary cell.
When a lead-acid cell is charged by an external current
source, the chemical reactions experienced during discharge
are reversed:
Lead-acid cell charging
Pb electrode
At (+) electrode: Pb(Il)\SO, + 2HO —> Pb(IV)O, + 3H* + HSO, + 2e€
At (-) electrode: Pb(II\SO,+H*+2e ~* Pb+HSO,
Overall cell: 2PbSO, + 2H,O —*™ PbO, + Pb + 2H,SO,
e REVIEW:
e Atoms bound together by electrons are called molecules.
e /onic bonds are molecular unions formed when an
electron-deficient atom (a positive ion) joins with an
electron-excessive atom (a negative ion).
e Electrochemical reactions involve the transfer of
electrons between atoms. This transfer can be harnessed
to form an electric current.
e A ce//is a device constructed to harness such chemical
reactions to generate electric current.
e Acell is said to be discharged when its internal chemical
reserves have been depleted through use.
e A secondary cell's chemistry can be reversed
(recharged) by forcing current backwards through it.
e A primary cell cannot be practically recharged.
e Lead-acid cell charge can be assessed with an
instrument called a hydrometer, which measures the
density of the electrolyte liquid. The denser the
electrolyte, the stronger the acid concentration, and the
greater charge state of the cell.
Battery construction
The word battery simply means a group of similar
components. In military vocabulary, a "battery" refers toa
cluster of guns. In electricity, a "battery" is a set of voltaic
cells designed to provide greater voltage and/or current
than is possible with one cell alone.
The symbol for a cell is very simple, consisting of one long
line and one short line, parallel to each other, with
connecting wires:
Cell
“L
T
The symbol for a battery is nothing more than a couple of
cell symbols stacked in series:
Battery
As was Stated before, the voltage produced by any particular
kind of cell is determined strictly by the chemistry of that
cell type. The size of the cell is irrelevant to its voltage. To
obtain greater voltage than the output of a single cell,
multiple cells must be connected in series. The total voltage
of a battery is the sum of all cell voltages. A typical
automotive lead-acid battery has six cells, for a nominal
voltage output of 6 x 2.0 or 12.0 volts:
2.0V 20V 2.0V 2.0 2.0V 2.0
Pap Pp ae
—___ 0} >
The cells in an automotive battery are contained within the
same hard rubber housing, connected together with thick,
lead bars instead of wires. The electrodes and electrolyte
solutions for each cell are contained in separate, partitioned
sections of the battery case. In large batteries, the
electrodes commonly take the shape of thin metal grids or
plates, and are often referred to as plates instead of
electrodes.
For the sake of convenience, battery symbols are usually
limited to four lines, alternating long/short, although the real
battery it represents may have many more cells than that.
On occasion, however, you might come across a symbol for a
battery with unusually high voltage, intentionally drawn
with extra lines. The lines, of course, are representative of
the individual cell plates:
symbol for a battery with
an unusually high voltage
SHAT
If the physical size of a cell has no impact on its voltage,
then what does it affect? The answer is resistance, which in
turn affects the maximum amount of current that a cell can
provide. Every voltaic cell contains some amount of internal
resistance due to the electrodes and the electrolyte. The
larger a cell is constructed, the greater the electrode contact
area with the electrolyte, and thus the less internal
resistance it will have.
Although we generally consider a cell or battery in a circuit
to be a perfect source of voltage (absolutely constant), the
current through it dictated solely by the external! resistance
of the circuit to which it is attached, this is not entirely true
in real life. Since every cell or battery contains some internal
resistance, that resistance must affect the current in any
given circuit:
Real battery
Ideal battery (with internal resistance)
—<— 8.333 A
=— 10A
LQ
10V: Eyq = 8.333 V
The real battery shown above within the dotted lines has an
internal resistance of 0.2 OQ, which affects its ability to
supply current to the load resistance of 1 QO. The ideal
battery on the left has no internal resistance, and so our
Ohm's Law calculations for current (I=E/R) give us a perfect
value of 10 amps for current with the 1 ohm load and 10 volt
supply. The real battery, with its built-in resistance further
impeding the flow of electrons, can only supply 8.333 amps
to the same resistance load.
The ideal battery, in a short circuit with O Q resistance,
would be able to supply an infinite amount of current. The
real battery, on the other hand, can only supply 50 amps (10
volts / 0.2 Q) to a short circuit of 0 O resistance, due to its
internal resistance. The chemical reaction inside the cell
may still be providing exactly 10 volts, but voltage is
dropped across that internal resistance as electrons flow
through the battery, which reduces the amount of voltage
available at the battery terminals to the load.
Since we live in an imperfect world, with imperfect batteries,
we need to understand the implications of factors such as
internal resistance. Typically, batteries are placed in
applications where their internal resistance is negligible
compared to that of the circuit load (where their short-circuit
current far exceeds their usual load current), and so the
performance is very close to that of an ideal voltage source.
If we need to construct a battery with lower resistance than
what one cell can provide (for greater current capacity), we
will have to connect the cells together in parallel:
equivalent to
Essentially, what we have done here is determine the
Thevenin equivalent of the five cells in parallel (an
equivalent network of one voltage source and one series
resistance). The equivalent network has the same source
voltage but a fraction of the resistance of any individual cell
in the original network. The overall effect of connecting cells
in parallel is to decrease the equivalent internal resistance,
just as resistors in parallel diminish in total resistance. The
equivalent internal resistance of this battery of 5 cells is 1/5
that of each individual cell. The overall voltage stays the
same: 2.0 volts. If this battery of cells were powering a
circuit, the current through each cell would be 1/5 of the
total circuit current, due to the equal split of current through
equal-resistance parallel branches.
e REVIEW:
e A battery is a cluster of cells connected together for
greater voltage and/or current capacity.
e Cells connected together in series (polarities aiding)
results in greater total voltage.
e Physical cell size impacts cell resistance, which in turn
impacts the ability for the cell to supply current to a
circuit. Generally, the larger the cell, the less its internal
resistance.
Cells connected together in parallel results in less total
resistance, and potentially greater total current.
Battery ratings
Because batteries create electron flow in a circuit by
exchanging electrons in ionic chemical reactions, and there
is a limited number of molecules in any charged battery
available to react, there must be a limited amount of total
electrons that any battery can motivate through a circuit
before its energy reserves are exhausted. Battery capacity
could be measured in terms of total number of electrons, but
this would be a huge number. We could use the unit of the
coulomb (equal to 6.25 x 108 electrons, or
6,250,000,000,000,000,000 electrons) to make the
quantities more practical to work with, but instead a new
unit, the amp-hour, was made for this purpose. Since 1 amp
is actually a flow rate of 1 coulomb of electrons per second,
and there are 3600 seconds in an hour, we can state a direct
proportion between coulombs and amp-hours: 1 amp-hour =
3600 coulombs. Why make up a new unit when an old would
have done just fine? To make your lives as students and
technicians more difficult, of course!
A battery with a capacity of 1 amp-hour should be able to
continuously supply a current of 1 amp to a load for exactly
1 hour, or 2 amps for 1/2 hour, or 1/3 amp for 3 hours, etc.,
before becoming completely discharged. In an ideal battery,
this relationship between continuous current and discharge
time is stable and absolute, but real batteries don't behave
exactly as this simple linear formula would indicate.
Therefore, when amp-hour capacity is given for a battery, it
is specified at either a given current, given time, or assumed
to be rated for a time period of 8 hours (if no limiting factor
IS given).
For example, an average automotive battery might have a
Capacity of about 70 amp-hours, specified at a current of 3.5
amps. This means that the amount of time this battery could
continuously supply a current of 3.5 amps to a load would
be 20 hours (70 amp-hours / 3.5 amps). But let's suppose
that a lower-resistance load were connected to that battery,
drawing 70 amps continuously. Our amp-hour equation tells
us that the battery should hold out for exactly 1 hour (70
amp-hours /70 amps), but this might not be true in real life.
With higher currents, the battery will dissipate more heat
across its internal resistance, which has the effect of altering
the chemical reactions taking place within. Chances are, the
battery would fully discharge some time before the
calculated time of 1 hour under this greater load.
Conversely, if a very light load (1 mA) were to be connected
to the battery, our equation would tell us that the battery
should provide power for 70,000 hours, or just under 8 years
(70 amp-hours / 1 milliamp), but the odds are that much of
the chemical energy in a real battery would have been
drained due to other factors (evaporation of electrolyte,
deterioration of electrodes, leakage current within battery)
long before 8 years had elapsed. Therefore, we must take
the amp-hour relationship as being an ideal approximation
of battery life, the amp-hour rating trusted only near the
specified current or timespan given by the manufacturer.
Some manufacturers will provide amp-hour derating factors
specifying reductions in total capacity at different levels of
current and/or temperature.
For secondary cells, the amp-hour rating provides a rule for
necessary charging time at any given level of charge
current. For example, the 70 amp-hour automotive battery
in the previous example should take 10 hours to charge from
a fully-discharged state at a constant charging current of 7
amps (70 amp-hours /7 amps).
Approximate amp-hour capacities of some common batteries
are given here:
e Typical automotive battery: 70 amp-hours @ 3.5A
(secondary cell)
e D-size carbon-zinc battery: 4.5 amp-hours @ 100 mA
(primary cell)
e 9 volt carbon-zinc battery: 400 milliamp-hours @ 8 mA
(primary cell)
As a battery discharges, not only does it diminish its internal
store of energy, but its internal resistance also increases (as
the electrolyte becomes less and less conductive), and its
open-circuit cell voltage decreases (as the chemicals
become more and more dilute). The most deceptive change
that a discharging battery exhibits is increased resistance.
The best check for a battery's condition is a voltage
measurement under load, while the battery is supplying a
substantial current through a circuit. Otherwise, a simple
voltmeter check across the terminals may falsely indicate a
healthy battery (adequate voltage) even though the internal
resistance has increased considerably. What constitutes a
“substantial current" is determined by the battery's design
parameters. A voltmeter check revealing too low of a
voltage, of course, would positively indicate a discharged
battery:
Fully charged battery:
Scenario for a fully charged battery
ons
ri arte 010 3 + Sass
Voltmeter indication: Voltmeter indication:
100 2 W) see
! 13.187 V
13.2¥ — -
32% a 2
No load Under load
Now, if the battery discharges a bit...
Scenario for a slightly discharged battery
sa
3.0 V¥ —
+ Voltmeter indication:
BboOV
+ leis ecard ieaia
Voltmeter indication:
$100.0 W) 12.381 V
+.
No load Under load
...and discharges a bit further...
Scenario for a moderately discharged battery
na , ee na
V Voltmeter indication:
+ vind beetle
Voltmeter indication:
ao
= = ea $100.0 W) 9.583 V
Lis = LS ¥ -
+.
No load Under load
.. and a bit further until its dead.
Scenario for a dead battery
~ Voltmeter indicati 00 3 ~ Voltmeter indicati
oltmeter indication: rn oltmeter indtation:
75¥ $1000 (V) SV
a
>
No load Under load
Notice how much better the battery's true condition is
revealed when its voltage is checked under load as opposed
to without a load. Does this mean that its pointless to check
a battery with just a voltmeter (no load)? Well, no. Ifa
simple voltmeter check reveals only 7.5 volts for a 13.2 volt
battery, then you know without a doubt that its dead.
However, if the voltmeter were to indicate 12.5 volts, it may
be near full charge or somewhat depleted -- you couldn't tell
without a load check. Bear in mind also that the resistance
used to place a battery under load must be rated for the
amount of power expected to be dissipated. For checking
large batteries such as an automobile (12 volt nominal)
lead-acid battery, this may mean a resistor with a power
rating of several hundred watts.
e REVIEW:
e The amp-hour is a unit of battery energy capacity, equal
to the amount of continuous current multiplied by the
discharge time, that a battery can supply before
exhausting its internal store of chemical energy.
; ; Amp-hour rating
Continuous current (in Amps) =
Charge/discharge time (in hours)
Amp-hour rating
Charge/discharge time (in hours) = . :
g g ; Continuous current (in Amps)
An amp-hour battery rating is only an approximation of
the battery's charge capacity, and should be trusted
only at the current level or time specified by the
manufacturer. Such a rating cannot be extrapolated for
very high currents or very long times with any accuracy.
Discharged batteries lose voltage and increase in
resistance. The best check for a dead battery isa
voltage test under load.
Special-purpose batteries
Back in the early days of electrical measurement
technology, a special type of battery known as a mercury
standard cell was popularly used as a voltage calibration
standard. The output of a mercury cell was 1.0183 to 1.0194
volts DC (depending on the specific design of cell), and was
extremely stable over time. Advertised drift was around
0.004 percent of rated voltage per year. Mercury standard
cells were sometimes known as Weston cells or cadmium
cells.
Mercury "standard" cell
glass bulb
Cdso,
cadmium
sulphate
solution
cadmium
sulphate solution
Hg2SO, Caso,
mercury cadmium amalgam
Unfortunately, mercury cells were rather intolerant of any
Current drain and could not even be measured with an
analog voltmeter without compromising accuracy.
Manufacturers typically called for no more than 0.1 mA of
current through the cell, and even that figure was
considered a momentary, or surge maximum! Consequently,
standard cells could only be measured with a potentiometric
(null-balance) device where current drain is almost zero.
Short-circuiting a mercury cell was prohibited, and once
short-circuited, the cell could never be relied upon again as
a standard device.
Mercury standard cells were also susceptible to slight
changes in voltage if physically or thermally disturbed. Two
different types of mercury standard cells were developed for
different calibration purposes: saturated and unsaturated.
Saturated standard cells provided the greatest voltage
stability over time, at the expense of thermal instability. In
other words, their voltage drifted very little with the passage
of time (just a few microvolts over the span of a decade!),
but tended to vary with changes in temperature (tens of
microvolts per degree Celsius). These cells functioned best
in temperature-controlled laboratory environments where
long-term stability is paramount. Unsaturated cells provided
thermal stability at the expense of stability over time, the
voltage remaining virtually constant with changes in
temperature but decreasing steadily by about 100 UV every
year. These cells functioned best as "field" calibration
devices where ambient temperature is not precisely
controlled. Nominal voltage for a saturated cell was 1.0186
volts, and 1.019 volts for an unsaturated cell.
Modern semiconductor voltage (zener diode regulator)
references have superseded standard cell batteries as
laboratory and field voltage standards.
A fascinating device closely related to primary-cell batteries
is the fuel cell, so-called because it harnesses the chemical
reaction of combustion to generate an electric current. The
process of chemical oxidation (oxygen ionically bonding
with other elements) is capable of producing an electron
flow between two electrodes just as well as any combination
of metals and electrolytes. A fuel cell can be thought of as a
battery with an externally supplied chemical energy source.
Hydrogen/Oxygen fuel cell
_ load ,
hydrogen in . n oxygen in
[ Hoo O,
tw]
e a
electrolyte
tps —+'}
8 O
| Ht —>
; O
-
membranes
water out
To date, the most successful fuel cells constructed are those
which run on hydrogen and oxygen, although much research
has been done on cells using hydrocarbon fuels. While
"burning" hydrogen, a fuel cell's only waste byproducts are
water and a small amount of heat. When operating on
carbon-containing fuels, carbon dioxide is also released as a
byproduct. Because the operating temperature of modern
fuel cells is far below that of normal combustion, no oxides
of nitrogen (NO,) are formed, making it far less polluting, all
other factors being equal.
The efficiency of energy conversion in a fuel cell from
chemical to electrical far exceeds the theoretical Carnot
efficiency limit of any internal-combustion engine, which is
an exciting prospect for power generation and hybrid
electric automobiles.
Another type of "battery" is the so/ar cell, a by-product of
the semiconductor revolution in electronics. The
photoelectric effect, whereby electrons are dislodged from
atoms under the influence of light, has been known in
physics for many decades, but it has only been with recent
advances in semiconductor technology that a device existed
capable of harnessing this effect to any practical degree.
Conversion efficiencies for silicon solar cells are still quite
low, but their benefits as power sources are legion: no
moving parts, no noise, no waste products or pollution (aside
from the manufacture of solar cells, which is still a fairly
"dirty" industry), and indefinite life.
Solar cell
; a
wires thin, round vafer of
crystalline silicon
1¢
—
schematic symbol
Specific cost of solar cell technology (dollars per kilowatt) is
still very high, with little prospect of significant decrease
barring some kind of revolutionary advance in technology.
Unlike electronic components made from semiconductor
material, which can be made smaller and smaller with less
scrap as a result of better quality control, a single solar cell
still takes the same amount of ultra-pure silicon to make as
it did thirty years ago. Superior quality control fails to yield
the same production gain seen in the manufacture of chips
and transistors (where isolated specks of impurity can ruin
many microscopic circuits on one wafer of silicon). The same
number of impure inclusions does little to impact the overall
efficiency of a 3-inch solar cell.
Yet another type of special-purpose "battery" is the chemical
detection cell. Simply put, these cells chemically react with
specific substances in the air to create a voltage directly
proportional to the concentration of that substance. A
common application for a chemical detection cell is in the
detection and measurement of oxygen concentration. Many
portable oxygen analyzers have been designed around
these small cells. Cell chemistry must be designed to match
the specific substance(s) to be detected, and the cells do
tend to "wear out," as their electrode materials deplete or
become contaminated with use.
e REVIEW:
e mercury standard cells are special types of batteries
which were once used as voltage calibration standards
before the advent of precision semiconductor reference
devices.
A fuel cellis a kind of battery that uses a combustible
fuel and oxidizer as reactants to generate electricity.
They are promising sources of electrical power in the
future, "burning" fuels with very low emissions.
A solar cell uses ambient light energy to motivate
electrons from one electrode to the other, producing
voltage (and current, providing an external circuit).
e A chemical detection cell is a special type of voltaic cell
which produces voltage proportional to the
concentration of an applied substance (usually a specific
gas in ambient air).
Practical considerations
When connecting batteries together to form larger "banks"
(a battery of batteries?), the constituent batteries must be
matched to each other so as to not cause problems. First we
will consider connecting batteries in series for greater
voltage:
FF || —IIF
load
We know that the current is equal at all points in a series
circuit, so whatever amount of current there is in any one of
the series-connected batteries must be the same for all the
others as well. For this reason, each battery must have the
same amp-hour rating, or else some of the batteries will
become depleted sooner than others, compromising the
capacity of the whole bank. Please note that the total amp-
hour capacity of this series battery bank is not affected by
the number of batteries.
Next, we will consider connecting batteries in parallel for
greater current capacity (lower internal resistance), or
greater amp-hour capacity:
We know that the voltage is equal across all branches of a
parallel circuit, so we must be sure that these batteries are
of equal voltage. If not, we will have relatively large currents
circulating from one battery through another, the higher-
voltage batteries overpowering the lower-voltage batteries.
This is not good.
On this same theme, we must be sure that any overcurrent
protection (circuit breakers or fuses) are installed in such a
way as to be effective. For our series battery bank, one fuse
will suffice to protect the wiring from excessive current,
since any break in a series circuit stops current through all
parts of the circuit:
4] |
fuse
load
With a parallel battery bank, one fuse is adequate for
protecting the wiring against load overcurrent (between the
parallel-connected batteries and the load), but we have
other concerns to protect against as well. Batteries have
been known to internally short-circuit, due to electrode
separator failure, causing a problem not unlike that where
batteries of unequal voltage are connected in parallel: the
good batteries will overpower the failed (lower voltage)
battery, causing relatively large currents within the
batteries' connecting wires. To guard against this
eventuality, we should protect each and every battery
against overcurrent with individual battery fuses, in addition
to the load fuse:
main
fuse
+
load
When dealing with secondary-cell batteries, particular
attention must be paid to the method and timing of
charging. Different types and construction of batteries have
different charging needs, and the manufacturer's
recommendations are probably the best guide to follow
when designing or maintaining a system. Two distinct
concerns of battery charging are cycling and overcharging.
Cycling refers to the process of charging a battery to a "full"
condition and then discharging it to a lower state. All
batteries have a finite (limited) cycle life, and the allowable
"depth" of cycle (how far it should be discharged at any
time) varies from design to design. Overcharging is the
condition where current continues to be forced backwards
through a secondary cell beyond the point where the cell
has reached full charge. With lead-acid cells in particular,
overcharging leads to electrolysis of the water ("boiling" the
water out of the battery) and shortened life.
Any battery containing water in the electrolyte is subject to
the production of hydrogen gas due to electrolysis. This is
especially true for overcharged lead-acid cells, but not
exclusive to that type. Hydrogen is an extremely flammable
gas (especially in the presence of free oxygen created by the
same electrolysis process), odorless and colorless. Such
batteries pose an explosion threat even under normal
operating conditions, and must be treated with respect. The
author has been a firsthand witness to a lead-acid battery
explosion, where a spark created by the removal of a battery
charger (small DC power supply) from an automotive battery
ignited hydrogen gas within the battery case, blowing the
top off the battery and splashing sulfuric acid everywhere.
This occurred in a high school automotive shop, no less. If it
were not for all the students nearby wearing safety glasses
and buttoned-collar overalls, significant injury could have
occurred.
When connecting and disconnecting charging equipment to
a battery, always make the last connection (or first
disconnection) at a location away from the battery itself
(such as at a point on one of the battery cables, at least a
foot away from the battery), so that any resultant spark has
little or no chance of igniting hydrogen gas.
In large, permanently installed battery banks, batteries are
equipped with vent caps above each cell, and hydrogen gas
is vented outside of the battery room through hoods
immediately over the batteries. Hydrogen gas is very light
and rises quickly. The greatest danger is when it is allowed
to accumulate in an area, awaiting ignition.
More modern lead-acid battery designs are sealed,
fabricated to re-combine the electrolyzed hydrogen and
oxygen back into water, inside the battery case itself.
Adequate ventilation might still be a good idea, just in case
a battery were to develop a leak. [JOM]
e REVIEW:
e Connecting batteries in series increases voltage, but
does not increase overall amp-hour capacity.
e All batteries in a series bank must have the same amp-
hour rating.
e Connecting batteries in parallel increases total current
Capacity by decreasing total resistance, and it also
increases overall amp-hour capacity.
e All batteries in a parallel bank must have the same
voltage rating.
e Batteries can be damaged by excessive cycling and
overcharging.
e Water-based electrolyte batteries are capable of
generating explosive hydrogen gas, which must not be
allowed to accumulate in an area.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
John Anhalt (December 2008): Updated Lead-acid cell
chemistry..
Bibliography
1. [DOE]“DOE Handbook, Primer on Lead-Acid Storage
Batteries”, DOE-HDBK-1084-95, September 1995, pp.
13. at
http://www.hss.energy.gov/NuclearSafety/techstds/stand
ard/hdbk1084/hdbk1084.pdf
2. [JOM]Robert Nelson, “The Basic Chemistry of Gas
Recombination in Lead-Acid Batteries”, JOM, 53 (1)
(2001), pp. 28-33. at
http://www.tms.org/pubs/journals/JOM/0101/Nelson-
0101.html
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
|| 4] l_—
—| | +4/l—
Lessons In Electric Circuits
-- Volume |
Chapter 12
PHYSICS OF CONDUCTORS
AND INSULATORS
e Introduction
e Conductor size
e Conductor ampacity
e Fuses
e Specific resistance
e Temperature coefficient of resistance
e Superconductivity
e Insulator breakdown voltage
e Data
e Contributors
Introduction
By now you should be well aware of the correlation between
electrical conductivity and certain types of materials. Those
materials allowing for easy passage of free electrons are
called conductors, while those materials impeding the
passage of free electrons are called insulators.
Unfortunately, the scientific theories explaining why certain
materials conduct and others don't are quite complex, rooted
in quantum mechanical explanations in how electrons are
arranged around the nuclei of atoms. Contrary to the well-
known "planetary" model of electrons whirling around an
atom's nucleus as well-defined chunks of matter in circular or
elliptical orbits, electrons in "orbit" don't really act like pieces
of matter at all. Rather, they exhibit the characteristics of
both particle and wave, their behavior constrained by
placement within distinct zones around the nucleus referred
to as "Shells" and "subshells." Electrons can occupy these
zones only in a limited range of energies depending on the
particular zone and how occupied that zone is with other
electrons. If electrons really did act like tiny planets held in
orbit around the nucleus by electrostatic attraction, their
actions described by the same laws describing the motions of
real planets, there could be no real distinction between
conductors and insulators, and chemical bonds between
atoms would not exist in the way they do now. It is the
discrete, "quantitized" nature of electron energy and
placement described by quantum physics that gives these
phenomena their regularity.
When an electron is free to assume higher energy states
around an atom's nucleus (due to its placement ina
particular "shell"), it may be free to break away from the
atom and comprise part of an electric current through the
substance. If the quantum limitations imposed on an electron
deny it this freedom, however, the electron is considered to
be "bound" and cannot break away (at least not easily) to
constitute a current. The former scenario is typical of
conducting materials, while the latter is typical of insulating
materials.
Some textbooks will tell you that an element's conductivity
or nonconductivity is exclusively determined by the number
of electrons residing in the atoms' outer "shell" (called the
valence shell), but this is an oversimplification, as any
examination of conductivity versus valence electrons in a
table of elements will confirm. The true complexity of the
situation is further revealed when the conductivity of
molecules (collections of atoms bound to one another by
electron activity) is considered.
A good example of this is the element carbon, which
comprises materials of vastly differing conductivity: graphite
and diamond. Graphite is a fair conductor of electricity, while
diamond is practically an insulator (stranger yet, it is
technically classified as a semiconductor, which in its pure
form acts as an insulator, but can conduct under high
temperatures and/or the influence of impurities). Both
graphite and diamond are composed of the exact same types
of atoms: carbon, with 6 protons, 6 neutrons and 6 electrons
each. The fundamental difference between graphite and
diamond being that graphite molecules are flat groupings of
carbon atoms while diamond molecules are tetrahedral
(pyramid-shaped) groupings of carbon atoms.
If atoms of carbon are joined to other types of atoms to form
compounds, electrical conductivity becomes altered once
again. Silicon carbide, a compound of the elements silicon
and carbon, exhibits nonlinear behavior: its electrical
resistance decreases with increases in applied voltage!
Hydrocarbon compounds (such as the molecules found in
oils) tend to be very good insulators. As you can see, a
simple count of valence electrons in an atom is a poor
indicator of a substance's electrical conductivity.
All metallic elements are good conductors of electricity, due
to the way the atoms bond with each other. The electrons of
the atoms comprising a mass of metal are so uninhibited in
their allowable energy states that they float freely between
the different nuclei in the substance, readily motivated by
any electric field. The electrons are so mobile, in fact, that
they are sometimes described by scientists as an e/ectron
gas, or even an electron sea in which the atomic nuclei rest.
This electron mobility accounts for some of the other
common properties of metals: good heat conductivity,
malleability and ductility (easily formed into different
shapes), and a lustrous finish when pure.
Thankfully, the physics behind all this is mostly irrelevant to
our purposes here. Suffice it to say that some materials are
good conductors, some are poor conductors, and some are in
between. For now it is good enough to simply understand
that these distinctions are determined by the configuration
of the electrons around the constituent atoms of the material.
An important step in getting electricity to do our bidding is
to be able to construct paths for electrons to flow with
controlled amounts of resistance. It is also vitally important
that we be able to prevent electrons from flowing where we
don't want them to, by using insulating materials. However,
not all conductors are the same, and neither are all
insulators. We need to understand some of the
characteristics of common conductors and insulators, and be
able to apply these characteristics to specific applications.
Almost all conductors possess a certain, measurable
resistance (special types of materials called superconductors
possess absolutely no electrical resistance, but these are not
ordinary materials, and they must be held in special
conditions in order to be super conductive). Typically, we
assume the resistance of the conductors in a circuit to be
zero, and we expect that current passes through them
without producing any appreciable voltage drop. In reality,
however, there will almost always be a voltage drop along
the (normal) conductive pathways of an electric circuit,
whether we want a voltage drop to be there or not:
wire resistance
Source —
Load something less than
source Voltage
wire resistance
In order to calculate what these voltage drops will be in any
particular circuit, we must be able to ascertain the resistance
of ordinary wire, knowing the wire size and diameter. Some of
the following sections of this chapter will address the details
of doing this.
e REVIEW:
e Electrical conductivity of a material is determined by the
configuration of electrons in that materials atoms and
molecules (groups of bonded atoms).
e All normal conductors possess resistance to some degree.
e Electrons flowing through a conductor with (any)
resistance will produce some amount of voltage drop
across the length of that conductor.
Conductor size
It should be common-sense knowledge that liquids flow
through large-diameter pipes easier than they do through
small-diameter pipes (if you would like a practical
illustration, try drinking a liquid through straws of different
diameters). The same general principle holds for the flow of
electrons through conductors: the broader the cross-sectional
area (thickness) of the conductor, the more room for
electrons to flow, and consequently, the easier it is for flow to
occur (less resistance).
Electrical wire is usually round in cross-section (although
there are some unique exceptions to this rule), and comes in
two basic varieties: solid and stranded. Solid copper wire is
just as it sounds: a single, solid strand of copper the whole
length of the wire. Stranded wire is composed of smaller
strands of solid copper wire twisted together to form a single,
larger conductor. The greatest benefit of stranded wire is its
mechanical flexibility, being able to withstand repeated
bending and twisting much better than solid copper (which
tends to fatigue and break after time).
Wire size can be measured in several ways. We could speak
of a wire's diameter, but since its really the cross-sectional
area that matters most regarding the flow of electrons, we
are better off designating wire size in terms of area.
Cross-sectional area
end-view of is 0.008155 square inches
solid round wire
~<_——. 0.1019
inches
The wire cross-section picture shown above is, of course, not
drawn to scale. The diameter is shown as being 0.1019
inches. Calculating the area of the cross-section with the
formula Area = mtr?, we get an area of 0.008155 square
inches:
A=tr
0.1019 inches \
A=(3.1416) —
A = 0.008155 square inches
These are fairly small numbers to work with, so wire sizes are
often expressed in measures of thousandths-of-an-inch, or
mils. For the illustrated example, we would say that the
diameter of the wire was 101.9 mils (0.1019 inch times
1000). We could also, if we wanted, express the area of the
wire in the unit of square mils, calculating that value with the
same circle-area formula, Area = Ttr?:
Cross-sectional area
end-view of is 8155.27 square mils
solid round wire
«— 101.9
mils
A=tr
101.9 mils\
A = (3.1416); ——$——
A = 8155.27 square mils
However, electricians and others frequently concerned with
wire size use another unit of area measurement tailored
specifically for wire's circular cross-section. This special unit
is called the circular mil (sometimes abbreviated cmi/). The
sole purpose for having this special unit of measurement is to
eliminate the need to invoke the factor tt (3.1415927 ...) in
the formula for calculating area, plus the need to figure wire
radius when you've been given diameter. The formula for
calculating the circular-mil area of a circular wire is very
simple:
Circular Wire Area Formula
Aad
Because this is a unit of area measurement, the
mathematical power of 2 is still in effect (doubling the width
of a circle will a/ways quadruple its area, no matter what
units are used, or if the width of that circle is expressed in
terms of radius or diameter). To illustrate the difference
between measurements in square mils and measurements in
circular mils, | will compare a circle with a square, showing
the area of each shape in both unit measures:
Area = 0.7854 square mils Area = 1 square mil
Area = 1 circular mil Area = 1.273 circular mils
- Li
And for another size of wire:
Area = 3.1416 square mils Area = 4 square mils
Area = 4 circular mils Area = 5.0930 circular mils
ol ree
Obviously, the circle of a given diameter has less cross-
sectional area than a square of width and height equal to the
circle's diameter: both units of area measurement reflect
that. However, it should be clear that the unit of "square mil"
is really tailored for the convenient determination of a
square's area, while "circular mil" is tailored for the
convenient determination of a circle's area: the respective
formula for each is simpler to work with. It must be
understood that both units are valid for measuring the area
of a shape, no matter what shape that may be. The
conversion between circular mils and square mils is a simple
ratio: there are m (3.1415927 ...) square mils to every 4
circular mils.
Another measure of cross-sectional wire area is the gauge.
The gauge scale is based on whole numbers rather than
fractional or decimal inches. The larger the gauge number,
the skinnier the wire; the smaller the gauge number, the
fatter the wire. For those acquainted with shotguns, this
inversely-proportional measurement scale should sound
familiar.
The table at the end of this section equates gauge with inch
diameter, circular mils, and square inches for solid wire. The
larger sizes of wire reach an end of the common gauge scale
(which naturally tops out at a value of 1), and are
represented by a series of zeros. "3/0" is another way to
represent "000," and is pronounced "triple-ought." Again,
those acquainted with shotguns should recognize the
terminology, strange as it may sound. To make matters even
more confusing, there is more than one gauge "standard" in
use around the world. For electrical conductor sizing, the
American Wire Gauge (AWG), also Known as the Brown and
Sharpe (B&S) gauge, is the measurement system of choice.
In Canada and Great Britain, the British Standard Wire Gauge
(SWG) is the legal measurement system for electrical
conductors. Other wire gauge systems exist in the world for
classifying wire diameter, such as the Stubs steel wire gauge
and the Stee/ Music Wire Gauge (MWG), but these
measurement systems apply to non-electrical wire use.
The American Wire Gauge (AWG) measurement system,
despite its oddities, was designed with a purpose: for every
three steps in the gauge scale, wire area (and weight per unit
length) approximately doubles. This is a handy rule to
remember when making rough wire size estimations!
For very large wire sizes (fatter than 4/0), the wire gauge
system is typically abandoned for cross-sectional area
measurement in thousands of circular mils (MCM), borrowing
the old Roman numeral "M" to denote a multiple of
"thousand" in front of "CM" for "circular mils." The following
table of wire sizes does not show any sizes bigger than 4/0
gauge, because so/id copper wire becomes impractical to
handle at those sizes. Stranded wire construction is favored,
instead.
Soild copper wire table: below
Soild copper wire table:
[Size |Diameter|Cross-sectional area
[AWG| inches | cir. mils sq. inches Ilb/1000 ft’
4/0 |0.4600 {211,600 .1662 40.5
3/0 0.4096 {167,800 1318 07.9
2/0 0.3648 133,100 1045 02.8
1/0 [0.3249 [105,500 08289 19.5
1 [0.2893 [83,690 06573 53.5
2 0.2576 |66,370 05213 00.9
3 0.2294 [52,630 04134 59.3
i
om
iii
@jo20a3_an7ao
5 0.1819 (83,100
6 0.1620 26,250
70.1443 20,820.
7 0.1443 20,820.
8 0.1285 16510
90.1144 13,090
100.1019 10,360
11 0.09074 8,234
12 0.08081 6,530
13 0.07196 5,178
14 0.06408 (4,107
15 (0.05707 8.257
16 _0.05082_2,583
17 0.04526 2,048
18 0.04030 1,624
19 0.03589 1,268
20 (0.03196 [1,022
21 0.02846 [610.1
22 0.02535 6425
23 0.02257 509.5
23 0.02257 509.5.
24 0.02010 404.0
25 (0.01790 [3204
26 0.01594 2541
27 0.01420 2015
28 0.01264 159.8
29 (0.01126 1267
30 (0.01003_[100.5.
.03278
.02600
.02062
.01635
.01635
.01297
.01028
.008155
.006467
.005129
.004067
.003225
.002558
.002028
.001609
.001276
.001012
.0008023
.0006363
.0005046
.0004001
.0004001
.0003173
.0002517
.0001996
.0001583
.0001255
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For some high-current applications, conductor sizes beyond
the practical size limit of round wire are required. In these
instances, thick bars of solid metal called busbars are used
as conductors. Busbars are usually made of copper or
aluminum, and are most often uninsulated. They are
physically supported away from whatever framework or
structure is holding them by insulator standoff mounts.
Although a square or rectangular cross-section is very
common for busbar shape, other shapes are used as well.
Cross-sectional area for busbars is typically rated in terms of
circular mils (even for square and rectangular bars!), most
likely for the convenience of being able to directly equate
busbar size with round wire.
¢ REVIEW:
Electrons flow through large-diameter wires easier than
small-diameter wires, due to the greater cross-sectional
area they have in which to move.
Rather than measure small wire sizes in inches, the unit
of "mil" (1/1000 of an inch) is often employed.
The cross-sectional area of a wire can be expressed in
terms of square units (Square inches or square mils),
circular mils, or "gauge" scale.
Calculating square-unit wire area for a circular wire
involves the circle area formula:
A=ar (Square units)
Calculating circular-mil wire area for a circular wire is
much simpler, due to the fact that the unit of "circular
mil" was sized just for this purpose: to eliminate the "pi"
and the d/2 (radius) factors in the formula.
A= (Circular units)
There are mt (3.1416) square mils for every 4 circular mils.
The gauge system of wire sizing is based on whole
numbers, larger numbers representing smaller-area wires
and vice versa. Wires thicker than 1 gauge are
represented by zeros: 0, 00, 000, and 0000 (spoken
"single-ought," "double-ought," "triple-ought," and
“quadruple-ought."
Very large wire sizes are rated in thousands of circular
mils (MCM's), typical for busbars and wire sizes beyond
4/0.
Busbars are solid bars of copper or aluminum used in
high-current circuit construction. Connections made to
busbars are usually welded or bolted, and the busbars
are often bare (uninsulated), Supported away from metal
frames through the use of insulating standoffs.
Conductor ampacity
The smaller the wire, the greater the resistance for any given
length, all other factors being equal. A wire with greater
resistance will dissipate a greater amount of heat energy for
any given amount of current, the power being equal to
P=I2R.
Dissipated power in a resistance manifests itself in the form
of heat, and excessive heat can be damaging to a wire (not
to mention objects near the wire!), especially considering the
fact that most wires are insulated with a plastic or rubber
coating, which can melt and burn. Thin wires will, therefore,
tolerate less current than thick wires, all other factors being
equal. A conductor's current-carrying limit is known as its
ampacity.
Primarily for reasons of safety, certain standards for electrical
wiring have been established within the United States, and
are specified in the National Electrical Code (NEC). Typical
NEC wire ampacity tables will show allowable maximum
currents for different sizes and applications of wire. Though
the melting point of copper theoretically imposes a limit on
wire ampacity, the materials commonly employed for
insulating conductors melt at temperatures far below the
melting point of copper, and so practical ampacity ratings
are based on the thermal limits of the insulation. Voltage
dropped as a result of excessive wire resistance Is also a
factor in sizing conductors for their use in circuits, but this
consideration is better assessed through more complex
means (which we will cover in this chapter). A table derived
from an NEC listing is shown for example:
Ampacities of copper wire: below
Ampacities of copper wire, in free air at 30° C:
INSULATION
TYPE:
| | RUW, T THW, THWN FEP, FEPB
Tw |—sRUH__| THHN, XHHW
Current Rating) Current Rating |Current Ratin
AWG @ 60 eee @ 75 degrees C @ 90 es
ee
as fis +| ies
16 le. SSSCS~—~—SS
ih
* = estimated values; normally, these small wire sizes are not
manufactured with these insulation types, above.
Notice the substantial ampacity differences between same-
size wires with different types of insulation. This is due,
again, to the thermal limits (60°, 75°, 90°) of each type of
insulation material.
These ampacity ratings are given for copper conductors in
"free air" (maximum typical air circulation), as opposed to
wires placed in conduit or wire trays. As you will notice, the
table fails to specify ampacities for small wire sizes. This is
because the NEC concerns itself primarily with power wiring
(large currents, big wires) rather than with wires common to
low-current electronic work.
There is meaning in the letter sequences used to identify
conductor types, and these letters usually refer to properties
of the conductor's insulating layer(s). Some of these letters
symbolize individual properties of the wire while others are
simply abbreviations. For example, the letter "T" by itself
means "thermoplastic" as an insulation material, as in "TW"
or "THHN." However, the three-letter combination "MTW" is
an abbreviation for Machine Too! Wire, a type of wire whose
insulation is made to be flexible for use in machines
experiencing significant motion or vibration.
Wire insulation codes: below
Soild copper wire table:
2
Code Insulation Material
IC | Cotton
ol
Fluorinated Ethylene Propylene
Mineral (magnesium oxide)
erfluoroalkoxy
Rubber (sometimes Neoprene) __|
ilicone "rubber"
ilicone-asbestos
hermoplastic-asbestos
olytetrafluoroethylene ("Teflon")
Modified ethylene tetrafluoroethylene|
5 degrees Celsius
Outer covering ("jacket")
ylon
Wet
Ul] </| 7
:
|
we
WO
Therefore, a "THWN" conductor has Thermoplastic insulation,
is Heat resistant to 75° Celsius, is rated for Wet conditions,
and comes with a Nylon outer jacketing.
Letter codes like these are only used for general-purpose
wires such as those used in households and businesses. For
high-power applications and/or severe service conditions, the
complexity of conductor technology defies classification
according to a few letter codes. Overhead power line
conductors are typically bare metal, suspended from towers
by glass, porcelain, or ceramic mounts known as insulators.
Even so, the actual construction of the wire to withstand
physical forces both static (dead weight) and dynamic (wind)
loading can be complex, with multiple layers and different
types of metals wound together to form a single conductor.
Large, underground power conductors are sometimes
insulated by paper, then enclosed in a steel pipe filled with
pressurized nitrogen or oil to prevent water intrusion. Such
conductors require support equipment to maintain fluid
pressure throughout the pipe.
Other insulating materials find use in small-scale
applications. For instance, the small-diameter wire used to
make electromagnets (coils producing a magnetic field from
the flow of electrons) are often insulated with a thin layer of
enamel. The enamel is an excellent insulating material and is
very thin, allowing many "turns" of wire to be wound in a
small space.
e REVIEW:
e Wire resistance creates heat in operating circuits. This
heat is a potential fire ignition hazard.
e Skinny wires have a lower allowable current ("ampacity")
than fat wires, due to their greater resistance per unit
length, and consequently greater heat generation per
unit current.
e The National Electrical Code (NEC) specifies ampacities
for power wiring based on allowable insulation
temperature and wire application.
Fuses
Normally, the ampacity rating of a conductor is a circuit
design limit never to be intentionally exceeded, but there is
an application where ampacity exceedence is expected: in
the case of fuses.
A fuse is nothing more than a short length of wire designed
to melt and separate in the event of excessive current. Fuses
are always connected in series with the component(s) to be
protected from overcurrent, so that when the fuse blows
(opens) it will open the entire circuit and stop current
through the component(s). A fuse connected in one branch of
a parallel circuit, of course, would not affect current through
any of the other branches.
Normally, the thin piece of fuse wire is contained within a
safety sheath to minimize hazards of arc blast if the wire
burns open with violent force, as can happen in the case of
severe overcurrents. In the case of small automotive fuses,
the sheath is transparent so that the fusible element can be
visually inspected. Residential wiring used to commonly
employ screw-in fuses with glass bodies and a thin, narrow
metal foil strip in the middle. A photograph showing both
types of fuses is shown here:
a
Glass cartridge type fuses
Screw-im type fuse
Cartridge type fuses are popular in automotive applications,
and in industrial applications when constructed with sheath
materials other than glass. Because fuses are designed to
"fail" open when their current rating is exceeded, they are
typically designed to be replaced easily in a circuit. This
means they will be inserted into some type of holder rather
than being directly soldered or bolted to the circuit
conductors. The following is a photograph showing a couple
of glass cartridge fuses in a multi-fuse holder:
The fuses are held by spring metal clips, the clips themselves
being permanently connected to the circuit conductors. The
base material of the fuse holder (or fuse block as they are
sometimes called) is chosen to be a good insulator.
Another type of fuse holder for cartridge-type fuses is
commonly used for installation in equipment control panels,
where it is desirable to conceal all electrical contact points
from human contact. Unlike the fuse block just shown, where
all the metal clips are openly exposed, this type of fuse
holder completely encloses the fuse in an insulating housing:
Disassembled
CARTRIDGE FUSE HOLDER
Assembled
—
The most common device in use for overcurrent protection in
high-current circuits today is the circuit breaker. Circuit
breakers are specially designed switches that automatically
open to stop current in the event of an overcurrent condition.
Small circuit breakers, such as those used in residential,
commercial and light industrial service are thermally
operated. They contain a bimetallic strip (a thin strip of two
metals bonded back-to-back) carrying circuit current, which
bends when heated. When enough force is generated by the
bimetallic strip (due to overcurrent heating of the strip), the
trip mechanism is actuated and the breaker will open. Larger
circuit breakers are automatically actuated by the strength of
the magnetic field produced by current-carrying conductors
within the breaker, or can be triggered to trip by external
devices monitoring the circuit current (those devices being
called protective relays).
Because circuit breakers don't fail when subjected to
overcurrent conditions -- rather, they merely open and can be
re-closed by moving a lever -- they are more likely to be
found connected to a circuit in a more permanent manner
than fuses. A photograph of a small circuit breaker is shown
here:
a rl
From outside appearances, it looks like nothing more than a
switch. Indeed, it could be used as such. However, its true
function is to operate as an overcurrent protection device.
It should be noted that some automobiles use inexpensive
devices known as fusible links for overcurrent protection in
the battery charging circuit, due to the expense of a
properly-rated fuse and holder. A fusible link is a primitive
fuse, being nothing more than a short piece of rubber-
insulated wire designed to melt open in the event of
overcurrent, with no hard sheathing of any kind. Such crude
and potentially dangerous devices are never used in industry
or even residential power use, mainly due to the greater
voltage and current levels encountered. As far as this author
is concerned, their application even in automotive circuits is
questionable.
The electrical schematic drawing symbol for a fuse Is an S-
Shaped curve:
Fuse
Fuses are primarily rated, as one might expect, in the unit for
current: amps. Although their operation depends on the self-
generation of heat under conditions of excessive current by
means of the fuse's own electrical resistance, they are
engineered to contribute a negligible amount of extra
resistance to the circuits they protect. This is largely
accomplished by making the fuse wire as short as is
practically possible. Just as a normal wire's ampacity is not
related to its length (10-gauge solid copper wire will handle
40 amps of current in free air, regardless of how long or short
of a piece it is), a fuse wire of certain material and gauge will
blow at a certain current no matter how long it is. Since
length is not a factor in current rating, the shorter it can be
made, the less resistance it will have end-to-end.
However, the fuse designer also has to consider what
happens after a fuse blows: the melted ends of the once-
continuous wire will be separated by an air gap, with full
supply voltage between the ends. If the fuse isn't made long
enough on a high-voltage circuit, a soark may be able to
jump from one of the melted wire ends to the other,
completing the circuit again:
480 V
drop
>
480 V
Load
When the fuse "blows," full
supply voltage will be dropped
across it and there will be no
current in the circuit.
vatage | Sel «
vatage | Lal
Load
480 V —
If the voltage across the blown
fuse is high enough, a spark may
jump the gap, allowing some
current in the circuit. THIS WOULD
NOT BE GOOD!!!
Consequently, fuses are rated in terms of their voltage
Capacity as well as the current level at which they will blow.
Some large industrial fuses have replaceable wire elements,
to reduce the expense. The body of the fuse is an opaque,
reusable cartridge, shielding the fuse wire from exposure and
shielding surrounding objects from the fuse wire.
There's more to the current rating of a fuse than a single
number. If a current of 35 amps is sent through a 30 amp
fuse, it may blow suddenly or delay before blowing,
depending on other aspects of its design. Some fuses are
intended to blow very fast, while others are designed for
more modest "opening" times, or even for a delayed action
depending on the application. The latter fuses are sometimes
called s/ow-b/ow fuses due to their intentional time-delay
characteristics.
A classic example of a slow-blow fuse application is in electric
motor protection, where /nrush currents of up to ten times
normal operating current are commonly experienced every
time the motor is started from a dead stop. If fast-blowing
fuses were to be used in an application like this, the motor
could never get started because the normal inrush current
levels would blow the fuse(s) immediately! The design of a
slow-blow fuse is such that the fuse element has more mass
(but no more ampacity) than an equivalent fast-blow fuse,
meaning that it will heat up slower (but to the same ultimate
temperature) for any given amount of current.
On the other end of the fuse action spectrum, there are so-
called semiconductor fuses designed to open very quickly in
the event of an overcurrent condition. Semiconductor
devices such as transistors tend to be especially intolerant of
overcurrent conditions, and as such require fast-acting
protection against overcurrents in high-power applications.
Fuses are always supposed to be placed on the "hot" side of
the load in systems that are grounded. The intent of this is
for the load to be completely de-energized in all respects
after the fuse opens. To see the difference between fusing the
“hot" side versus the "neutral" side of a load, compare these
two circuits:
"Hot”
- blown fuse
no voltage between either side =
of load and ground
"Neutral"
- voltage present between either side
of load and ground!
In either case, the fuse successfully interrupted current to
the load, but the lower circuit fails to interrupt potentially
dangerous voltage from either side of the load to ground,
where a person might be standing. The first circuit design is
much safer.
As it was said before, fuses are not the only type of
overcurrent protection device in use. Switch-like devices
called circuit breakers are often (and more commonly) used
to open circuits with excessive current, their popularity due
to the fact that they don't destroy themselves in the process
of breaking the circuit as fuses do. In any case, though,
placement of the overcurrent protection device in a circuit
will follow the same general guidelines listed above: namely,
to "fuse" the side of the power supply not connected to
ground.
Although overcurrent protection placement in a circuit may
determine the relative shock hazard of that circuit under
various conditions, it must be understood that such devices
were never intended to guard against electric shock. Neither
fuses nor circuit breakers were designed to open in the event
of a person getting shocked; rather, they are intended to
open only under conditions of potential conductor
overheating. Overcurrent devices primarily protect the
conductors of a circuit from overtemperature damage (and
the fire hazards associated with overly hot conductors), and
secondarily protect specific pieces of equipment such as
loads and generators (some fast-acting fuses are designed to
protect electronic devices particularly susceptible to current
surges). Since the current levels necessary for electric shock
or electrocution are much lower than the normal current
levels of common power loads, a condition of overcurrent is
not indicative of shock occurring. There are other devices
designed to detect certain shock conditions (ground-fault
detectors being the most popular), but these devices strictly
serve that one purpose and are uninvolved with protection of
the conductors against overheating.
e REVIEW:
e A fuseis a small, thin conductor designed to melt and
separate into two pieces for the purpose of breaking a
circuit in the event of excessive current.
e A circuit breaker is a specially designed switch that
automatically opens to interrupt circuit current in the
event of an overcurrent condition. They can be "tripped"
(opened) thermally, by magnetic fields, or by external
devices called "protective relays," depending on the
design of breaker, its size, and the application.
e Fuses are primarily rated in terms of maximum current,
but are also rated in terms of how much voltage drop
they will safely withstand after interrupting a circuit.
e Fuses can be designed to blow fast, slow, or anywhere in
between for the same maximum level of current.
e The best place to install a fuse in a grounded power
system is on the ungrounded conductor path to the load.
That way, when the fuse blows there will only be the
grounded (safe) conductor still connected to the load,
making it safer for people to be around.
Specific resistance
Conductor ampacity rating is a crude assessment of
resistance based on the potential for current to create a fire
hazard. However, we may come across situations where the
voltage drop created by wire resistance in a circuit poses
concerns other than fire avoidance. For instance, we may be
designing a circuit where voltage across a component is
critical, and must not fall below a certain limit. If this is the
case, the voltage drops resulting from wire resistance may
Cause an engineering problem while being well within safe
(fire) limits of ampacity:
— 2300 feet ———
wire resistance
Load
(requires at least 220 V)
wire resistance
If the load in the above circuit will not tolerate less than 220
volts, given a source voltage of 230 volts, then we'd better
be sure that the wiring doesn't drop more than 10 volts along
the way. Counting both the supply and return conductors of
this circuit, this leaves a maximum tolerable drop of 5 volts
along the length of each wire. Using Ohm's Law (R=E/I), we
can determine the maximum allowable resistance for each
piece of wire:
E
R= —
1
= 5V
25 A
R=0.2Q9
We know that the wire length is 2300 feet for each piece of
wire, but how do we determine the amount of resistance for a
specific size and length of wire? To do that, we need another
formula:
5=—-—
This formula relates the resistance of a conductor with its
specific resistance (the Greek letter "rho" (p), which looks
similar to a lower-case letter "p"), its length ("I"), and its
cross-sectional area ("A"). Notice that with the length
variable on the top of the fraction, the resistance value
increases as the length increases (analogy: it is more difficult
to force liquid through a long pipe than a short one), and
decreases as cross-sectional area increases (analogy: liquid
flows easier through a fat pipe than through a skinny one).
Specific resistance is a constant for the type of conductor
material being calculated.
The specific resistances of several conductive materials can
be found in the following table. We find copper near the
bottom of the table, second only to silver in having low
specific resistance (good conductivity):
Specific resistance table: below
Specific resistance at 20° C:
Nichrome [Alloy
Nichrome V_ |Alloy
Manganin [Alloy
Constantan [Alloy
Stee [Alloy
Element
Element
Element
Element
Element
Tungsten Element
13
50
90
72.97
00
3.16
7.81
1.69
5.49
2.42
1.76
nil
TMI
Aluminum [Element (15.94
Gold Element 13.32
Copper___Element [10.09
Silver [Element __ 9.546
= = Steel alloy at 99.5% iron, 0.5%
carbon
il
Notice that the figures for specific resistance in the above
table are given in the very strange unit of "ohms-cmil/ft" (Q-
cmil/ft), This unit indicates what units we are expected to use
in the resistance formula (R=pIl/A). In this case, these figures
for specific resistance are intended to be used when length is
measured in feet and cross-sectional area is measured in
circular mils.
The metric unit for specific resistance is the ohm-meter (Q-
m), or ohm-centimeter (Q-cm), with 1.66243 x 10°9 Q-meters
per Q-cmil/ft (1.66243 x 10° O-cm per Q-cmil/ft). In the Q-cm
column of the table, the figures are actually scaled as WO-cm
due to their very small magnitudes. For example, iron is
listed as 9.61 WOQ-cm, which could be represented as 9.61 x
10° O-cm.
When using the unit of Q-meter for specific resistance in the
R=pl/A formula, the length needs to be in meters and the
area in square meters. When using the unit of Q-centimeter
(Q-cm) in the same formula, the length needs to be in
centimeters and the area in square centimeters.
All these units for specific resistance are valid for any
material (Q-cmil/ft, Q-m, or Q-cm). One might prefer to use Q-
cmil/ft, however, when dealing with round wire where the
cross-sectional area is already known in circular mils.
Conversely, when dealing with odd-shaped busbar or custom
busbar cut out of metal stock, where only the linear
dimensions of length, width, and height are known, the
specific resistance units of Q-meter or Q-cm may be more
appropriate.
Going back to our example circuit, we were looking for wire
that had 0.2 O or less of resistance over a length of 2300
feet. Assuming that we're going to use copper wire (the most
common type of electrical wire manufactured), we can set up
our formula as such:
|
R= p —
ae
... Solving for unknown area A...
A= p —
gs )
A =(10.09 Q-emilstty (2300 feet_)
0.22
A= 116,035 cmils
Algebraically solving for A, we get a value of 116,035 circular
mils. Referencing our solid wire size table, we find that
“double-ought" (2/0) wire with 133,100 cmils is adequate,
whereas the next lower size, "single-ought" (1/0), at 105,500
cmils is too small. Bear in mind that our circuit current is a
modest 25 amps. According to our ampacity table for copper
wire in free air, 14 gauge wire would have sufficed (as far as
not starting a fire is concerned). However, from the
standpoint of voltage drop, 14 gauge wire would have been
very unacceptable.
Just for fun, let's see what 14 gauge wire would have done to
our power circuit's performance. Looking at our wire size
table, we find that 14 gauge wire has a cross-sectional area
of 4,107 circular mils. If we're still using copper as a wire
material (a good choice, unless we're rea//y rich and can
afford 4600 feet of 14 gauge silver wire!), then our specific
resistance will still be 10.09 Q-cmil/ft:
|
R-=p—
P A
2300 feet
R = (10.09 Q-cmil/ft) ee
4107 cmil
R=5.651Q
Remember that this is 5.651 OQ per 2300 feet of 14-gauge
copper wire, and that we have two runs of 2300 feet in the
entire circuit, so each wire piece in the circuit has 5.651 QO of
resistance:
—— 2300 feet ———
wire resistance
Load
(requires at least 220 V)
wire resistance
Our total circuit wire resistance is 2 times 5.651, or 11.301 Q.
Unfortunately, this is fartoo much resistance to allow 25
amps of current with a source voltage of 230 volts. Even if
our load resistance was 0 Q, our wiring resistance of 11.301 OQ
would restrict the circuit current to a mere 20.352 amps! As
you can see, a "Small" amount of wire resistance can make a
big difference in circuit performance, especially in power
circuits where the currents are much higher than typically
encountered in electronic circuits.
Let's do an example resistance problem for a piece of
custom-cut busbar. Suppose we have a piece of solid
aluminum bar, 4 centimeters wide by 3 centimeters tall by
125 centimeters long, and we wish to figure the end-to-end
resistance along the long dimension (125 cm). First, we
would need to determine the cross-sectional area of the bar:
Area = Width x Height
A=(4cm)(3 cm)
A= 12 square cm
We also need to know the specific resistance of aluminum, in
the unit proper for this application (Q-cm). From our table of
specific resistances, we see that this is 2.65 x 10° Q-cm.
Setting up our R=ol/A formula, we have:
|
R=p—
P A
125
R = (2.65 x 10° Q-cm) ay
12 cm-
As you can see, the sheer thickness of a busbar makes for
very low resistances compared to that of standard wire sizes,
even when using a material with a greater specific
resistance.
The procedure for determining busbar resistance Is not
fundamentally different than for determining round wire
resistance. We just need to make sure that cross-sectional
area is calculated properly and that all the units correspond
to each other as they should.
¢ REVIEW:
e Conductor resistance increases with increased length and
decreases with increased cross-sectional area, all other
factors being equal.
e Specific Resistance ("p") is a property of any conductive
material, a figure used to determine the end-to-end
resistance of a conductor given length and area in this
formula: R = pl/A
e Specific resistance for materials are given in units of Q-
cmil/ft or O-meters (metric). Conversion factor between
these two units is 1.66243 x 10°9 Q-meters per Q-cmil/ft,
or 1.66243 x 10°77 Q-cm per Q-cmil/ft.
e If wiring voltage drop in a circuit is critical, exact
resistance calculations for the wires must be made before
wire size is chosen.
Temperature coefficient of resistance
You might have noticed on the table for specific resistances
that all figures were specified at a temperature of 20°
Celsius. If you suspected that this meant specific resistance
of a material may change with temperature, you were right!
Resistance values for conductors at any temperature other
than the standard temperature (usually specified at 20
Celsius) on the specific resistance table must be determined
through yet another formula:
R= Ref [l + ou T = Tee) ]
Where,
R= Conductor resistance at temperature "T"
R,-¢ = Conductor resistance at reference temperature
T,.,, usually 20° C, but sometimes 0° C.
a= Temperature coefficient of resistance for the
conductor material.
ref?
T= Conductor temperature in degrees Celcius.
T,.<= Reference temperature that o is specified at
for the conductor material.
The "alpha" (a) constant is Known as the temperature
coefficient of resistance, and symbolizes the resistance
change factor per degree of temperature change. Just as all
materials have a certain specific resistance (at 20° C), they
also change resistance according to temperature by certain
amounts. For pure metals, this coefficient is a positive
number, meaning that resistance /ncreases with increasing
temperature. For the elements carbon, silicon, and
germanium, this coefficient is a negative number, meaning
that resistance decreases with increasing temperature. For
some metal alloys, the temperature coefficient of resistance
is very close to zero, meaning that the resistance hardly
changes at all with variations in temperature (a good
property if you want to build a precision resistor out of metal
wire!). The following table gives the temperature coefficients
of resistance for several common metals, both pure and alloy:
Temperature coefficient table: below
Temperature coefficient (a) per degree C:
SSS SSSSSS===_a=_SSSSSSS—S=s
Material
ickel
ron
Molybdenu
ungsten
luminum
opper
ilver
latinum
old
INC
teel*
ichrome
ichrome V
Manganin
onstantan
PK
mM
= Steel alloy at 99.5%
Element/Alloy
Element
Element
Element
Element
Element
Element
Element
Element
Element
Element
Alloy
Alloy
Alloy
Alloy
Alloy
iron, 0.5%
Temp. coefficient
.005866
.005671
.004579
.004403
.004308
.004041
.003819
.003729
.003715
.003847
.003
.00017
.00013
.000015
+0.00007 4
carbon
Let's take a look at an example circuit to see how
temperature can affect wire resistance, and consequently
circuit performance:
R viret =15
14V — 2502
R
15 Q
wire#2 ~~
This circuit has a total wire resistance (wire 1 + wire 2) of 30
Q at standard temperature. Setting up a table of voltage,
current, and resistance values we get:
Wire, Wire, Load Total
Volts
Amps
250 Ohms
At 20° Celsius, we get 12.5 volts across the load and a total
of 1.5 volts (0.75 + 0.75) dropped across the wire resistance.
If the temperature were to rise to 35° Celsius, we could easily
determine the change of resistance for each piece of wire.
Assuming the use of copper wire (a = 0.004041) we get:
ao —- m
k= Ref [l + au(T = T.ee)]
R = (15 Q)[1 + 0.00404 1(35° - 20°]
R= 15.909 Q
Recalculating our circuit values, we see what changes this
increase in temperature will bring:
Wire, Wire, Load Total
.
Volts
19.677m | Amps
As you can see, voltage across the load went down (from
12.5 volts to 12.42 volts) and voltage drop across the wires
went up (from 0.75 volts to 0.79 volts) as a result of the
temperature increasing. Though the changes may seem
small, they can be significant for power lines stretching miles
between power plants and substations, substations and
a3 —- m
loads. In fact, power utility companies often have to take line
resistance changes resulting from seasonal temperature
variations into account when calculating allowable system
loading.
REVIEW:
Most conductive materials change specific resistance
with changes in temperature. This is why figures of
specific resistance are always specified at a standard
temperature (usually 20° or 25° Celsius).
The resistance-change factor per degree Celsius of
temperature change is called the temperature coefficient
of resistance. This factor is represented by the Greek
lower-case letter "alpha" (a).
A positive coefficient for a material means that its
resistance increases with an increase in temperature.
Pure metals typically have positive temperature
coefficients of resistance. Coefficients approaching zero
can be obtained by alloying certain metals.
A negative coefficient for a material means that its
resistance decreases with an increase in temperature.
Semiconductor materials (carbon, silicon, germanium)
typically have negative temperature coefficients of
resistance.
The formula used to determine the resistance of a
conductor at some temperature other than what is
specified in a resistance table is as follows:
R=R,. [1 + Q(T - T,,¢)]
Where,
R= Conductor resistance at temperature "T"
R,-¢ = Conductor resistance at reference temperature
T,.¢, usually 20° C, but sometimes 0° C.
a= Temperature coefficient of resistance for the
conductor material.
T= Conductor temperature in degrees Celcius.
T,.<= Reference temperature that o is specified at
for the conductor material.
Superconductivity
Conductors lose all of their electrical resistance when cooled
to super-low temperatures (near absolute zero, about -27 3°
Celsius). It must be understood that superconductivity is not
merely an extrapolation of most conductors' tendency to
gradually lose resistance with decreasing temperature;
rather, it is a sudden, quantum leap in resistivity from finite
to nothing. A superconducting material has absolutely zero
electrical resistance, not just some small amount.
Superconductivity was first discovered by H. Kamerlingh
Onnes at the University of Leiden, Netherlands in 1911. Just
three years earlier, in 1908, Onnes had developed a method
of liquefying helium gas, which provided a medium with
which to supercool experimental objects to just a few
degrees above absolute zero. Deciding to investigate
changes in electrical resistance of mercury when cooled to
this low of a temperature, he discovered that its resistance
dropped to nothing just below the boiling point of helium.
There is some debate over exactly how and why
superconducting materials superconduct. One theory holds
that electrons group together and travel in pairs (called
Cooper pairs) within a superconductor rather than travel
independently, and that has something to do with their
frictionless flow. Interestingly enough, another phenomenon
of super-cold temperatures, superfluidity, happens with
certain liquids (especially liquid helium), resulting in
frictionless flow of molecules.
Superconductivity promises extraordinary capabilities for
electric circuits. If conductor resistance could be eliminated
entirely, there would be no power losses or inefficiencies in
electric power systems due to Stray resistances. Electric
motors could be made almost perfectly (100%) efficient.
Components such as capacitors and inductors, whose ideal
characteristics are normally spoiled by inherent wire
resistances, could be made ideal in a practical sense.
Already, some practical superconducting conductors, motors,
and capacitors have been developed, but their use at this
present time is limited due to the practical problems intrinsic
to maintaining super-cold temperatures.
The threshold temperature for a superconductor to switch
from normal conduction to superconductivity is called the
transition temperature. Transition temperatures for "classic"
Superconductors are in the cryogenic range (near absolute
zero), but much progress has been made in developing
"high-temperature" superconductors which superconduct at
warmer temperatures. One type is a ceramic mixture of
yttrium, barium, copper, and oxygen which transitions ata
relatively balmy -160° Celsius. Ideally, a superconductor
should be able to operate within the range of ambient
temperatures, or at least within the range of inexpensive
refrigeration equipment.
The critical temperatures for a few common substances are
shown here in this table. Temperatures are given in kelvins,
which has the same incremental span as degrees Celsius (an
increase or decrease of 1 kelvin is the same amount of
temperature change as 1° Celsius), only offset so that 0 Kis
absolute zero. This way, we don't have to deal with a lot of
negative figures.
Critical temperature, superconductors below
Critical temperatures given in Kelvins
rome | PGi | sense
Alloy temperature(K)
luminum .
admium
ead 2
Mercury .
iobium Element ;
horium .
.
itanium Element
ranium 0
inc .
Cupric
Superconducting materials also interact in interesting ways
with magnetic fields. While in the superconducting state, a
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superconducting material will tend to exclude all magnetic
fields, a phenomenon known as the Meissner effect.
However, if the magnetic field strength intensifies beyond a
critical level, the superconducting material will be rendered
non-superconductive. In other words, superconducting
materials will lose their superconductivity (no matter how
cold you make them) if exposed to too strong of a magnetic
field. In fact, the presence of any magnetic field tends to
lower the critical temperature of any superconducting
material: the more magnetic field present, the colder you
have to make the material before it will superconduct.
This is another practical limitation to superconductors in
circuit design, since electric current through any conductor
produces a magnetic field. Even though a superconducting
wire would have zero resistance to oppose current, there will
still be a /imit of how much current could practically go
through that wire due to its critical magnetic field limit.
There are already a few industrial applications of
superconductors, especially since the recent (1987) advent
of the yttrium-barium-copper-oxygen ceramic, which only
requires liquid nitrogen to cool, as opposed to liquid helium.
It is even possible to order superconductivity kits from
educational suppliers which can be operated in high school
labs (liquid nitrogen not included). Typically, these kits
exhibit superconductivity by the Meissner effect, suspending
a tiny magnet in mid-air over a Superconducting disk cooled
by a bath of liquid nitrogen.
The zero resistance offered by superconducting circuits leads
to unique consequences. In a superconducting short-circuit,
it is possible to maintain large currents indefinitely with zero
applied voltage!
electrons will flow unimpeded by
resistance, continuing to flow
forever!
Rings of superconducting material have been experimentally
proven to sustain continuous current for years with no
applied voltage. So far as anyone knows, there is no
theoretical time limit to how long an unaided current could
be sustained in a superconducting circuit. If you're thinking
this appears to be a form of perpetual motion, you're correct!
Contrary to popular belief, there is no law of physics
prohibiting perpetual motion; rather, the prohibition stands
against any machine or system generating more energy than
it consumes (what would be referred to as an over-unity
device). At best, all a perpetual motion machine (like the
superconducting ring) would be good for is to store energy,
not generate it freely!
Superconductors also offer some strange possibilities having
nothing to do with Ohm's Law. One such possibility is the
construction of a device called a Josephson Junction, which
acts as a relay of sorts, controlling one current with another
current (with no moving parts, of course). The small size and
fast switching time of Josephson Junctions may lead to new
computer circuit designs: an alternative to using
semiconductor transistors.
e REVIEW:
e Superconductors are materials which have absolutely
zero electrical resistance.
e All presently known superconductive materials need to
be cooled far below ambient temperature to
Superconduct. The maximum temperature at which they
do so is called the transition temperature.
Insulator breakdown voltage
The atoms in insulating materials have very tightly-bound
electrons, resisting free electron flow very well. However,
insulators cannot resist indefinite amounts of voltage. With
enough voltage applied, any insulating material will
eventually succumb to the electrical "pressure" and electron
flow will occur. However, unlike the situation with conductors
where current is in a linear proportion to applied voltage
(given a fixed resistance), current through an insulator is
quite nonlinear: for voltages below a certain threshold level,
virtually no electrons will flow, but if the voltage exceeds
that threshold, there will be a rush of current.
Once current is forced through an insulating material,
breakdown of that material's molecular structure has
occurred. After breakdown, the material may or may not
behave as an insulator any more, the molecular structure
having been altered by the breach. There is usually a
localized "puncture" of the insulating medium where the
electrons flowed during breakdown.
Thickness of an insulating material plays a role in
determining its breakdown voltage, otherwise known as
dielectric strength. Specific dielectric strength is sometimes
listed in terms of volts per mil (1/1000 of an inch), or
kilovolts per inch (the two units are equivalent), but in
practice it has been found that the relationship between
breakdown voltage and thickness is not exactly linear. An
insulator three times as thick has a dielectric strength
Slightly less than 3 times as much. However, for rough
estimation use, volt-per-thickness ratings are fine.
Dielectric strength: below
Dielectric strength in kilovolts per inch (kV/in):
i Rotors
orcelain 40 to 200
araffin Wax _|200to300_
ransformer Oi 400 |
akelite__—(B00to550_—
ubber 450to700
hellac poo id
aper 250
eflon 1500
lass 2000 to 3000
5000
fF
=
Ol
oO
a)
ie)
|
* = Materials listed are specially prepared for electrical use,
above.
REVIEW:
With a high enough applied voltage, electrons can be
freed from the atoms of insulating materials, resulting in
current through that material.
The minimum voltage required to "violate" an insulator
by forcing current through it is called the breakdown
voltage, or dielectric strength.
The thicker a piece of insulating material, the higher the
breakdown voltage, all other factors being equal.
Specific dielectric strength is typically rated in one of two
equivalent units: volts per mil, or kilovolts per inch.
Data
Tables of specific resistance and temperature coefficient of
resistance for elemental materials (not alloys) were derived
from figures found in the 78" edition of the CRC Handbook of
Chemistry and Physics.
Table of superconductor critical temperatures derived from
figures found in the 215* volume of Collier's Encyclopedia,
1968.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Aaron Forster (February 18, 2003): Typographical error
correction.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—| | +4/\—
—| | +4/l—
Lessons In Electric Circuits
-- Volume |
Chapter 13
CAPACITORS
Electric fields and capacitance
Capacitors and calculus
Factors affecting capacitance
Series and parallel capacitors
Practical considerations
Contributors
Electric fields and capacitance
Whenever an electric voltage exists between two separated
conductors, an electric field is present within the space
between those conductors. In basic electronics, we study the
interactions of voltage, current, and resistance as they
pertain to circuits, which are conductive paths through which
electrons may travel. When we talk about fields, however,
we're dealing with interactions that can be spread across
empty space.
Admittedly, the concept of a "field" is somewhat abstract. At
least with electric current it isn't too difficult to envision tiny
particles called electrons moving their way between the
nuclei of atoms within a conductor, but a "field" doesn't even
have mass, and need not exist within matter at all.
Despite its abstract nature, almost every one of us has direct
experience with fields, at least in the form of magnets. Have
you ever played with a pair of magnets, noticing how they
attract or repel each other depending on their relative
orientation? There is an undeniable force between a pair of
magnets, and this force is without "substance." It has no
mass, no color, no odor, and if not for the physical force
exerted on the magnets themselves, it would be utterly
insensible to our bodies. Physicists describe the interaction
of magnets in terms of magnetic fields in the space between
them. If iron filings are placed near a magnet, they orient
themselves along the lines of the field, visually indicating its
presence.
The subject of this chapter is e/ectric fields (and devices
called capacitors that exploit them), not magnetic fields, but
there are many similarities. Most likely you have experienced
electric fields as well. Chapter 1 of this book began with an
explanation of static electricity, and how materials such as
wax and wool -- when rubbed against each other -- produced
a physical attraction. Again, physicists would describe this
interaction in terms of e/ectric fields generated by the two
objects as a result of their electron imbalances. Suffice it to
say that whenever a voltage exists between two points, there
will be an electric field manifested in the space between
those points.
Fields have two measures: a field force and a field flux. The
field force is the amount of "push" that a field exerts over a
certain distance. The field f/ux is the total quantity, or effect,
of the field through space. Field force and flux are roughly
analogous to voltage ("push") and current (flow) through a
conductor, respectively, although field flux can exist in
totally empty space (without the motion of particles such as
electrons) whereas current can only take place where there
are free electrons to move. Field flux can be opposed in
Space, just as the flow of electrons can be opposed by
resistance. The amount of field flux that will develop in space
IS proportional to the amount of field force applied, divided
by the amount of opposition to flux. Just as the type of
conducting material dictates that conductor's specific
resistance to electric current, the type of insulating material
separating two conductors dictates the specific opposition to
field flux.
Normally, electrons cannot enter a conductor unless there is
a path for an equal amount of electrons to exit (remember
the marble-in-tube analogy?). This is why conductors must
be connected together in a circular path (a circuit) for
continuous current to occur. Oddly enough, however, extra
electrons can be "squeezed" into a conductor without a path
to exit if an electric field is allowed to develop in space
relative to another conductor. The number of extra free
electrons added to the conductor (or free electrons taken
away) is directly proportional to the amount of field flux
between the two conductors.
Capacitors are components designed to take advantage of
this phenomenon by placing two conductive plates (usually
metal) in close proximity with each other. There are many
different styles of capacitor construction, each one suited for
particular ratings and purposes. For very small capacitors,
two circular plates sandwiching an insulating material will
suffice. For larger capacitor values, the "plates" may be strips
of metal foil, sandwiched around a flexible insulating
medium and rolled up for compactness. The highest
Capacitance values are obtained by using a microscopic-
thickness layer of insulating oxide separating two conductive
surfaces. In any case, though, the general idea is the same:
two conductors, separated by an insulator.
The schematic symbol for a capacitor is quite simple, being
little more than two short, parallel lines (representing the
plates) separated by a gap. Wires attach to the respective
plates for connection to other components. An older,
obsolete schematic symbol for capacitors showed interleaved
plates, which is actually a more accurate way of representing
the real construction of most capacitors:
Capacitor symbols
+
When a voltage is applied across the two plates of a
Capacitor, a concentrated field flux is created between them,
allowing a significant difference of free electrons (a charge)
to develop between the two plates:
deficiency of electrons
excess free electrons
As the electric field is established by the applied voltage,
extra free electrons are forced to collect on the negative
conductor, while free electrons are "robbed" from the positive
conductor. This differential charge equates to a storage of
energy in the capacitor, representing the potential charge of
the electrons between the two plates. The greater the
difference of electrons on opposing plates of a capacitor, the
greater the field flux, and the greater "charge" of energy the
Capacitor will store.
Because capacitors store the potential energy of
accumulated electrons in the form of an electric field, they
behave quite differently than resistors (which simply
dissipate energy in the form of heat) in a circuit. Energy
storage in a capacitor is a function of the voltage between
the plates, as well as other factors which we will discuss later
in this chapter. A capacitor's ability to store energy asa
function of voltage (potential difference between the two
leads) results in a tendency to try to maintain voltage ata
constant level. In other words, capacitors tend to resist
changes in voltage drop. When voltage across a capacitor is
increased or decreased, the capacitor "resists" the change by
drawing current from or supplying current to the source of
the voltage change, in opposition to the change.
To store more energy in a capacitor, the voltage across it
must be increased. This means that more electrons must be
added to the (-) plate and more taken away from the (+)
plate, necessitating a current in that direction. Conversely, to
release energy from a capacitor, the voltage across it must be
decreased. This means some of the excess electrons on the
(-) plate must be returned to the (+) plate, necessitating a
current in the other direction.
Just as Isaac Newton's first Law of Motion ("an object in
motion tends to stay in motion; an object at rest tends to
stay at rest") describes the tendency of a mass to oppose
changes in velocity, we can state a capacitor's tendency to
oppose changes in voltage as such: "A charged capacitor
tends to stay charged; a discharged capacitor tends to stay
discharged." Hypothetically, a capacitor left untouched will
indefinitely maintain whatever state of voltage charge that
its been left it. Only an outside source (or drain) of current
can alter the voltage charge stored by a perfect capacitor:
+,
iL voitage (charge) sustained wit
C T- the capaci -circui
pacitor open-circuited
Practically speaking, however, capacitors will eventually lose
their stored voltage charges due to internal leakage paths for
electrons to flow from one plate to the other. Depending on
the specific type of capacitor, the time it takes for a stored
voltage charge to self-dissipate can be a /ong time (several
years with the capacitor sitting on a shelf!).
When the voltage across a capacitor is increased, it draws
current from the rest of the circuit, acting as a power load. In
this condition the capacitor is said to be charging, because
there is an increasing amount of energy being stored in its
electric field. Note the direction of electron current with
regard to the voltage polarity:
Energy being absorbed by
the capacitor from the rest
of the circuit.
—<« |
...to the rest of C ile increasing
the circuit - voltage
| —~
The capacitor acts as a LOAD
Conversely, when the voltage across a capacitor is
decreased, the capacitor supplies current to the rest of the
circuit, acting as a power source. In this condition the
Capacitor is said to be discharging. \ts store of energy -- held
in the electric field -- is decreasing now as energy is released
to the rest of the circuit. Note the direction of electron
current with regard to the voltage polarity:
Energy being released by the
capacitor to the rest of the circuit
|—>
... to the rest of C i decreasing
the circuit voltage
—~——_— |
The capacitor acts as a SOURCE
If a source of voltage is suddenly applied to an uncharged
Capacitor (a sudden increase of voltage), the capacitor will
draw current from that source, absorbing energy from it, until
the capacitor's voltage equals that of the source. Once the
Capacitor voltage reached this final (charged) state, its
current decays to zero. Conversely, if a load resistance is
connected to a charged capacitor, the capacitor will supply
current to the load, until it has released all its stored energy
and its voltage decays to zero. Once the capacitor voltage
reaches this final (discharged) state, its current decays to
zero. In their ability to be charged and discharged, capacitors
can be thought of as acting somewhat like secondary-cell
batteries.
The choice of insulating material between the plates, as was
mentioned before, has a great impact upon how much field
flux (and therefore how much charge) will develop with any
given amount of voltage applied across the plates. Because
of the role of this insulating material in affecting field flux, it
has a special name: dielectric. Not all dielectric materials are
equal: the extent to which materials inhibit or encourage the
formation of electric field flux is called the permittivity of the
dielectric.
The measure of a capacitor's ability to store energy fora
given amount of voltage drop is called capacitance. Not
surprisingly, capacitance is also a measure of the intensity of
opposition to changes in voltage (exactly how much current
it will produce for a given rate of change in voltage).
Capacitance is symbolically denoted with a capital "C," and is
measured in the unit of the Farad, abbreviated as "F."
Convention, for some odd reason, has favored the metric
prefix "micro" in the measurement of large capacitances, and
SO many capacitors are rated in terms of confusingly large
microFarad values: for example, one large capacitor | have
seen was rated 330,000 microFarads!! Why not state it as
330 milliFarads? | don't know.
An obsolete name for a capacitor is condenser or condensor.
These terms are not used in any new books or schematic
diagrams (to my knowledge), but they might be encountered
in older electronics literature. Perhaps the most well-known
usage for the term "condenser" is in automotive engineering,
where a small capacitor called by that name was used to
mitigate excessive sparking across the switch contacts
(called "points") in electromechanical ignition systems.
e REVIEW:
e Capacitors react against changes in voltage by supplying
or drawing current in the direction necessary to oppose
the change.
e When a capacitor is faced with an increasing voltage, it
acts aS a /Joad: drawing current as it absorbs energy
(current going in the negative side and out the positive
side, like a resistor).
e When a capacitor is faced with a decreasing voltage, it
acts as a source: supplying current as it releases stored
energy (current going out the negative side and in the
positive side, like a battery).
e The ability of a capacitor to store energy in the form of an
electric field (and consequently to oppose changes in
voltage) is called capacitance. It is measured in the unit
of the Farad (F).
e Capacitors used to be commonly known by another term:
condenser (alternatively spelled "condensor").
Capacitors and calculus
Capacitors do not have a stable "resistance" as conductors
do. However, there is a definite mathematical relationship
between voltage and current for a capacitor, as follows:
"Ohm's Law" for a capacitor
dv
dt
: ee &
Where,
i= Instantaneous current through the capacitor
C = Capacitance in Farads
\
—— = Instantaneous rate of voltage change
dt —_ (volts per second)
The lower-case letter "i" symbolizes instantaneous current,
which means the amount of current at a specific point in
time. This stands in contrast to constant current or average
current (capital letter "I") over an unspecified period of time.
The expression "dv/dt" is one borrowed from calculus,
meaning the instantaneous rate of voltage change over time,
or the rate of change of voltage (volts per second increase or
decrease) at a specific point in time, the same specific point
in time that the instantaneous current is referenced at. For
whatever reason, the letter vis usually used to represent
instantaneous voltage rather than the letter e. However, it
would not be incorrect to express the instantaneous voltage
rate-of-change as "de/dt" instead.
In this equation we see something novel to our experience
thusfar with electric circuits: the variable of time. When
relating the quantities of voltage, current, and resistance toa
resistor, it doesn't matter if we're dealing with measurements
taken over an unspecified period of time (E=IR; V=IR), or ata
specific moment in time (e=ir; v=ir). The same basic formula
holds true, because time is irrelevant to voltage, current, and
resistance in a component like a resistor.
In a capacitor, however, time is an essential variable,
because current is related to how rapidly voltage changes
over time. To fully understand this, a few illustrations may be
necessary. Suppose we were to connect a capacitor toa
variable-voltage source, constructed with a potentiometer
and a battery:
Ammeter
(zero-center)
If the potentiometer mechanism remains in a single position
(wiper is stationary), the voltmeter connected across the
capacitor will register a constant (unchanging) voltage, and
the ammeter will register 0 amps. In this scenario, the
instantaneous rate of voltage change (dv/dt) is equal to zero,
because the voltage is unchanging. The equation tells us
that with 0 volts per second change for a dv/dt, there must
be zero instantaneous current (i). From a physical
perspective, with no change in voltage, there is no need for
any electron motion to add or subtract charge from the
Capacitor's plates, and thus there will be no current.
Capacitor
voltage
E,
Time -—>
Potentiometer wiper not moving
Capacitor
current
I,
Time —>
Now, if the potentiometer wiper is moved slowly and steadily
In the "up" direction, a greater voltage will gradually be
imposed across the capacitor. Thus, the voltmeter indication
will be increasing at a slow rate:
Potentiometer wiper moving
slowly in the "up" direction
Steady current
+ .
(V) Increasing
voltage
If we assume that the potentiometer wiper is being moved
such that the rate of voltage increase across the capacitor is
steady (for example, voltage increasing at a constant rate of
2 volts per second), the dv/dt term of the formula will bea
fixed value. According to the equation, this fixed value of
dv/dt, multiplied by the capacitor's capacitance in Farads
(also fixed), results in a fixed current of some magnitude.
From a physical perspective, an increasing voltage across the
Capacitor demands that there be an increasing charge
differential between the plates. Thus, for a slow, steady
voltage increase rate, there must be a slow, steady rate of
charge building in the capacitor, which equates to a slow,
steady flow rate of electrons, or current. In this scenario, the
Capacitor is acting as a /oad, with electrons entering the
negative plate and exiting the positive, accumulating energy
in the electric field.
em |
Capacitor ._t Voltage
voltage rv change
E,
Time —-—>
Potentiometer wiper moving slowly "up"
Capacitor
current
1,
Time —>~
If the potentiometer is moved in the same direction, but ata
faster rate, the rate of voltage change (dv/dt) will be greater
and so will be the capacitor's current:
Potentiometer wiper moving
quickly in the "up" direction
(greater)
Steady current
(faster)
O Increasing
voltage
_ Time —}|
Capacitor pt
voltage
Ec
Time —-
Potentiometer wiper moving quickly "up"
Capacitor
current
1,
Time —>
When mathematics students first study calculus, they begin
by exploring the concept of rates of change for various
mathematical functions. The derivative, which is the first and
most elementary calculus principle, is an expression of one
variable's rate of change in terms of another. Calculus
students have to learn this principle while studying abstract
equations. You get to learn this principle while studying
something you can relate to: electric circuits!
To put this relationship between voltage and current ina
capacitor in calculus terms, the current through a capacitor is
the derivative of the voltage across the capacitor with
respect to time. Or, stated in simpler terms, a capacitor's
Current is directly proportional to how quickly the voltage
across it is changing. In this circuit where capacitor voltage is
set by the position of a rotary knob on a potentiometer, we
can say that the capacitor's current is directly proportional to
how quickly we turn the knob.
If we were to move the potentiometer's wiper in the same
direction as before ("up"), but at varying rates, we would
obtain graphs that looked like this:
Capacitor
voltage
E-
Time —>
Potentiometer wiper moving "up" at
different rates
Capacitor
current
I,
Time —>
Note how that at any given point in time, the capacitor's
current is proportional to the rate-of-change, or s/ope of the
capacitor's voltage plot. When the voltage plot line is rising
quickly (steep slope), the current will likewise be great.
Where the voltage plot has a mild slope, the current is small.
At one place in the voltage plot where it levels off (zero
slope, representing a period of time when the potentiometer
wasn't moving), the current falls to zero.
If we were to move the potentiometer wiper in the "down"
direction, the capacitor voltage would decrease rather than
increase. Again, the capacitor will react to this change of
voltage by producing a current, but this time the current will
be in the opposite direction. A decreasing capacitor voltage
requires that the charge differential between the capacitor's
plates be reduced, and the only way that can happen is if the
electrons reverse their direction of flow, the capacitor
discharging rather than charging. In this condition, with
electrons exiting the negative plate and entering the
positive, the capacitor will act as a source, like a battery,
releasing its stored energy to the rest of the circuit.
Potentiometer wiper moving
in the "down" direction
+ ;
(Vv) Decreasing
voltage
Again, the amount of current through the capacitor is directly
proportional to the rate of voltage change across it. The only
difference between the effects of a decreasing voltage and
an increasing voltage is the direction of electron flow. For the
same rate of voltage change over time, either increasing or
decreasing, the current magnitude (amps) will be the same.
Mathematically, a decreasing voltage rate-of-change Is
expressed as a negative dv/dt quantity. Following the formula
| = C(dv/dt), this will result in a current figure (i) that is
likewise negative in sign, indicating a direction of flow
corresponding to discharge of the capacitor.
Factors affecting capacitance
There are three basic factors of capacitor construction
determining the amount of capacitance created. These
factors all dictate capacitance by affecting how much electric
field flux (relative difference of electrons between plates) will
develop for a given amount of electric field force (voltage
between the two plates):
PLATE AREA: All other factors being equal, greater plate
area gives greater capacitance; less plate area gives less
Capacitance.
Explanation: Larger plate area results in more field flux
(charge collected on the plates) for a given field force
(voltage across the plates).
less capacitance more capacitance
PLATE SPACING: All other factors being equal, further plate
Spacing gives less capacitance; closer plate spacing gives
greater capacitance.
Explanation: Closer spacing results in a greater field force
(voltage across the capacitor divided by the distance
between the plates), which results in a greater field flux
(charge collected on the plates) for any given voltage
applied across the plates.
less capacitance more capacitance
eles sail
=i iar an
DIELECTRIC MATERIAL: All other factors being equal,
greater permittivity of the dielectric gives greater
Capacitance; less permittivity of the dielectric gives less
Capacitance.
Explanation: Although its complicated to explain, some
materials offer less opposition to field flux for a given amount
of field force. Materials with a greater permittivity allow for
more field flux (offer less opposition), and thus a greater
collected charge, for any given amount of field force (applied
voltage).
less capacitance more capacitance
air —> mu <— glass
(relative permittivity (relative permittivity
= 1.0006) = 7.0)
"Relative" permittivity means the permittivity of a material,
relative to that of a pure vacuum. The greater the number,
the greater the permittivity of the material. Glass, for
instance, with a relative permittivity of 7, has seven times
the permittivity of a pure vacuum, and consequently will
allow for the establishment of an electric field flux seven
times stronger than that of a vacuum, all other factors being
equal.
The following is a table listing the relative permittivities (also
known as the "dielectric constant") of various common
substances:
Temperature coefficient table: below
Temperature coefficient (a) per degree C:
constant)
acum 00007
ir M0006
TFE,FEP ("Teflon") 2.0
olypropylene_2.20t0228
BS resin 2.4 to 3.2
olystyrene 2.45 to 4.0
Waxedpaper 2.5
ransformeroil __-2.5to4
ard Rubber —2.5to480. SSS
Wood(Oak) 3:3
: i
3.
ilicones 3.4 to 4.3
akelite - 3.5 to 6.0
vartz fused BB
Wood (Maple) (44. SS™S~S
lass 4.9to7.5
|
astor oil
wood (Birch)
Mica, muscovite
Barium-strontium- 7500
titanite
An approximation of capacitance for any pair of separated
conductors can be found with this formula:
EA
d
Where,
C=
C= Capacitance in Farads
€= Permittivity of dielectric (absolute, not
relative)
A= Area of plate overlap in square meters
d= Distance between plates in meters
A formula for capacitance in picofarads using practical
dimensions:
_ 0.0885K(n-1) A — 0.225K(n-1)A’
d d’ }
ae
ee a
C= Capacitance in picofarads +
C
K = Dielectric constant
A= Areaof one plate in square centimeters
A’= Area of one plate in square inches
d= _ Thickness in centimeters
d’= Thickness in inches
n= Number of plates
A capacitor can be made variable rather than fixed in value
by varying any of the physical factors determining
Capacitance. One relatively easy factor to vary in capacitor
construction is that of plate area, or more properly, the
amount of plate overlap.
The following photograph shows an example of a variable
Capacitor using a set of interleaved metal plates and an air
gap as the dielectric material:
A VARIABLE CAPACITOR (AIR DIELECTRIC)
As the shaft is rotated, the degree to which the sets of plates
overlap each other will vary, changing the effective area of
the plates between which a concentrated electric field can be
established. This particular capacitor has a capacitance in
the picofarad range, and finds use in radio circuitry.
Series and parallel capacitors
When capacitors are connected in series, the total
Capacitance is less than any one of the series capacitors'
individual capacitances. If two or more capacitors are
connected in series, the overall effect is that of a single
(equivalent) capacitor having the sum total of the plate
spacings of the individual capacitors. As we've just seen, an
increase in plate spacing, with all other factors unchanged,
results in decreased capacitance.
C;
equivalent to —> C
Ty “total
C,
an
Thus, the total capacitance is less than any one of the
individual capacitors’ capacitances. The formula for
calculating the series total capacitance is the same form as
for calculating parallel resistances:
Series Capacitances
When capacitors are connected in parallel, the total
Capacitance is the sum of the individual capacitors'
capacitances. If two or more capacitors are connected in
parallel, the overall effect is that of a single equivalent
capacitor having the sum total of the plate areas of the
individual capacitors. As we've just seen, an increase in plate
area, with all other factors unchanged, results in increased
Capacitance.
equivalentto —> C
a total
Thus, the total capacitance is more than any one of the
individual capacitors' capacitances. The formula for
calculating the parallel total capacitance is the same form as
for calculating series resistances:
Paralle! Capacitances
C
‘tota
y= C+ G+... C,
As you will no doubt notice, this is exactly opposite of the
phenomenon exhibited by resistors. With resistors, series
connections result in additive values while parallel
connections result in diminished values. With capacitors, its
the reverse: parallel connections result in additive values
while series connections result in diminished values.
e REVIEW:
e Capacitances diminish in series.
e Capacitances add in parallel.
Practical considerations
Capacitors, like all electrical components, have limitations
which must be respected for the sake of reliability and proper
circuit operation.
Working voltage: Since capacitors are nothing more than two
conductors separated by an insulator (the dielectric), you
must pay attention to the maximum voltage allowed across
it. If too much voltage is applied, the "breakdown" rating of
the dielectric material may be exceeded, resulting in the
capacitor internally short-circuiting.
Polarity: Some capacitors are manufactured so they can only
tolerate applied voltage in one polarity but not the other.
This is due to their construction: the dielectric isa
microscopically thin layer of insulation deposited on one of
the plates by a DC voltage during manufacture. These are
called e/ectro/ytic capacitors, and their polarity is clearly
marked.
Electrolytic ("polarized")
capacitor
+L curved side of symbol is
always negative!
Reversing voltage polarity to an electrolytic capacitor may
result in the destruction of that super-thin dielectric layer,
thus ruining the device. However, the thinness of that
dielectric permits extremely high values of capacitance ina
relatively small package size. For the same reason,
electrolytic capacitors tend to be low in voltage rating as
compared with other types of capacitor construction.
Equivalent circuit: Since the plates in a capacitor have some
resistance, and since no dielectric is a perfect insulator, there
is no such thing as a "perfect" capacitor. In real life, a
capacitor has both a series resistance and a parallel
(leakage) resistance interacting with its purely capacitive
characteristics:
Capacitor equivalent circuit
R
series
Ri eakage
Cideal
Fortunately, it is relatively easy to manufacture capacitors
with very small series resistances and very high leakage
resistances!
Physical Size: For most applications in electronics, minimum
size is the goal for component engineering. The smaller
components can be made, the more circuitry can be built
into a smaller package, and usually weight is saved as well.
With capacitors, there are two major limiting factors to the
minimum size of a unit: working voltage and capacitance.
And these two factors tend to be in opposition to each other.
For any given choice in dielectric materials, the only way to
increase the voltage rating of a capacitor is to increase the
thickness of the dielectric. However, as we have seen, this
has the effect of decreasing capacitance. Capacitance can be
brought back up by increasing plate area. but this makes for
a larger unit. This is why you cannot judge a capacitor's
rating in Farads simply by size. A capacitor of any given size
may be relatively high in capacitance and low in working
voltage, vice versa, or Some compromise between the two
extremes. Take the following two photographs for example:
A HIGH-VOLTAGE OIL CAPACITOR
2000-Volt DCreting
This is a fairly large capacitor in physical size, but it has quite
a low capacitance value: only 2 uF. However, its working
voltage is quite high: 2000 volts! If this capacitor were re-
engineered to have a thinner layer of dielectric between its
plates, at least a hundredfold increase in capacitance might
be achievable, but at a cost of significantly lowering its
working voltage. Compare the above photograph with the
one below. The capacitor shown in the lower picture is an
electrolytic unit, similar in size to the one above, but with
very different values of capacitance and working voltage:
The thinner dielectric layer gives it a much greater
Capacitance (20,000 UF) and a drastically reduced working
voltage (35 volts continuous, 45 volts intermittent).
Here are some samples of different capacitor types, all
smaller than the units shown previously:
The electrolytic and tantalum capacitors are polarized
(polarity sensitive), and are always labeled as such. The
electrolytic units have their negative (-) leads distinguished
by arrow symbols on their cases. Some polarized capacitors
have their polarity designated by marking the positive
terminal. The large, 20,000 uF electrolytic unit shown in the
upright position has its positive (+) terminal labeled with a
"plus" mark. Ceramic, mylar, plastic film, and air capacitors
do not have polarity markings, because those types are
nonpolarized (they are not polarity sensitive).
Capacitors are very common components in electronic
circuits. Take a close look at the following photograph --
every component marked with a "C" designation on the
printed circuit board is a capacitor:
COME:A,C.D
COM?2:8,0,E
COMS:A,C.F
COM4:8,EF
Some of the capacitors shown on this circuit board are
standard electrolytic: C3, (top of board, center) and C3¢ (left
side, 1/3 from the top). Some others are a special kind of
electrolytic capacitor called tanta/um, because this is the
type of metal used to make the plates. Tantalum capacitors
have relatively high capacitance for their physical size. The
following capacitors on the circuit board shown above are
tantalum: Cy, (just to the lower-left of C39), Cy9 (directly
below Rj, which is below C39), Co, (lower-left corner of
board), and C55 (lower-right).
Examples of even smaller capacitors can be seen in this
photograph:
—. oO a
moe SEA =
c
o
8 a ee |
ye em NT
1th ‘Bkeccro
pL
i270 ae
The capacitors on this circuit board are "surface mount
devices" as are all the resistors, for reasons of saving space.
Following component labeling convention, the capacitors can
be identified by labels beginning with the letter "C".
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Warren Young (August 2002): Photographs of different
Capacitor types.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—||4]t—
—/ | 4]
Lessons In Electric Circuits
-- Volume |
Chapter 14
MAGNETISM AND
ELECTROMAGNETISM
Permanent magnets
Electromagnetism
Magnetic units of measurement
Permeability and saturation
Electromagnetic induction
Mutual inductance
Contributors
Permanent magnets
Centuries ago, it was discovered that certain types of
mineral rock possessed unusual properties of attraction to
the metal iron. One particular mineral, called /odestone, or
magnetite, is found mentioned in very old historical records
(about 2500 years ago in Europe, and much earlier in the
Far East) as a subject of curiosity. Later, it was employed in
the aid of navigation, as it was found that a piece of this
unusual rock would tend to orient itself in a north-south
direction if left free to rotate (Suspended on a string orona
float in water). A scientific study undertaken in 1269 by
Peter Peregrinus revealed that steel could be similarly
"charged" with this unusual property after being rubbed
against one of the "poles" of a piece of lodestone.
Unlike electric charges (such as those observed when amber
is rubbed against cloth), magnetic objects possessed two
poles of opposite effect, denoted "north" and "south" after
their self-orientation to the earth. As Peregrinus found, it was
impossible to isolate one of these poles by itself by cutting a
piece of lodestone in half: each resulting piece possessed its
own pair of poles:
... after breaking in half...
Like electric charges, there were only two types of poles to
be found: north and south (by analogy, positive and
negative). Just as with electric charges, same poles repel one
another, while opposite poles attract. This force, like that
caused by static electricity, extended itself invisibly over
Space, and could even pass through objects such as paper
and wood with little effect upon strength.
The philosopher-scientist Rene Descartes noted that this
invisible "field" could be mapped by placing a magnet
underneath a flat piece of cloth or wood and sprinkling iron
filings on top. The filings will align themselves with the
magnetic field, "mapping" its shape. The result shows how
the field continues unbroken from one pole of a magnet to
the other:
As with any kind of field (electric, magnetic, gravitational),
the total quantity, or effect, of the field is referred toasa
flux, while the "push" causing the flux to form in space is
called a force. Michael Faraday coined the term "tube" to
refer to a string of magnetic flux in space (the term "line" is
more commonly used now). Indeed, the measurement of
magnetic field flux is often defined in terms of the number of
flux lines, although it is doubtful that such fields exist in
individual, discrete lines of constant value.
Modern theories of magnetism maintain that a magnetic
field is produced by an electric charge in motion, and thus it
is theorized that the magnetic field of a so-called
"permanent" magnets such as lodestone is the result of
electrons within the atoms of iron spinning uniformly in the
Same direction. Whether or not the electrons in a material's
atoms are subject to this kind of uniform spinning is dictated
by the atomic structure of the material (not unlike how
electrical conductivity is dictated by the electron binding in
a material's atoms). Thus, only certain types of substances
react with magnetic fields, and even fewer have the ability
to permanently sustain a magnetic field.
lron is one of those types of substances that readily
magnetizes. If a piece of iron is brought near a permanent
magnet, the electrons within the atoms in the iron orient
their spins to match the magnetic field force produced by
the permanent magnet, and the iron becomes "magnetized."
The iron will magnetize in such a way as to incorporate the
magnetic flux lines into its shape, which attracts it toward
the permanent magnet, no matter which pole of the
permanent magnet is offered to the iron:
(unmagnetized)
The previously unmagnetized iron becomes magnetized as it
is brought closer to the permanent magnet. No matter what
pole of the permanent magnet is extended toward the iron,
the iron will magnetize in such a way as to be attracted
toward the magnet:
Referencing the natural magnetic properties of iron (Latin =
"ferrum"), a ferromagnetic material is one that readily
magnetizes (its constituent atoms easily orient their electron
spins to conform to an external magnetic field force). All
materials are magnetic to some degree, and those that are
not considered ferromagnetic (easily magnetized) are
classified as either paramagnetic (slightly magnetic) or
diamagnetic (tend to exclude magnetic fields). Of the two,
diamagnetic materials are the strangest. In the presence of
an external magnetic field, they actually become slightly
magnetized in the opposite direction, so as to repel the
external field!
44 ‘ '
! ‘ oe * ‘
1 4 - Fy ee ‘ i
‘ - 4% .
»s $j“ “Sseses* .* ~ '
~ ae + Ms ig, - '
"enenee* . . aot To --* ’
a ae aa ~~ a, o's Leacee™ Ps
*. ws ~seanneeoee a
.
~=—— Se al -
repulsion Shins -
ieee cee
If a ferromagnetic material tends to retain its magnetization
after an external field is removed, it is said to have good
retentivity. This, of course, is a necessary quality fora
permanent magnet.
e REVIEW:
e Lodestone (also called Magnetite) is a naturally-
occurring "permanent" magnet mineral. By
"permanent," it is meant that the material maintains a
magnetic field with no external help. The characteristic
of any magnetic material to do so is called retentivity.
e Ferromagnetic materials are easily magnetized.
e Paramagnetic materials are magnetized with more
difficulty.
e Diamagnetic materials actually tend to repel external
magnetic fields by magnetizing in the opposite
direction.
Electromagnetism
The discovery of the relationship between magnetism and
electricity was, like so many other scientific discoveries,
stumbled upon almost by accident. The Danish physicist
Hans Christian Oersted was lecturing one day in 1820 on the
possibility of electricity and magnetism being related to one
another, and in the process demonstrated it conclusively by
experiment in front of his whole class! By passing an electric
current through a metal wire suspended above a magnetic
compass, Oersted was able to produce a definite motion of
the compass needle in response to the current. What began
as conjecture at the start of the class session was confirmed
as fact at the end. Needless to say, Oersted had to revise his
lecture notes for future classes! His serendipitous discovery
paved the way for a whole new branch of science:
electromagnetics.
Detailed experiments showed that the magnetic field
produced by an electric current is always oriented
perpendicular to the direction of flow. A simple method of
showing this relationship is called the /eft-hand rule. Simply
stated, the left-hand rule says that the magnetic flux lines
produced by a current-carrying wire will be oriented the
Same direction as the curled fingers of a person's left hand
(in the "hitchhiking" position), with the thumb pointing in
the direction of electron flow:
The "left-hand" rule
IE ai
=— | «— |
The magnetic field encircles this straight piece of current-
carrying wire, the magnetic flux lines having no definite
"north" or "south' poles.
While the magnetic field surrounding a current-carrying wire
is indeed interesting, it is quite weak for common amounts
of current, able to deflect a compass needle and not much
more. To create a stronger magnetic field force (and
consequently, more field flux) with the same amount of
electric current, we can wrap the wire into a coil shape,
where the circling magnetic fields around the wire will join
to create a larger field with a definite magnetic (north and
south) polarity:
magnetic field
The amount of magnetic field force generated by a coiled
wire iS proportional to the current through the wire
multiplied by the number of "turns" or "wraps" of wire in the
coil. This field force is called magnetomotive force (mmf),
and is very much analogous to electromotive force (E) in an
electric circuit.
An electromagnet is a piece of wire intended to generate a
magnetic field with the passage of electric current through
it. Though all current-carrying conductors produce magnetic
fields, an electromagnet is usually constructed in such a way
as to maximize the strength of the magnetic field it
produces for a special purpose. Electromagnets find frequent
application in research, industry, medical, and consumer
products.
As an electrically-controllable magnet, electromagnets find
application in a wide variety of "electromechanical" devices:
machines that effect mechanical force or motion through
electrical power. Perhaps the most obvious example of such
a machine is the e/ectric motor.
Another example is the re/ay, an electrically-controlled
switch. If a switch contact mechanism is built so that it can
be actuated (opened and closed) by the application of a
magnetic field, and an electromagnet coil is placed in the
near vicinity to produce that requisite field, it will be
possible to open and close the switch by the application of a
current through the coil. In effect, this gives us a device that
enables elelctricity to control electricity:
Relay
—g— ';‘;
Au wt
v v
4 '
a { )
‘ t
" a
7i Wy
my 1, 1 5
Applying current through the coil
causes the switch to close.
Relays can be constructed to actuate multiple switch
contacts, or operate them in "reverse" (energizing the coil
will open the switch contact, and unpowering the coil will
allow it to spring closed again).
Multiple-contact
relay
ee Ae Relay with "normally-
} ae closed" contact
Ber ae a oe oe
¢ REVIEW:
When electrons flow through a conductor, a magnetic
field will be produced around that conductor.
e The left-hand rule states that the magnetic flux lines
produced by a current-carrying wire will be oriented the
Same direction as the curled fingers of a person's left
hand (in the "hitchhiking" position), with the thumb
pointing in the direction of electron flow.
e The magnetic field force produced by a current-carrying
wire can be greatly increased by shaping the wire into a
coil instead of a straight line. If wound in a coil shape,
the magnetic field will be oriented along the axis of the
coil's length.
e The magnetic field force produced by an electromagnet
(called the magnetomotive force, or mmf), is
proportional to the product (multiplication) of the
current through the electromagnet and the number of
complete coil "turns" formed by the wire.
Magnetic units of measurement
If the burden of two systems of measurement for common
quantities (English vs. metric) throws your mind into
confusion, this is not the place for you! Due to an early lack
of standardization in the science of magnetism, we have
been plagued with no less than three complete systems of
measurement for magnetic quantities.
First, we need to become acquainted with the various
quantities associated with magnetism. There are quite a few
more quantities to be dealt with in magnetic systems than
for electrical systems. With electricity, the basic quantities
are Voltage (E), Current (I), Resistance (R), and Power (P).
The first three are related to one another by Ohm's Law
(E=IR ; I=E/R ; R=E/I), while Power is related to voltage,
current, and resistance by Joule's Law (P=IE ; P=I?R ;
P=E2/R).
With magnetism, we have the following quantities to deal
with:
Magnetomotive Force -- The quantity of magnetic field
force, or "push." Analogous to electric voltage (electromotive
force).
Field Flux -- The quantity of total field effect, or
"substance" of the field. Analogous to electric current.
Field Intensity -- The amount of field force (mmf)
distributed over the length of the electromagnet. Sometimes
referred to as Magnetizing Force.
Flux Density -- The amount of magnetic field flux
concentrated in a given area.
Reluctance -- The opposition to magnetic field flux through
a given volume of space or material. Analogous to electrical
resistance.
Permeability -- The specific measure of a material's
acceptance of magnetic flux, analogous to the specific
resistance of a conductive material (p), except inverse
(greater permeability means easier passage of magnetic
flux, whereas greater specific resistance means more
difficult passage of electric current).
But wait... the fun is just beginning! Not only do we have
more quantities to keep track of with magnetism than with
electricity, but we have several different systems of unit
measurement for each of these quantities. As with common
quantities of length, weight, volume, and temperature, we
have both English and metric systems. However, there is
actually more than one metric system of units, and multiple
metric systems are used in magnetic field measurements!
One is called the cgs, which stands for Centimeter-Gram-
Second, denoting the root measures upon which the whole
system is based. The other was originally Known as the mks
system, which stood for Meter-Kilogram-Second, which was
later revised into another system, called rmks, standing for
Rationalized Meter-Kilogram-Second. This ended up being
adopted as an international standard and renamed S/
(Systeme International).
Unit of Measurement
and abbreviation
Field Force Gilbert (Gb)
Field Flux | = | Maxwell (Mx) Weber (Wb)
Field Amp-turns Amp-turns
Quantity | Symbol
Flux Lines per
Density pa Gauss (G) Tesla (T) square neh
> Gilberts per | Amp-turns Amp-turns
ui Gauss per Tesla-meters | -Lines per
And yes, the u symbol is really the same as the metric prefix
"micro." | find this especially confusing, using the exact
same alphabetical character to symbolize both a specific
quantity and a general metric prefix!
As you might have guessed already, the relationship
between field force, field flux, and reluctance is much the
same as that between the electrical quantities of
electromotive force (E), current (Il), and resistance (R). This
provides something akin to an Ohm's Law for magnetic
circuits:
A comparison of "Ohm’s Law" for
electric and magnetic circuits:
E=I1R mmf = DR
Electrical Magnetic
And, given that permeability is inversely analogous to
specific resistance, the equation for finding the reluctance of
a magnetic material is very similar to that for finding the
resistance of a conductor:
A comparison of electrical
and magnetic opposition:
R= p s RK = ae
A WA
Electrical Magnetic
In either case, a longer piece of material provides a greater
opposition, all other factors being equal. Also, a larger cross-
sectional area makes for less opposition, all other factors
being equal.
The major caveat here is that the reluctance of a material to
magnetic flux actually changes with the concentration of
flux going through it. This makes the "Ohm's Law" for
magnetic circuits nonlinear and far more difficult to work
with than the electrical version of Ohm's Law. It would be
analogous to having a resistor that changed resistance as
the current through it varied (a circuit composed of varistors
instead of resistors).
Permeability and saturation
The nonlinearity of material permeability may be graphed
for better understanding. We'll place the quantity of field
intensity (H), equal to field force (mmf) divided by the
length of the material, on the horizontal axis of the graph.
On the vertical axis, we'll place the quantity of flux density
(B), equal to total flux divided by the cross-sectional area of
the material. We will use the quantities of field intensity (H)
and flux density (B) instead of field force (mmf) and total
flux (®) so that the shape of our graph remains independent
of the physical dimensions of our test material. What we're
trying to do here is show a mathematical relationship
between field force and flux for any chunk of a particular
substance, in the same spirit as describing a material's
specific resistance in ohm-cmil/ft instead of its actual
resistance in ohms.
sheet steel
“Cast steel
Flux density
(B)
, cast iron
Field intensity (H)
This is called the normal magnetization curve, or B-H curve,
for any particular material. Notice how the flux density for
any of the above materials (cast iron, cast steel, and sheet
steel) levels off with increasing amounts of field intensity.
This effect is known as saturation. When there is little
applied magnetic force (low H), only a few atoms are in
alignment, and the rest are easily aligned with additional
force. However, as more flux gets crammed into the same
cross-sectional area of a ferromagnetic material, fewer atoms
are available within that material to align their electrons
with additional force, and so it takes more and more force
(H) to get less and less "help" from the material in creating
more flux density (B). To put this in economic terms, we're
seeing a case of diminishing returns (B) on our investment
(H). Saturation is a phenomenon limited to iron-core
electromagnets. Air-core electromagnets don't saturate, but
on the other hand they don't produce nearly as much
magnetic flux as a ferromagnetic core for the same number
of wire turns and current.
Another quirk to confound our analysis of magnetic flux
versus force is the phenomenon of magnetic hysteresis. As a
general term, hysteresis means a lag between input and
output in a system upon a change in direction. Anyone
who's ever driven an old automobile with "loose" steering
knows what hysteresis is: to change from turning left to
turning right (or vice versa), you have to rotate the steering
wheel an additional amount to overcome the built-in "lag" in
the mechanical linkage system between the steering wheel
and the front wheels of the car. In a magnetic system,
hysteresis is seen in a ferromagnetic material that tends to
stay magnetized after an applied field force has been
removed (see "retentivity” in the first section of this
chapter), if the force is reversed in polarity.
Let's use the same graph again, only extending the axes to
indicate both positive and negative quantities. First we'll
apply an increasing field force (current through the coils of
our electromagnet). We should see the flux density increase
(go up and to the right) according to the normal
magnetization curve:
Flux density
(B)
<e-ss= O
Field intensity (H)
Next, we'll stop the current going through the coil of the
electromagnet and see what happens to the flux, leaving the
first curve still on the graph:
Flux density
(B)
Field intensity (H)
Due to the retentivity of the material, we still have a
magnetic flux with no applied force (no current through the
coil). Our electromagnet core is acting as a permanent
magnet at this point. Now we will slowly apply the same
amount of magnetic field force in the opposite direction to
our sample:
Flux density
(B)
j Field intensity (H)
The flux density has now reached a point equivalent to what
it was with a full positive value of field intensity (H), except
in the negative, or opposite, direction. Let's stop the current
going through the coil again and see how much flux
remains:
Flux density
(B)
Field intensity (H)
Once again, due to the natural retentivity of the material, it
will hold a magnetic flux with no power applied to the coil,
except this time its in a direction opposite to that of the last
time we stopped current through the coil. If we re-apply
power in a positive direction again, we should see the flux
density reach its prior peak in the upper-right corner of the
graph again:
Flux density
(B)
f Field intensity (H)
/ —»
The "S"-shaped curve traced by these steps form what is
called the hysteresis curve of a ferromagnetic material for a
given set of field intensity extremes (-H and +H). If this
doesn't quite make sense, consider a hysteresis graph for
the automobile steering scenario described earlier, one
graph depicting a "tight" steering system and one depicting
a "loose" system:
An ideal steering system
angle of front wheels
(right) ”
rotation of
(CW) steering wheel
(left)
A "loose" steering system
angle of front wheels
(right) Se 7
rotation of
(CW) steering wheel
(left)
_amount of "looseness"
in the steering mechanism
Just as in the case of automobile steering systems,
hysteresis can be a problem. If you're designing a system to
produce precise amounts of magnetic field flux for given
amounts of current, hysteresis may hinder this design goal
(due to the fact that the amount of flux density would
depend on the current and how strongly it was magnetized
before!). Similarly, a loose steering system is unacceptable
in a race car, where precise, repeatable steering response is
a necessity. Also, having to overcome prior magnetization in
an electromagnet can be a waste of energy if the current
used to energize the coil is alternating back and forth (AC).
The area within the hysteresis curve gives a rough estimate
of the amount of this wasted energy.
Other times, magnetic hysteresis is a desirable thing. Such
is the case when magnetic materials are used as a means of
storing information (computer disks, audio and video tapes).
In these applications, it is desirable to be able to magnetize
a speck of iron oxide (ferrite) and rely on that material's
retentivity to "remember" its last magnetized state. Another
productive application for magnetic hysteresis is in filtering
high-frequency electromagnetic "noise" (rapidly alternating
surges of voltage) from signal wiring by running those wires
through the middle of a ferrite ring. The energy consumed in
overcoming the hysteresis of ferrite attenuates the strength
of the "noise" signal. Interestingly enough, the hysteresis
curve of ferrite is quite extreme:
Hysteresis curve for ferrite
Flux density
(B)
Field intensity (H)
e REVIEW:
e The permeability of a material changes with the amount
of magnetic flux forced through it.
The specific relationship of force to flux (field intensity H
to flux density B) is graphed in a form called the normal
magnetization curve.
It is possible to apply so much magnetic field force to a
ferromagnetic material that no more flux can be
crammed into it. This condition is Known as magnetic
saturation.
When the retentivity of a ferromagnetic substance
interferes with its re-magnetization in the opposite
direction, a condition known as hysteresis occurs.
Electromagnetic induction
While Oersted's surprising discovery of electromagnetism
paved the way for more practical applications of electricity,
it was Michael Faraday who gave us the key to the practical
generation of electricity: electromagnetic induction. Faraday
discovered that a voltage would be generated across a
length of wire if that wire was exposed to a perpendicular
magnetic field flux of changing intensity.
An easy way to create a magnetic field of changing intensity
is to move a permanent magnet next to a wire or coil of wire.
Remember: the magnetic field must increase or decrease in
intensity perpendicular to the wire (so that the lines of flux
“cut across" the conductor), or else no voltage will be
induced:
Electromagnetic induction
_ current changes direction Jo
with change in magnet motion
jf
+ voltage changes polarity
with change in magnet motion
~~
magnet moved
back and forth
—_<p
Faraday was able to mathematically relate the rate of
change of the magnetic field flux with induced voltage (note
the use of a lower-case letter "e" for voltage. This refers to
instantaneous voltage, or voltage at a specific point in time,
rather than a steady, stable voltage.):
e= N——
dt
Where,
e= (Instantaneous) induced voltage in volts
N= Number of turns in wire coil (straight wire = 1)
® = Magnetic flux in Webers
t= Time in seconds
The "d" terms are standard calculus notation, representing
rate-of-change of flux over time. "N" stands for the number
of turns, or wraps, in the wire coil (assuming that the wire is
formed in the shape of a coil for maximum electromagnetic
efficiency).
This phenomenon is put into obvious practical use in the
construction of electrical generators, which use mechanical
power to move a magnetic field past coils of wire to generate
voltage. However, this is by no means the only practical use
for this principle.
If we recall that the magnetic field produced by a current-
carrying wire was always perpendicular to that wire, and
that the flux intensity of that magnetic field varied with the
amount of current through it, we can see that a wire is
capable of inducing a voltage along its own length simply
due to a change in current through it. This effect is called
self-induction: a changing magnetic field produced by
changes in current through a wire inducing voltage along
the length of that same wire. If the magnetic field flux is
enhanced by bending the wire into the shape of a coil,
and/or wrapping that coil around a material of high
permeability, this effect of self-induced voltage will be more
intense. A device constructed to take advantage of this
effect is called an inductor, and will be discussed in greater
detail in the next chapter.
REVIEW:
A magnetic field of changing intensity perpendicular to
a wire will induce a voltage along the length of that wire.
The amount of voltage induced depends on the rate of
change of the magnetic field flux and the number of
turns of wire (if coiled) exposed to the change in flux.
Faraday's equation for induced voltage: e = N(d@/dt)
A current-carrying wire will experience an induced
voltage along its length if the current changes (thus
changing the magnetic field flux perpendicular to the
wire, thus inducing voltage according to Faraday's
formula). A device built specifically to take advantage of
this effect is called an inductor.
Mutual inductance
If two coils of wire are brought into close proximity with each
other so the magnetic field from one links with the other, a
voltage will be generated in the second coil as a result. This
is called mutual inductance: when voltage impressed upon
one coil induces a voltage in another.
A device specifically designed to produce the effect of
mutual inductance between two or more coils is called a
transformer.
A MUTUAL INDUCTANCE STANDARD
—
3
r4
- 7
—
foe
f
Ya
_
fA
is
Ae hits
+ MINE
The device shown in the above photograph is a kind of
transformer, with two concentric wire coils. It is actually
intended as a precision standard unit for mutual inductance,
but for the purposes of illustrating what the essence of a
transformer is, it will suffice. The two wire coils can be
distinguished from each other by color: the bulk of the
tube's length is wrapped in green-insulated wire (the first
coil) while the second coil (wire with bronze-colored
insulation) stands in the middle of the tube's length. The
wire ends run down to connection terminals at the bottom of
the unit. Most transformer units are not built with their wire
coils exposed like this.
Because magnetically-induced voltage only happens when
the magnetic field flux is changing in strength relative to the
wire, mutual inductance between two coils can only happen
with alternating (changing -- AC) voltage, and not with
direct (steady -- DC) voltage. The only applications for
mutual inductance in a DC system is where some means is
available to switch power on and off to the coil (thus
creating a pulsing DC voltage), the induced voltage peaking
at every pulse.
A very useful property of transformers is the ability to
transform voltage and current levels according to a simple
ratio, determined by the ratio of input and output coil turns.
If the energized coil of a transformer is energized by an AC
voltage, the amount of AC voltage induced in the
unpowered coil will be equal to the input voltage multiplied
by the ratio of output to input wire turns in the coils.
Conversely, the current through the windings of the output
coil compared to the input coil will follow the opposite ratio:
if the voltage is increased from input coil to output coil, the
current will be decreased by the same proportion. This
action of the transformer is analogous to that of mechanical
gear, belt sheave, or chain sprocket ratios:
Torque-reducing geartrain
Large gear
(many teeth)
Small gear
(few teeth)
low torque. high speed
high torque. low speed
"Step-down" transformer
AC voltage
source
low current
A transformer designed to output more voltage than it takes
in across the input coil is called a "step-up" transformer,
while one designed to do the opposite is called a "step-
down," in reference to the transformation of voltage that
takes place. The current through each respective coil, of
course, follows the exact opposite proportion.
e REVIEW:
e Mutual inductance is where the magnetic field
generated by a coil of wire induces voltage in an
adjacent coil of wire.
e A transformer is a device constructed of two or more
coils in close proximity to each other, with the express
purpose of creating a condition of mutual inductance
between the coils.
e Transformers only work with changing voltages, not
steady voltages. Thus, they may be classified as an AC
device and not a DC device.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4]
—| | +4/l—
Lessons In Electric Circuits
-- Volume |
Chapter 15
INDUCTORS
e Magnetic fields and inductance
Inductors and calculus
Factors affecting inductance
Series and parallel inductors
Practical considerations
Contributors
Magnetic fields and inductance
Whenever electrons flow through a conductor, a magnetic
field will develop around that conductor. This effect is called
electromagnetism. Magnetic fields effect the alignment of
electrons in an atom, and can cause physical force to develop
between atoms across space just as with electric fields
developing force between electrically charged particles. Like
electric fields, magnetic fields can occupy completely empty
Space, and affect matter at a distance.
Fields have two measures: a field force and a field flux. The
field force is the amount of "push" that a field exerts over a
certain distance. The field f/ux is the total quantity, or effect,
of the field through space. Field force and flux are roughly
analogous to voltage ("push") and current (flow) through a
conductor, respectively, although field flux can exist in
totally empty space (without the motion of particles such as
electrons) whereas current can only take place where there
are free electrons to move. Field flux can be opposed in
Space, just as the flow of electrons can be opposed by
resistance. The amount of field flux that will develop in space
IS proportional to the amount of field force applied, divided
by the amount of opposition to flux. Just as the type of
conducting material dictates that conductor's specific
resistance to electric current, the type of material occupying
the space through which a magnetic field force is impressed
dictates the specific opposition to magnetic field flux.
Whereas an electric field flux between two conductors allows
for an accumulation of free electron charge within those
conductors, a magnetic field flux allows for a certain "inertia"
to accumulate in the flow of electrons through the conductor
producing the field.
Inductors are components designed to take advantage of this
phenomenon by shaping the length of conductive wire in the
form of a coil. This shape creates a stronger magnetic field
than what would be produced by a straight wire. Some
inductors are formed with wire wound in a self-supporting
coil. Others wrap the wire around a solid core material of
some type. Sometimes the core of an inductor will be
straight, and other times it will be joined in a loop (square,
rectangular, or circular) to fully contain the magnetic flux.
These design options all have an effect on the performance
and characteristics of inductors.
The schematic symbol for an inductor, like the capacitor, is
quite simple, being little more than a coil symbol
representing the coiled wire. Although a simple coil shape is
the generic symbol for any inductor, inductors with cores are
sometimes distinguished by the addition of parallel lines to
the axis of the coil. A newer version of the inductor symbol
dispenses with the coil shape in favor of several "humps" in a
row:
Inductor symbols
; 3|
generic, or air-core iron core
iron core generic
(alternative) (néwer symbol)
As the electric current produces a concentrated magnetic
field around the coil, this field flux equates to a storage of
energy representing the kinetic motion of the electrons
through the coil. The more current in the coil, the stronger
the magnetic field will be, and the more energy the inductor
will store.
‘ magn etic
is— field
Because inductors store the kinetic energy of moving
electrons in the form of a magnetic field, they behave quite
differently than resistors (which simply dissipate energy in
the form of heat) in a circuit. Energy storage in an inductor is
a function of the amount of current through it. An inductor's
ability to store energy as a function of current results in a
tendency to try to maintain current at a constant level. In
other words, inductors tend to resist changes in current.
When current through an inductor is increased or decreased,
the inductor "resists" the change by producing a voltage
between its leads in opposing polarity to the change.
To store more energy in an inductor, the current through it
must be increased. This means that its magnetic field must
increase in strength, and that change in field strength
produces the corresponding voltage according to the
principle of electromagnetic self-induction. Conversely, to
release energy from an inductor, the current through it must
be decreased. This means that the inductor's magnetic field
must decrease in strength, and that change in field strength
self-induces a voltage drop of just the opposite polarity.
Just as Isaac Newton's first Law of Motion ("an object in
motion tends to stay in motion; an object at rest tends to
stay at rest") describes the tendency of a mass to oppose
changes in velocity, we can state an inductor's tendency to
oppose changes in current as such: "Electrons moving
through an inductor tend to stay in motion; electrons at rest
In an inductor tend to stay at rest." Hypothetically, an
inductor left short-circuited will maintain a constant rate of
current through it with no external assistance:
—_—
— >
current sustained with
the inductor short-circuited
Practically soeaking, however, the ability for an inductor to
self-sustain current is realized only with superconductive
wire, as the wire resistance in any normal inductor is enough
to cause current to decay very quickly with no external
source of power.
When the current through an inductor is increased, it drops a
voltage opposing the direction of electron flow, acting asa
power load. In this condition the inductor is said to be
charging, because there is an increasing amount of energy
being stored in its magnetic field. Note the polarity of the
voltage with regard to the direction of current:
Energy being absorbed by
the inductor from the rest
of the circuit.
~— increasing current
+ ‘\
os vallage Gop
increasing current —>
The inductor acts as a LOAD
Conversely, when the current through the inductor is
decreased, it drops a voltage aiding the direction of electron
flow, acting aS a power source. In this condition the inductor
is said to be discharging, because its store of energy is
decreasing as it releases energy from its magnetic field to the
rest of the circuit. Note the polarity of the voltage with regard
to the direction of current.
Energy being released by
the inductor to the rest
of the circuit.
—— decreasing current
...to the rest of ; voltage dro
the circuit
a 4
decreasing current —>
The inductor acts as a SOURCE
If a source of electric power is suddenly applied to an
unmagnetized inductor, the inductor will initially resist the
flow of electrons by dropping the full voltage of the source.
As current begins to increase, a stronger and stronger
magnetic field will be created, absorbing energy from the
source. Eventually the current reaches a maximum level, and
stops increasing. At this point, the inductor stops absorbing
energy from the source, and is dropping minimum voltage
across its leads, while the current remains at a maximum
level. As an inductor stores more energy, its current level
increases, while its voltage drop decreases. Note that this is
precisely the opposite of capacitor behavior, where the
storage of energy results in an increased voltage across the
component! Whereas capacitors store their energy charge by
maintaining a static voltage, inductors maintain their energy
"charge" by maintaining a steady current through the coil.
The type of material the wire is coiled around greatly impacts
the strength of the magnetic field flux (and therefore the
amount of stored energy) generated for any given amount of
current through the coil. Coil cores made of ferromagnetic
materials (such as soft iron) will encourage stronger field
fluxes to develop with a given field force than nonmagnetic
substances such as aluminum or air.
The measure of an inductor's ability to store energy fora
given amount of current flow is called inductance. Not
surprisingly, inductance Is also a measure of the intensity of
opposition to changes in current (exactly how much self-
induced voltage will be produced for a given rate of change
of current). Inductance is symbolically denoted with a capital
"L," and is measured in the unit of the Henry, abbreviated as
a
An obsolete name for an inductor is choke, so called for its
common usage to block ("choke") high-frequency AC signals
in radio circuits. Another name for an inductor, still used in
modern times, is reactor, especially when used in large
power applications. Both of these names will make more
sense after you've studied alternating current (AC) circuit
theory, and especially a principle Known as inductive
reactance.
e REVIEW:
e Inductors react against changes in current by dropping
voltage in the polarity necessary to oppose the change.
e When an inductor is faced with an increasing current, it
acts as a load: dropping voltage as it absorbs energy
(negative on the current entry side and positive on the
current exit side, like a resistor).
e When an inductor is faced with a decreasing current, it
acts as a source: creating voltage as it releases stored
energy (positive on the current entry side and negative
on the current exit side, like a battery).
e The ability of an inductor to store energy in the form of a
magnetic field (and consequently to oppose changes in
current) is called inductance. It is measured in the unit of
the Henry (H).
e Inductors used to be commonly known by another term:
Choke. In large power applications, they are sometimes
referred to as reactors.
Inductors and calculus
Inductors do not have a stable "resistance" as conductors do.
However, there is a definite mathematical relationship
between voltage and current for an inductor, as follows:
"Ohm's Law” for an inductor
= Ll
dt
Where,
v = Instantaneous voltage across the inductor
L = Inductance in Henrys
di
—— = Instantaneous rate of current change
dt = (amps per second)
You should recognize the form of this equation from the
capacitor chapter. It relates one variable (in this case,
inductor voltage drop) to a rate of change of another variable
(in this case, inductor current). Both voltage (v) and rate of
current change (di/dt) are instantaneous: that is, in relation
to a specific point in time, thus the lower-case letters "v" and
"i", AS with the capacitor formula, it is convention to express
instantaneous voltage as vrather than e, but using the latter
designation would not be wrong. Current rate-of-change
(di/dt) is expressed in units of amps per second, a positive
number representing an increase and a negative number
representing a decrease.
Like a capacitor, an inductor's behavior is rooted in the
variable of time. Aside from any resistance intrinsic to an
inductor's wire coil (which we will assume is zero for the sake
of this section), the voltage dropped across the terminals of
an inductor is purely related to how quickly its current
changes over time.
Suppose we were to connect a perfect inductor (one having
zero ohms of wire resistance) to a circuit where we could vary
the amount of current through it with a potentiometer
connected as a variable resistor:
Voltmeter
(zero-center)
If the potentiometer mechanism remains in a single position
(wiper is stationary), the series-connected ammeter will
register a constant (unchanging) current, and the voltmeter
connected across the inductor will register 0 volts. In this
scenario, the instantaneous rate of current change (di/dt) is
equal to zero, because the current is stable. The equation
tells us that with 0 amps per second change for a di/dt, there
must be zero instantaneous voltage (v) across the inductor.
From a physical perspective, with no current change, there
will be a steady magnetic field generated by the inductor.
With no change in magnetic flux (d®/dt = 0 Webers per
second), there will be no voltage dropped across the length
of the coil due to induction.
Inductor
current
1,
Time —
Potentiometer wiper not moving
Inductor
voltage
E,
Time —>
If we move the potentiometer wiper slowly in the "up"
direction, its resistance from end to end will slowly decrease.
This has the effect of increasing current in the circuit, so the
ammeter indication should be increasing at a slow rate:
Potentiometer wiper moving
slowly in the "up" direction
Steady
voltage
Increasing
current
Assuming that the potentiometer wiper is being moved such
that the rate of current increase through the inductor is
steady, the di/dt term of the formula will be a fixed value.
This fixed value, multiplied by the inductor's inductance in
Henrys (also fixed), results in a fixed voltage of some
magnitude. From a physical perspective, the gradual increase
in current results in a magnetic field that is likewise
increasing. This gradual increase in magnetic flux causes a
voltage to be induced in the coil as expressed by Michael
Faraday's induction equation e = N(d@®/dt). This self-induced
voltage across the coil, as a result of a gradual change in
current magnitude through the coil, happens to be of a
polarity that attempts to oppose the change in current. In
other words, the induced voltage polarity resulting from an
increase in current will be oriented in such a way as to push
against the direction of current, to try to keep the current at
its former magnitude. This phenomenon exhibits a more
general principle of physics known as Lenz's Law, which
states that an induced effect will always be opposed to the
cause producing it.
In this scenario, the inductor will be acting as a /oad, with the
negative side of the induced voltage on the end where
electrons are entering, and the positive side of the induced
voltage on the end where electrons are exiting.
~
Inductor i i Current
current rs \ change
1
Time —>
Potentiometer wiper moving slowly "up"
Inductor
voltage
E,
Time —~
Changing the rate of current increase through the inductor
by moving the potentiometer wiper "up" at different speeds
results in different amounts of voltage being dropped across
the inductor, all with the same polarity (opposing the
increase in current):
Inductor
current
1
Time —>
Potentiometer wiper moving "up" at
different rates
Inductor
voltage
E,
Time —>
Here again we see the derivative function of calculus
exhibited in the behavior of an inductor. In calculus terms,
we would say that the induced voltage across the inductor is
the derivative of the current through the inductor: that is,
proportional to the current's rate-of-change with respect to
time.
Reversing the direction of wiper motion on the potentiometer
(going "down" rather than "up") will result in its end-to-end
resistance increasing. This will result in circuit current
decreasing (a negative figure for di/dt). The inductor, always
opposing any change in current, will produce a voltage drop
opposed to the direction of change:
Potentiometer wiper moving
in the "down" direction
Decreasing
current
How much voltage the inductor will produce depends, of
course, on how rapidly the current through it is decreased. As
described by Lenz's Law, the induced voltage will be opposed
to the change in current. With a decreasing current, the
voltage polarity will be oriented so as to try to keep the
current at its former magnitude. In this scenario, the inductor
will be acting as a source, with the negative side of the
induced voltage on the end where electrons are exiting, and
the positive side of the induced voltage on the end where
electrons are entering. The more rapidly current is decreased,
the more voltage will be produced by the inductor, in its
release of stored energy to try to keep current constant.
Again, the amount of voltage across a perfect inductor is
directly proportional to the rate of current change through it.
The only difference between the effects of a decreasing
Current and an increasing current is the polarity of the
induced voltage. For the same rate of current change over
time, either increasing or decreasing, the voltage magnitude
(volts) will be the same. For example, a di/dt of -2 amps per
second will produce the same amount of induced voltage
drop across an inductor as a di/dt of +2 amps per second,
just in the opposite polarity.
If current through an inductor is forced to change very
rapidly, very high voltages will be produced. Consider the
following circuit:
Neon lamp
Switch
In this circuit, a lamp is connected across the terminals of an
inductor. A switch is used to control current in the circuit, and
power is supplied by a 6 volt battery. When the switch is
closed, the inductor will briefly oppose the change in current
from zero to some magnitude, but will drop only a small
amount of voltage. It takes about 70 volts to ionize the neon
gas inside a neon bulb like this, so the bulb cannot be lit on
the 6 volts produced by the battery, or the low voltage
momentarily dropped by the inductor when the switch is
closed:
no light
When the switch is opened, however, it suddenly introduces
an extremely high resistance into the circuit (the resistance
of the air gap between the contacts). This sudden
introduction of high resistance into the circuit causes the
circuit current to decrease almost instantly. Mathematically,
the di/dt term will be a very large negative number. Such a
rapid change of current (from some magnitude to zero in
very little time) will induce a very high voltage across the
inductor, oriented with negative on the left and positive on
the right, in an effort to oppose this decrease in current. The
voltage produced is usually more than enough to light the
neon lamp, if only for a brief moment until the current decays
to zero:
Light!
For maximum effect, the inductor should be sized as large as
possible (at least 1 Henry of inductance).
Factors affecting inductance
There are four basic factors of inductor construction
determining the amount of inductance created. These factors
all dictate inductance by affecting how much magnetic field
flux will develop for a given amount of magnetic field force
(current through the inductor's wire coil):
NUMBER OF WIRE WRAPS, OR "TURNS" IN THE COIL:
All other factors being equal, a greater number of turns of
wire in the coil results in greater inductance; fewer turns of
wire in the coil results in less inductance.
Explanation: More turns of wire means that the coil will
generate a greater amount of magnetic field force (measured
in amp-turns!), for a given amount of coil current.
less inductance more inductance
: a
COIL AREA: All other factors being equal, greater coil area
(as measured looking lengthwise through the coil, at the
cross-section of the core) results in greater inductance; less
coil area results in less inductance.
Explanation: Greater coil area presents less opposition to the
formation of magnetic field flux, for a given amount of field
force (amp-turns).
less inductance more inductance
: =
COIL LENGTH: All other factors being equal, the longer the
coil's length, the less inductance; the shorter the coil's
length, the greater the inductance.
Explanation: A longer path for the magnetic field flux to take
results in more opposition to the formation of that flux for
any given amount of field force (amp-turns).
less inductance more inductance
3
CORE MATERIAL: All other factors being equal, the greater
the magnetic permeability of the core which the coil is
wrapped around, the greater the inductance; the less the
permeability of the core, the less the inductance.
Explanation: A core material with greater magnetic
permeability results in greater magnetic field flux for any
given amount of field force (amp-turns).
less inductance more inductance
air core soft iron core
(permeability = 1) (permeability = 600)
An approximation of inductance for any coil of wire can be
found with this formula:
NBA
IL = Mello
ek
Where, a
L = Inductance of coil in Henrys
N= Number of turns in wire coil (straight wire = 1)
it= Permeability of core material (absolute, not relative)
Lt. = Relative permeability, dimensionless (1,=1 for air)
l= 1.26 x 10 ® T-m/At permeability of free space
A = Area of coil in square meters = mr
|= Average length of coil in meters
It must be understood that this formula yields approximate
figures only. One reason for this is the fact that permeability
changes as the field intensity varies (remember the nonlinear
"B/H" curves for different materials). Obviously, if
permeability (u) in the equation is unstable, then the
inductance (L) will also be unstable to some degree as the
current through the coil changes in magnitude. If the
hysteresis of the core material is significant, this will also
have strange effects on the inductance of the coil. Inductor
designers try to minimize these effects by designing the core
In such a way that its flux density never approaches
saturation levels, and so the inductor operates in a more
linear portion of the B/H curve.
If an inductor is designed so that any one of these factors
may be varied at will, its inductance will correspondingly
vary. Variable inductors are usually made by providing a way
to vary the number of wire turns in use at any given time, or
by varying the core material (a sliding core that can be
moved in and out of the coil). An example of the former
design is shown in this photograph:
7)
hall ' we Sy 1
Ih) H
MAD A OME IOAN ant
¥ \
UP? ty » a a)
\ ages iy Nish = eT!
te oy ’ heh Lek nay’
Y 0 " ‘ it} 1 ;
7 ! iy | Wt 1! Ay) if
| nye i HVT 1 |
vi Wl) Hh AMUOU WYNN CAAA Ml
* i i) Haiti tft } Wt fl } \ '
iI st | I} | |
}
ail AANA au |
kad aaa
Be mM ii } i l bie i "
This unit uses sliding copper contacts to tap into the coil at
different points along its length. The unit shown happens to
be an air-core inductor used in early radio work.
A fixed-value inductor is shown in the next photograph,
another antique air-core unit built for radios. The connection
terminals can be seen at the bottom, as well as the few turns
of relatively thick wire:
Here is another inductor (of greater inductance value), also
intended for radio applications. Its wire coil is wound around
a white ceramic tube for greater rigidity:
Inductors can also be made very small for printed circuit
board applications. Closely examine the following
photograph and see if you can identify two inductors near
each other:
Ww
The two inductors on this circuit board are labeled L, and L>,
and they are located to the right-center of the board. Two
nearby components are R3 (a resistor) and Cy. (a capacitor).
These inductors are called "toroidal" because their wire coils
are wound around donut-shaped ("torus") cores.
Like resistors and capacitors, inductors can be packaged as
“surface mount devices" as well. The following photograph
shows just how small an inductor can be when packaged as
such:
mc
8 8-0 we) 98
8 af we
a a a a |
; a0
eg oo TI 8S
q 7 Fi 4)
HVE ‘BRccro yc
pal A
2 g ows |
= ae
MMM «og
A pair of inductors can be seen on this circuit board, to the
right and center, appearing as small black chips with the
number "100" printed on both. The upper inductor's label
can be seen printed on the green circuit board as Ls. Of
course these inductors are very small in inductance value,
but it demonstrates just how tiny they can be manufactured
to meet certain circuit design needs.
Series and parallel inductors
When inductors are connected in series, the total inductance
is the sum of the individual inductors’ inductances. To
understand why this is so, consider the following: the
definitive measure of inductance is the amount of voltage
dropped across an inductor for a given rate of current change
through it. If inductors are connected together in series (thus
sharing the same current, and seeing the same rate of
change in current), then the total voltage dropped as the
result of a change in current will be additive with each
inductor, creating a greater total voltage than either of the
individual inductors alone. Greater voltage for the same rate
of change in current means greater inductance.
[total voltage drop | [total voltage drop | drop |+
Pe =
increase in ge | epee ener bos —-
Thus, the total inductance for series inductors is more than
any one of the individual inductors’ inductances. The formula
for calculating the series total inductance is the same form as
for calculating series resistances:
Series Inductances
Lota! = Lb; +b,+.-.-.L,
When inductors are connected in parallel, the total
inductance is less than any one of the parallel inductors'
inductances. Again, remember that the definitive measure of
inductance is the amount of voltage dropped across an
inductor for a given rate of current change through it. Since
the current through each parallel inductor will be a fraction
of the total current, and the voltage across each parallel
inductor will be equal, a change in total current will result in
less voltage dropped across the parallel array than for any
one of the inductors considered separately. In other words,
there will be less voltage dropped across parallel inductors
for a given rate of change in current than for any of those
inductors considered separately, because total current
divides among parallel branches. Less voltage for the same
rate of change in current means less inductance.
+ ~~
= |. |voltage
oo aE drop
i
O
increase In current ——>
Thus, the total inductance is less than any one of the
individual inductors' inductances. The formula for calculating
the parallel total inductance is the same form as for
calculating parallel resistances:
Parallel Inductances
Lirtal =
e REVIEW:
e Inductances add in series.
e Inductances diminish in parallel.
Practical considerations
Inductors, like all electrical components, have limitations
which must be respected for the sake of reliability and proper
circuit operation.
Rated current: Since inductors are constructed of coiled wire,
and any wire will be limited in its current-carrying capacity
by its resistance and ability to dissipate heat, you must pay
attention to the maximum current allowed through an
inductor.
Equivalent circuit: Since inductor wire has some resistance,
and circuit design constraints typically demand the inductor
be built to the smallest possible dimensions, there is no such
thing as a "perfect" inductor. Inductor coil wire usually
presents a substantial amount of series resistance, and the
close spacing of wire from one coil turn to another (separated
by insulation) may present measurable amounts of stray
Capacitance to interact with its purely inductive
characteristics. Unlike capacitors, which are relatively easy to
manufacture with negligible stray effects, inductors are
difficult to find in "pure" form. In certain applications, these
undesirable characteristics may present significant
engineering problems.
Inductor size: Inductors tend to be much larger, physically,
than capacitors are for storing equivalent amounts of energy.
This is especially true considering the recent advances in
electrolytic capacitor technology, allowing incredibly large
Capacitance values to be packed into a small package. If a
circuit designer needs to store a large amount of energy ina
small volume and has the freedom to choose either
capacitors or inductors for the task, he or she will most likely
choose a capacitor. A notable exception to this rule is in
applications requiring huge amounts of either capacitance or
inductance to store electrical energy: inductors made of
superconducting wire (zero resistance) are more practical to
build and safely operate than capacitors of equivalent value,
and are probably smaller too.
Interference: Inductors may affect nearby components on a
circuit board with their magnetic fields, which can extend
significant distances beyond the inductor. This is especially
true if there are other inductors nearby on the circuit board.
If the magnetic fields of two or more inductors are able to
"link" with each others’ turns of wire, there will be mutual
inductance present in the circuit as well as self-inductance,
which could very well cause unwanted effects. This is
another reason why circuit designers tend to choose
Capacitors over inductors to perform similar tasks: capacitors
inherently contain their respective electric fields neatly
within the component package and therefore do not typically
generate any "mutual" effects with other components.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
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oo + —>
— 4 —>
Lessons In Electric Circuits -
- Volume |
Chapter 16
RC AND L/R TIME
CONSTANTS
e Electrical transients
Capacitor transient response
Inductor transient response
Voltage and current calculations
Why L/R and not LR?
e Complex voltage and current calculations
e Complex circuits
e Solving for unknown time
¢« Contributors
Electrical transients
This chapter explores the response of capacitors and inductors
to sudden changes in DC voltage (called a transient voltage),
when wired in series with a resistor. Unlike resistors, which
respond instantaneously to applied voltage, capacitors and
inductors react over time as they absorb and release energy.
Capacitor transient response
Because capacitors store energy in the form of an electric field,
they tend to act like small secondary-cell batteries, being able
to store and release electrical energy. A fully discharged
Capacitor maintains zero volts across its terminals, and a
charged capacitor maintains a steady quantity of voltage across
its terminals, just like a battery. When capacitors are placed ina
circuit with other sources of voltage, they will absorb energy
from those sources, just as a secondary-cell battery will become
charged as a result of being connected to a generator. A fully
discharged capacitor, having a terminal voltage of zero, will
Initially act as a short-circuit when attached to a source of
voltage, drawing maximum current as it begins to build a
charge. Over time, the capacitor's terminal voltage rises to
meet the applied voltage from the source, and the current
through the capacitor decreases correspondingly. Once the
Capacitor has reached the full voltage of the source, it will stop
drawing current from it, and behave essentially as an open-
circuit.
Switch
R
| 10 kQ
av. Cc | 100 LF
When the switch is first closed, the voltage across the capacitor
(which we were told was fully discharged) is zero volts; thus, it
first behaves as though it were a short-circuit. Over time, the
capacitor voltage will rise to equal battery voltage, ending ina
condition where the capacitor behaves as an open-circuit.
Current through the circuit is determined by the difference in
voltage between the battery and the capacitor, divided by the
resistance of 10 kQ. As the capacitor voltage approaches the
battery voltage, the current approaches zero. Once the
Capacitor voltage has reached 15 volts, the current will be
exactly zero. Let's see how this works using real values:
Capacitor voltage
Gc 12 3 4 8 6 7 8 8. 10
Time (seconds)
| Time | Battery | Capacitor | Current |
|(seconds) | voltage | voltage | |
tee wea ee ae re
Pee te aa i ee ee ae |
ee ee ea ae ae
aaa mare ere we
aie aera eur ae ee rere
iar aaa ere a earn
ee ae ace i
Pegg a ae rae eee
The capacitor voltage's approach to 15 volts and the current's
approach to zero over time is what a mathematician would call
asymptotic: that is, they both approach their final values,
getting closer and closer over time, but never exactly reaches
their destinations. For all practical purposes, though, we can
say that the capacitor voltage will eventually reach 15 volts and
that the current will eventually equal zero.
Using the SPICE circuit analysis program, we can chart this
asymptotic buildup of capacitor voltage and decay of capacitor
current in a more graphical form (capacitor current is plotted in
terms of voltage drop across the resistor, using the resistor as a
shunt to measure current):
Capacitor charging
v1 10 dc 15
rl 12 10k
cl 2 © 100u ic=0
.tran .5 10 uic
.plot tran v(2,0) v(1,2)
end
legend:
*: v(2) Capacitor voltage
+: v(1,2) Capacitor current
time v(2)
PR aaeet See 0.000E+00 5. 000E+00 1.000E+01
.500E+01
| cel
.Q00E+00 5.976E-05 * ; ; +
.QOO0E-01 5.881E+00 . wie +.
.Q00E+00 9.474E+00 . or 6
.500E+00 1.166E+01 . + : : p
.Q00E+00 1.297E+01 . + ‘ ‘ a
NrRrRuUOo!:
2.500E+00 1.377E+01 . + ; : mys
3.000E+00 1.426E+01 . + : : ose
3.500E+00 1.455E+01 .+ : : be
4.000E+00 1.473E+01 .+ ‘ , bee
4.500E+00 1.484E+01 + =
5.Q000E+00 1.490E+01 + %
5.500E+00 1.494E+01 + *
6.000E+00 1.496E+01 + *
6.500E+00 1.498E+01 + *
7.Q000E+00 1.499E+01 + *
7.500E+00 1.499E+01 + *
8.000E+00 1.500E+01 + *
8.500E+00 1.500E+01 + *
9.000E+00 1.500E+01 + *
9.500E+00 1.500E+01 + *
1.000E+01 1.500E+01 + 5
As you can see, | have used the .plot command in the netlist
instead of the more familiar .print command. This generates a
pseudo-graphic plot of figures on the computer screen using
text characters. SPICE plots graphs in such a way that time is on
the vertical axis (going down) and amplitude (voltage/current)
is plotted on the horizontal (right=more; left=less). Notice how
the voltage increases (to the right of the plot) very quickly at
first, then tapering off as time goes on. Current also changes
very quickly at first then levels off as time goes on, but it is
approaching minimum (left of scale) while voltage approaches
maximum.
e REVIEW:
e Capacitors act somewhat like secondary-cell batteries when
faced with a sudden change in applied voltage: they
initially react by producing a high current which tapers off
over time.
e A fully discharged capacitor initially acts as a short circuit
(current with no voltage drop) when faced with the sudden
application of voltage. After charging fully to that level of
voltage, it acts as an open circuit (voltage drop with no
current).
e In a resistor-capacitor charging circuit, capacitor voltage
goes from nothing to full source voltage while current goes
from maximum to zero, both variables changing most
rapidly at first, approaching their final values slower and
slower as time goes on.
Inductor transient response
Inductors have the exact opposite characteristics of capacitors.
Whereas capacitors store energy in an e/ectric field (produced
by the voltage between two plates), inductors store energy ina
magnetic field (produced by the current through wire). Thus,
while the stored energy in a capacitor tries to maintain a
constant voltage across its terminals, the stored energy in an
inductor tries to maintain a constant current through its
windings. Because of this, inductors oppose changes in current,
and act precisely the opposite of capacitors, which oppose
changes in voltage. A fully discharged inductor (no magnetic
field), having zero current through it, will initially act as an
open-circuit when attached to a source of voltage (as it tries to
maintain zero current), dropping maximum voltage across its
leads. Over time, the inductor's current rises to the maximum
value allowed by the circuit, and the terminal voltage decreases
correspondingly. Once the inductor's terminal voltage has
decreased to a minimum (zero for a "perfect" inductor), the
current will stay at a maximum level, and it will behave
essentially as a short-circuit.
Switch
When the switch is first closed, the voltage across the inductor
will immediately jump to battery voltage (acting as though it
were an open-circuit) and decay down to zero over time
(eventually acting as though it were a short-circuit). Voltage
across the inductor is determined by calculating how much
voltage is being dropped across R, given the current through
the inductor, and subtracting that voltage value from the
battery to see what's left. When the switch is first closed, the
Current is zero, then it increases over time until it is equal to the
battery voltage divided by the series resistance of 1 Q. This
behavior is precisely opposite that of the series resistor-
Capacitor circuit, where current started at a maximum and
Capacitor voltage at zero. Let's see how this works using real
values:
Inductor voltage
0123 4 5 & f 8 8 10
Time (seconds)
| Time | Battery | Inductor | Current |
|(seconds) | voltage | voltage | |
eae oa ee ae ees ae re ae |
| 0 | 1°V | 15 V | 0 |
| 2 | 15 V | 2.030 V | 12.97 A |
ear ae cae cee
| 4 | as V | 0.275.V | 14.73 A
ae a ae i a
a ae A ee
Ne ara erage ere ara reC enya
Just as with the RC circuit, the inductor voltage's approach to 0
volts and the current's approach to 15 amps over time is
asymptotic. For all practical purposes, though, we can say that
the inductor voltage will eventually reach O volts and that the
current will eventually equal the maximum of 15 amps.
Again, we can use the SPICE circuit analysis program to chart
this asymptotic decay of inductor voltage and buildup of
inductor current in a more graphical form (inductor current is
plotted in terms of voltage drop across the resistor, using the
resistor as a Shunt to measure current):
inductor charging
v1 10 dc 15
riod <2 3)
ll 2 0 1 ic=0
.tran .5 10 uic
.plot tran v(2,0) v(1,2)
.end
legend:
*: v(2) Inductor voltage
+: v(1,2) Inductor current
time v(2)
(*+)------------ 0 .000E+00 5 .000E+00 1,.000E+01
1.500E+01
0.000E+00 1.500E+01 + : ; %
5.000E-01 9.119E+00 . + a 1s
1.000E+00 5.526E+00 . ue +.
1.500E+00 3.343E+00 . * , ; +
2.Q000E+00 2.026E+00 . * i : +
2.500E+00 1.226E+00 . * p p + ,
3.000E+00 7.429E-01 . * j j +,
3.500E+00 4.495E-01 .* ‘ . +,
4.000E+00 2.724E-01 .* j . +,
4.500E+00 1.648E-01 * +
5.000E+00 9.987E-02 * +
5.500E+00 6.042E-02 * +
6.000E+00 3.662E-02 * +
6.500E+00 2.215E-02 * +
7.000E+00 1.343E-02 * +
7.500E+00 8.123E-03 * +
8.000E+00 4.922E-03 * +
8.500E+00 2.978E-03 * +
9.000E+00 1.805E-03 * +
9.500E+00 1.092E-03 * +
1.000E+01 6.591E-04 * +
Notice how the voltage decreases (to the left of the plot) very
quickly at first, then tapering off as time goes on. Current also
changes very quickly at first then levels off as time goes on, but
it is approaching maximum (right of scale) while voltage
approaches minimum.
e REVIEW:
e A fully "discharged" inductor (no current through it) initially
acts as an open circuit (voltage drop with no current) when
faced with the sudden application of voltage. After
"charging" fully to the final level of current, it acts asa
Short circuit (current with no voltage drop).
e In a resistor-inductor "charging" circuit, inductor current
goes from nothing to full value while voltage goes from
maximum to zero, both variables changing most rapidly at
first, approaching their final values slower and slower as
time goes on.
Voltage and current calculations
There's a sure way to calculate any of the values in a reactive
DC circuit over time. The first step is to identify the starting and
final values for whatever quantity the capacitor or inductor
opposes change in; that is, whatever quantity the reactive
component is trying to hold constant. For capacitors, this
quantity is vo/tage; for inductors, this quantity is current. When
the switch in a circuit is closed (or opened), the reactive
component will attempt to maintain that quantity at the same
level as it was before the switch transition, so that value is to be
used for the "starting" value. The final value for this quantity is
whatever that quantity will be after an infinite amount of time.
This can be determined by analyzing a capacitive circuit as
though the capacitor was an open-circuit, and an inductive
circuit as though the inductor was a short-circuit, because that
is what these components behave as when they've reached "full
charge," after an infinite amount of time.
The next step is to calculate the time constant of the circuit: the
amount of time it takes for voltage or current values to change
approximately 63 percent from their starting values to their
final values in a transient situation. In a series RC circuit, the
time constant is equal to the total resistance in ohms multiplied
by the total capacitance in farads. For a series L/R circuit, it is
the total inductance in henrys divided by the total resistance in
ohms. In either case, the time constant is expressed in units of
seconds and symbolized by the Greek letter "tau" (T):
For resistor-capacitor circuits:
t=RE
For resistor-inductor circuits:
L
=
R
The rise and fall of circuit values such as voltage and current in
response to a transient is, as was mentioned before, asymptotic.
Being so, the values begin to rapidly change soon after the
transient and settle down over time. If plotted on a graph, the
approach to the final values of voltage and current form
exponential curves.
As was Stated before, one time constant is the amount of time it
takes for any of these values to change about 63 percent from
their starting values to their (ultimate) final values. For every
time constant, these values move (approximately) 63 percent
closer to their eventual goal. The mathematical formula for
determining the precise percentage is quite simple:
Percentage of change = ( - +) x 100%
et
The letter e stands for Euler's constant, which is approximately
2.182818. It is derived from calculus techniques, after
mathematically analyzing the asymptotic approach of the
circuit values. After one time constant's worth of time, the
percentage of change from starting value to final value is:
- z X 100% = 63.212%
e€
After two time constant's worth of time, the percentage of
change from starting value to final value is:
ce x 100% = 86.466%
e
After ten time constant's worth of time, the percentage is:
- H ) x 100% = 99.995%
e
The more time that passes since the transient application of
voltage from the battery, the larger the value of the
denominator in the fraction, which makes for a smaller value for
the whole fraction, which makes for a grand total (1 minus the
fraction) approaching 1, or 100 percent.
We can make a more universal formula out of this one for the
determination of voltage and current values in transient
circuits, by multiplying this quantity by the difference between
the final and starting circuit values:
Universal Time Constant Formula
t/t
Change = Final sea ( eee )
e
Where,
Final = Value of calculated variable after infinite time
(its ultimate value)
Start = Initial value of calculated variable
e= Euler’s number (=2.7182818)
t= Timein seconds
t= Timeconstant for circuit in seconds
Let's analyze the voltage rise on the series resistor-capacitor
circuit shown at the beginning of the chapter.
Switch
R
| 10kQ
15 V — Cc | 100 LF
Note that we're choosing to analyze voltage because that is the
quantity capacitors tend to hold constant. Although the formula
works quite well for current, the starting and final values for
Current are actually derived from the capacitor's voltage, so
calculating voltage is a more direct method. The resistance is
10 kQ, and the capacitance is 100 uF (microfarads). Since the
time constant (Tt) for an RC circuit is the product of resistance
and capacitance, we obtain a value of 1 second:
t=8C
t = (10 kQ)(100 LF)
t= 1 second
If the capacitor starts in a totally discharged state (0 volts),
then we can use that value of voltage for a "starting" value. The
final value, of course, will be the battery voltage (15 volts). Our
universal formula for capacitor voltage in this circuit looks like
this:
l
Change = ina sta ( ser )
fT
e
v1
Change = (15 V-0V) ( I )
e
So, after 7.25 seconds of applying voltage through the closed
switch, our capacitor voltage will have increased by:
Change = (15 V -0 V) : =)
e721
Change = (15 V - 0 V)(0.99929)
Change = 14.989 V
Since we started at a capacitor voltage of O volts, this increase
of 14.989 volts means that we have 14.989 volts after 7.25
seconds.
The same formula will work for determining current in that
circuit, too. Since we know that a discharged capacitor initially
acts like a short-circuit, the starting current will be the
maximum amount possible: 15 volts (from the battery) divided
by 10 kQ (the only opposition to current in the circuit at the
beginning):
IS V
Starting current =
10 kQ
Starting current = 1.5 mA
We also know that the final current will be zero, since the
capacitor will eventually behave as an open-circuit, meaning
that eventually no electrons will flow in the circuit. Now that we
know both the starting and final current values, we can use our
universal formula to determine the current after 7.25 seconds of
switch closure in the same RC circuit:
l
Change = (0 mA - 1.5 mA) ene )
e’* /
Change = (0 mA - 1.5 mA)(0.99929)
Change = - 1.4989 mA
Note that the figure obtained for change is negative, not
positive! This tells us that current has decreased rather than
increased with the passage of time. Since we started ata
current of 1.5 mA, this decrease (-1.4989 mA) means that we
have 0.001065 mA (1.065 UA) after 7.25 seconds.
We could have also determined the circuit current at time=7.25
seconds by subtracting the capacitor's voltage (14.989 volts)
from the battery's voltage (15 volts) to obtain the voltage drop
across the 10 kQ resistor, then figuring current through the
resistor (and the whole series circuit) with Ohm's Law (l=E/R).
Either way, we should obtain the same answer:
1_=-—
R
_ 15 V- 14.989 V
10 kQ
1= 1.065 tA
The universal time constant formula also works well for
analyzing inductive circuits. Let's apply it to our example L/R
circuit in the beginning of the chapter:
Switch
With an inductance of 1 henry and a series resistance of 1 QO,
our time constant is equal to 1 second:
L
t= —
R
t= 1 second
Because this is an inductive circuit, and we know that inductors
oppose change in current, we'll set up our time constant
formula for starting and final values of current. If we start with
the switch in the open position, the current will be equal to
zero, SO zero Is Our starting current value. After the switch has
been left closed for a long time, the current will settle out to its
final value, equal to the source voltage divided by the total
circuit resistance (I=E/R), or 15 amps in the case of this circuit.
If we desired to determine the value of current at 3.5 seconds,
we would apply the universal time constant formula as such:
Change = (15 A- 0 A) - 3)
en
Change = (15 A - 0 A)(0.9698)
Change = 14.547 A
Given the fact that our starting current was zero, this leaves us
at a circuit current of 14.547 amps at 3.5 seconds’ time.
Determining voltage in an inductive circuit is best
accomplished by first figuring circuit current and then
calculating voltage drops across resistances to find what's left
to drop across the inductor. With only one resistor in our
example circuit (having a value of 1 Q), this is rather easy:
Eg = (14.547 A\(1 Q)
E,= 14.547V
Subtracted from our battery voltage of 15 volts, this leaves
0.453 volts across the inductor at time=3.5 seconds.
E, = Epattery - Eg
E, = 15 V - 14.547 V
E, = 0.453 V
e REVIEW:
e Universal Time Constant Formula:
Universal Time Constant Formula
Change = Finale ( eee )
t/t
e
Where,
Final = Value of calculated variable after infinite time
(its ultimate value)
Start = Initial value of calculated variable
e= Euler's number (=2.7182818)
t= Timein seconds
. t= Time constant for circuit in seconds
e To analyze an RC or L/R circuit, follow these steps:
e (1): Determine the time constant for the circuit (RC or L/R).
e (2): Identify the quantity to be calculated (whatever
quantity whose change is directly opposed by the reactive
component. For capacitors this is voltage; for inductors this
IS Current).
e (3): Determine the starting and final values for that
quantity.
e (4): Plug all these values (Final, Start, time, time constant)
into the universal time constant formula and solve for
Change in quantity.
e (5): If the starting value was zero, then the actual value at
the specified time is equal to the calculated change given
by the universal formula. If not, add the change to the
starting value to find out where you're at.
Why L/R and not LR?
It is often perplexing to new students of electronics why the
time-constant calculation for an inductive circuit is different
from that of a capacitive circuit. For a resistor-capacitor circuit,
the time constant (in seconds) is calculated from the product
(multiplication) of resistance in ohms and capacitance in farads:
t=RC. However, for a resistor-inductor circuit, the time constant
is calculated from the quotient (division) of inductance in
henrys over the resistance in ohms: T=L/R.
This difference in calculation has a profound impact on the
qualitative analysis of transient circuit response. Resistor-
Capacitor circuits respond quicker with low resistance and
slower with high resistance; resistor-inductor circuits are just
the opposite, responding quicker with high resistance and
slower with low resistance. While capacitive circuits seem to
present no intuitive trouble for the new student, inductive
circuits tend to make less sense.
Key to the understanding of transient circuits is a firm grasp on
the concept of energy transfer and the electrical nature of it.
Both capacitors and inductors have the ability to store
quantities of energy, the capacitor storing energy in the
medium of an electric field and the inductor storing energy in
the medium of a magnetic field. A capacitor's electrostatic
energy storage manifests itself in the tendency to maintain a
constant voltage across the terminals. An inductor's
electromagnetic energy storage manifests itself in the tendency
to maintain a constant current through it.
Let's consider what happens to each of these reactive
components in a condition of discharge: that is, when energy is
being released from the capacitor or inductor to be dissipated
in the form of heat by a resistor:
Capacitor and inductor discharge
Stored =» ‘enely Stored =» ‘eneluy
en ergy >> energy en ergy >> energy
Za an heat 3 a heat
Time —» Time —»
In either case, heat dissipated by the resistor constitutes energy
leaving the circuit, and as a consequence the reactive
component loses its store of energy over time, resulting ina
measurable decrease of either voltage (capacitor) or current
(inductor) expressed on the graph. The more power dissipated
by the resistor, the faster this discharging action will occur,
because power is by definition the rate of energy transfer over
time.
Therefore, a transient circuit's time constant will be dependent
upon the resistance of the circuit. Of course, it is also
dependent upon the size (storage capacity) of the reactive
component, but since the relationship of resistance to time
constant is the issue of this section, we'll focus on the effects of
resistance alone. A circuit's time constant will be less (faster
discharging rate) if the resistance value is such that it
maximizes power dissipation (rate of energy transfer into heat).
For a capacitive circuit where stored energy manifests itself in
the form of a voltage, this means the resistor must have a low
resistance value so as to maximize current for any given
amount of voltage (given voltage times high current equals
high power). For an inductive circuit where stored energy
manifests itself in the form of a current, this means the resistor
must have a high resistance value so as to maximize voltage
drop for any given amount of current (given current times high
voltage equals high power).
This may be analogously understood by considering capacitive
and inductive energy storage in mechanical terms. Capacitors,
storing energy electrostatically, are reservoirs of potential
energy. Inductors, storing energy electromagnetically
(electrodynamically), are reservoirs of kinetic energy. In
mechanical terms, potential energy can be illustrated by a
suspended mass, while kinetic energy can be illustrated by a
moving mass. Consider the following illustration as an analogy
of a capacitor:
Potential en ergy storage
and release
Cart
| S
| 0,
gravity
The cart, sitting at the top of a slope, possesses potential
energy due to the influence of gravity and its elevated position
on the hill. If we consider the cart's braking system to be
analogous to the resistance of the system and the cart itself to
be the capacitor, what resistance value would facilitate rapid
release of that potential energy? Minimum resistance (no
brakes) would diminish the cart's altitude quickest, of course!
Without any braking action, the cart will freely roll downhill,
thus expending that potential energy as it loses height. With
maximum braking action (brakes firmly set), the cart will refuse
to roll (or it will roll very slowly) and it will hold its potential
energy for a long period of time. Likewise, a capacitive circuit
will discharge rapidly if its resistance is low and discharge
Slowly if its resistance is high.
Now let's consider a mechanical analogy for an inductor,
showing its stored energy in kinetic form:
Kinetic energy storage
and release
Cart
This time the cart is on level ground, already moving. Its energy
IS kinetic (motion), not potential (height). Once again if we
consider the cart's braking system to be analogous to circuit
resistance and the cart itself to be the inductor, what resistance
value would facilitate rapid release of that kinetic energy?
Maximum resistance (maximum braking action) would slow it
down quickest, of course! With maximum braking action, the
cart will quickly grind to a halt, thus expending its kinetic
energy as it slows down. Without any braking action, the cart
will be free to roll on indefinitely (barring any other sources of
friction like aerodynamic drag and rolling resistance), and it will
hold its kinetic energy for a long period of time. Likewise, an
inductive circuit will discharge rapidly if its resistance is high
and discharge slowly if its resistance is low.
Hopefully this explanation sheds more light on the subject of
time constants and resistance, and why the relationship
between the two is opposite for capacitive and inductive
circuits.
Complex voltage and current
calculations
There are circumstances when you may need to analyze a DC
reactive circuit when the starting values of voltage and current
are not respective of a fully "discharged" state. In other words,
the capacitor might start at a partially-charged condition
instead of starting at zero volts, and an inductor might start
with some amount of current already through it, instead of zero
as we have been assuming so far. Take this circuit as an
example, starting with the switch open and finishing with the
switch in the closed position:
Since this is an inductive circuit, we'll start our analysis by
determining the start and end values for current. This step is
vitally important when analyzing inductive circuits, as the
starting and ending vo/tage can only be known after the current
has been determined! With the switch open (starting
condition), there is a total (series) resistance of 3 Q, which limits
the final current in the circuit to 5 amps:
i
R
i= 15 V
3Q
=J A
1
So, before the switch is even closed, we have a current through
the inductor of 5 amps, rather than starting from 0 amps as in
the previous inductor example. With the switch closed (the final
condition), the 1 Q resistor is shorted across (bypassed), which
changes the circuit's total resistance to 2 QO. With the switch
closed, the final value for current through the inductor would
then be:
So, the inductor in this circuit has a starting current of 5 amps
and an ending current of 7.5 amps. Since the "timing" will take
place during the time that the switch is closed and R> is shorted
past, we need to calculate our time constant from L; and Rj: 1
Henry divided by 2 QO, or t = 1/2 second. With these values, we
can calculate what will happen to the current over time. The
voltage across the inductor will be calculated by multiplying
the current by 2 (to arrive at the voltage across the 2 QO
resistor), then subtracting that from 15 volts to see what's left.
If you realize that the voltage across the inductor starts at 5
volts (when the switch is first closed) and decays to 0 volts over
time, you can also use these figures for starting/ending values
in the general formula and derive the same results:
l
v0.5
e
Change = (7.5 A- 5 A) ( - ) Calculating current
.-Or...
l
wo.
e
Change = (0 V -5 V) - ) Calculating voltage
| Time | Battery | Inductor | Current |
|(seconds) | voltage | voltage | |
| 0.25 | 15V | 3.033 V | 5.984 A |
hie ae ee eae
ee ee ag a oe
ie ae ae ane eG
eg oe gee te gaan aa
Complex circuits
What do we do if we come across a circuit more complex than
the simple series configurations we've seen so far? Take this
circuit aS an example:
Switch
3kQ
The simple time constant formula (t=RC) is based on a simple
series resistance connected to the capacitor. For that matter,
the time constant formula for an inductive circuit (t=L/R) is also
based on the assumption of a simple series resistance. So, what
can we do in a situation like this, where resistors are connected
in a series-parallel fashion with the capacitor (or inductor)?
The answer comes from our studies in network analysis.
Thevenin's Theorem tells us that we can reduce any linear
circuit to an equivalent of one voltage source, one series
resistance, and a load component through a couple of simple
steps. To apply Thevenin's Theorem to our scenario here, we'll
regard the reactive component (in the above example circuit,
the capacitor) as the load and remove it temporarily from the
circuit to find the Thevenin voltage and Thevenin resistance.
Then, once we've determined the Thevenin equivalent circuit
values, we'll re-connect the capacitor and solve for values of
voltage or current over time as we've been doing so far.
After identifying the capacitor as the "load," we remove it from
the circuit and solve for voltage across the load terminals
(assuming, of course, that the switch is closed):
Switch
(closed) R,
500 © AT = 1S a2
R, R, Total
R,
Volts
Amps
55k __| Ohms
This step of the analysis tells us that the voltage across the load
terminals (same as that across resistor R>) will be 1.8182 volts
with no load connected. With a little reflection, it should be
clear that this will be our final voltage across the capacitor,
seeing as how a fully-charged capacitor acts like an open
circuit, drawing zero current. We will use this voltage value for
our Thevenin equivalent circuit source voltage.
a3 —- m
Now, to solve for our Thevenin resistance, we need to eliminate
all power sources in the original circuit and calculate resistance
as seen from the load terminals:
Switch
(closed) R,
Theveni
resistanc
A = 454.545 O
Rrhevenin = R, I (R, = R;)
Rrhevenin = 500 Q // (2 kQ + 3 kQ)
Rohevenin = 454.545 Q
Re-drawing our circuit as a Thevenin equivalent, we get this:
Switch
Thevenin
454.545 Q
Etheveni n— C
1.8182 V T
Our time constant for this circuit will be equal to the Thevenin
resistance times the capacitance (t=RC). With the above
values, we calculate:
100 LF
t=RC
t = (454.545 Q)( 100 LF)
t = 45.4545 milliseconds
Now, we can solve for voltage across the capacitor directly with
our universal time constant formula. Let's calculate for a value
of 60 milliseconds. Because this is a capacitive formula, we'll
set our calculations up for voltage:
Change = (Final - Start) ( - L )
Change = (1.8182 V -0 V) ( a" ne}
eoom/4s 4545m
Change = (1.8182 V (0.73286)
Change = 1.3325 V
Again, because our starting value for capacitor voltage was
assumed to be zero, the actual voltage across the capacitor at
60 milliseconds is equal to the amount of voltage change from
zero, or 1.3325 volts.
We could go a step further and demonstrate the equivalence of
the Thevenin RC circuit and the original circuit through
computer analysis. | will use the SPICE analysis program to
demonstrate this:
Comparison RC analysis
* first, the netlist for the original circuit:
v1 10 dc 20
rl 12 2k
r2 2 3 500
r3 3 0 3k
cl 2 3 100u ic=0
* then, the netlist for the thevenin equivalent:
v2 4 0 dc 1.818182
r4 4 5 454.545
c2 5 0 100u ic=0
* now, we analyze for a transient, sampling every .005 seconds
* over a time period of .37 seconds total, printing a list of
* values for voltage across the capacitor in the original
* circuit (between modes 2 and 3) and across the capacitor in
* the thevenin equivalent circuit (between nodes 5 and 0)
.tran .005 0.37 uic
.print tran v(2,3) v(5,0)
.end
time v(2,3) v(5)
0.000E+00 4.803E-06 4.803E-06
5.000E-03 1.890E-01 1.890E-01
1.Q00E-02 3.580E-01 3.580E-01
1.500E-02 5.082E-01 5.082E-01
2.000E-02 6.442E-01 6.442E-01
2.500E-02 7.689E-@01 7.689E-01
3.000E-02 8.772E-01 8.772E-01
3.500E-02 9.747E-01 9.747E-01
4.000E-02 1.064E+00 1.064E+00
4.500E-02 1.142E+00 1.142E+00
5.000E-02 1.212E+00 1.212E+00
5.500E-02 1.276E+00 1.276E+00
6.000E-02 1.333E+00 1.333E+00
6.500E-02 1.383E+00 1.383E+00
7.000E-02 1.429E+00 1.429E+00
7.500E-02 1.470E+00 1.470E+00
8.000E-02 1.505E+00 1.505E+00
8.500E-02 1.538E+00 1.538E+00
9.000E-02 1.568E+00 1.568E+00
9.500E-02 1.594E+00 1.594E+00
1.000E-01 1.617E+00 1.617E+00
1.050E-01 1.638E+00 1.638E+00
1.100E-01 1.657E+00 1.657E+00
1.150E-01 1.674E+00 1.674E+00
1.200E-01 1.689E+00 1.689E+00
1.250E-01 1.702E+00 1.702E+00
1.300E-01 1.714E+00 1.714E+00
1.350E-01 1.725E+00 1.725E+00
1.400E-01 1.735E+00 1.735E+00
WWWWWWWWWWWWWWWNNNNNNNNNNNNNNNNNNNNFPRPRPRPRPRPRPRRFRE
-450E-01
.500E-01
.550E-01
.600E-01
.650E-01
.700E-01
.750E-01
.800E-01
.850E-01
.900E-01
.950E-01
.Q00E-01
.Q50E-01
. 100E-01
.150E-01
.200E-01
.250E-01
.300E-01
.350E-01
-400E-01
-450E-01
.500E-01
.550E-01
.600E-01
.650E-01
.700E-01
.750E-01
.800E-01
.850E-01
.900E-01
.950E-01
.Q00E-01
.Q50E-01
. 100E-01
.150E-01
.200E-01
.250E-01
.300E-01
.350E-01
-400E-01
-450E-01
.500E-01
.550E-01
.600E-01
.650E-01
.700E-01
PPP RPP PPP PPP PPP PP PPP PPP PPP PP PP PP PPP PPP PP PP PP PPP
. 744E+00
.752E+00
. 758E+00
. 765E+00
. 770E+00
.775E+00
. 780E+00
. 784E+00
. 787E+00
.791E+00
. 793E+00
. 796E+00
. 798E+00
. 800E+00
.802E+00
.804E+00
.805E+00
.807E+00
. 808E+00
. 809E+00
.810E+00
.811E+00
.812E+00
.812E+00
.813E+00
.813E+00
.814E+00
.814E+00
.815E+00
.815E+00
.815E+00
.816E+00
.816E+00
.816E+00
.816E+00
.817E+00
.817E+00
.817E+00
.817E+00
.817E+00
.817E+00
.817E+00
.817E+00
.818E+00
.818E+00
.818E+00
PPP RPP PPP PPP PPP PP PPP PPP PPP PP PP PPP PP PPP PP PP PP PPP
. 744E+00
.752E+00
.758E+00
.765E+00
.770E+00
.775E+00
. 780E+00
. 784E+00
.787E+00
.791E+00
.793E+00
. 796E+00
.798E+00
. 800E+00
.802E+00
.804E+00
.805E+00
.807E+00
. 808E+00
. 809E+00
.810E+00
.811E+00
.812E+00
.812E+00
.813E+00
.813E+00
.814E+00
.814E+00
.815E+00
.815E+00
.815E+00
.816E+00
.816E+00
.816E+00
.816E+00
.817E+00
.817E+00
.817E+00
.817E+00
.817E+00
.817E+00
.817E+00
.817E+00
.818E+00
.818E+00
.818E+00
At every step along the way of the analysis, the capacitors in
the two circuits (original circuit versus Thevenin equivalent
circuit) are at equal voltage, thus demonstrating the
equivalence of the two circuits.
« REVIEW:
e To analyze an RC or L/R circuit more complex than simple
series, convert the circuit into a Thevenin equivalent by
treating the reactive component (capacitor or inductor) as
the "load" and reducing everything else to an equivalent
circuit of one voltage source and one series resistor. Then,
analyze what happens over time with the universal time
constant formula.
Solving for unknown time
Sometimes it is necessary to determine the length of time that
a reactive circuit will take to reach a predetermined value. This
IS especially true in cases where we're designing an RC or L/R
circuit to perform a precise timing function. To calculate this, we
need to modify our "Universal time constant formula." The
original formula looks like this:
Change = (Final-Start) - =) = Finals ( - -*)
e
However, we want to solve for time, not the amount of change.
To do this, we algebraically manipulate the formula so that time
is all by itself on one side of the equal sign, with all the rest on
the other side:
Change = (Final-Start) - *)
L- Change | _ et
Final-Start /
Change \ _ em
‘ ( ; ae Gtanee) = In(e"*)
t=—e finfr- —Coamgec__
Final - Start
The /n designation just to the right of the time constant term is
the natural logarithm function: the exact reverse of taking the
power of e. In fact, the two functions (powers of e and natural
logarithms) can be related as such:
If eX = a, then Ina = x.
If e* = a, then the natural logarithm of a will give you x: the
power that e must be was raised to in order to produce a.
Let's see how this all works on a real example circuit. Taking the
Same resistor-capacitor circuit from the beginning of the
chapter, we can work "backwards" from previously determined
values of voltage to find how long it took to get there.
Switch
R
| 10 kQ
15 V — Cc | 100 LF
The time constant is still the same amount: 1 second (10 kQ
times 100 uF), and the starting/final values remain unchanged
as well (Ec = 0 volts starting and 15 volts final). According to
our chart at the beginning of the chapter, the capacitor would
be charged to 12.970 volts at the end of 2 seconds. Let's plug
12.970 volts in as the "Change" for our new formula and see if
we arrive at an answer of 2 seconds:
:
t=-(1 xecond) in ft = 22) _
I5V-0V
t = -(1 second )(In 0.13534))
t = (1 second)(2)
t = 2 seconds
Indeed, we end up with a value of 2 seconds for the time it
takes to go from O to 12.970 volts across the capacitor. This
variation of the universal time constant formula will work for all
Capacitive and inductive circuits, both "charging" and
"discharging," provided the proper values of time constant,
Start, Final, and Change are properly determined beforehand.
Remember, the most important step in solving these problems
is the initial set-up. After that, its just a lot of button-pushing on
your calculator!
¢ REVIEW:
e To determine the time it takes for an RC or L/R circuit to
reach a certain value of voltage or current, you'll have to
modify the universal time constant formula to solve for time
instead of change.
t=—¢ finfx- —Comee__
Final - Start
e The mathematical function for reversing an exponent of "e"
is the natural logarithm (In), provided on any scientific
calculator.
Contributors
Contributors to this chapter are listed in chronological order of
their contributions, from most recent to first. See Appendix 2
(Contributor List) for dates and contact information.
Jason Starck (June 2000): HTML document formatting, which
led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design Science
License.
—|}|+4/{—
— 4 —
Appendix 1
ABOUT THIS BOOK
Purpose
They say that necessity is the mother of invention. At least
in the case of this book, that adage is true. As an industrial
electronics instructor, | was forced to use a sub-standard
textbook during my first year of teaching. My students were
daily frustrated with the many typographical errors and
obscure explanations in this book, having spent much time
at home struggling to comprehend the material within.
Worse yet were the many incorrect answers in the back of
the book to selected problems. Adding insult to injury was
the $100+ price.
Contacting the publisher proved to be an exercise in futility.
Even though the particular text | was using had been in
print and in popular use for a couple of years, they claimed
my complaint was the first they'd ever heard. My request to
review the draft for the next edition of their book was met
with disinterest on their part, and | resolved to find an
alternative text.
Finding a Suitable alternative was more difficult than | had
imagined. Sure, there were plenty of texts in print, but the
really good books seemed a bit too heavy on the math and
the less intimidating books omitted a lot of information | felt
was important. Some of the best books were out of print, and
those that were still being printed were quite expensive.
It was out of frustration that | compiled Lessons in Electric
Circuits from notes and ideas | had been collecting for years.
My primary goal was to put readable, high-quality
information into the hands of my students, but a secondary
goal was to make the book as affordable as possible. Over
the years, | had experienced the benefit of receiving free
instruction and encouragement in my pursuit of learning
electronics from many people, including several teachers of
mine in elementary and high school. Their selfless
assistance played a key role in my own studies, paving the
way for a rewarding career and fascinating hobby. If only |
could extend the gift of their help by giving to other people
what they gavetome...
So, | decided to make the book freely available. More than
that, | decided to make it "open," following the same
development model used in the making of free software
(most notably the various UNIX utilities released by the Free
Software Foundation, and the Linux operating system,
whose fame Is growing even as | write). The goal was to
copyright the text -- so as to protect my authorship -- but
expressly allow anyone to distribute and/or modify the text
to suit their own needs with a minimum of legal
encumbrance. This willful and formal revoking of standard
distribution limitations under copyright is whimsically
termed copyleft. Anyone can "copyleft" their creative work
simply by appending a notice to that effect on their work,
but several Licenses already exist, covering the fine legal
points in great detail.
The first such License | applied to my work was the GPL --
General Public License -- of the Free Software Foundation
(GNU). The GPL, however, is intended to copyleft works of
computer software, and although its introductory language
is broad enough to cover works of text, its wording is not as
clear as it could be for that application. When other, less
specific copyleft Licenses began appearing within the free
software community, | chose one of them (the Design
Science License, or DSL) as the official notice for my project.
In "copylefting" this text, | guaranteed that no instructor
would be limited by a text insufficient for their needs, as |
had been with error-ridden textbooks from major publishers.
I'm sure this book in its initial form will not satisfy everyone,
but anyone has the freedom to change it, leveraging my
efforts to suit variant and individual requirements. For the
beginning student of electronics, learn what you can from
this book, editing it as you feel necessary if you come across
a useful piece of information. Then, if you pass it on to
someone else, you will be giving them something better
than what you received. For the instructor or electronics
professional, feel free to use this as a reference manual,
adding or editing to your heart's content. The only "catch" is
this: if you plan to distribute your modified version of this
text, you must give credit where credit is due (to me, the
Original author, and anyone else whose modifications are
contained in your version), and you must ensure that
whoever you give the text to is aware of their freedom to
similarly share and edit the text. The next chapter covers
this process in more detail.
It must be mentioned that although | strive to maintain
technical accuracy in all of this book's content, the subject
matter is broad and harbors many potential dangers.
Electricity maims and kills without provocation, and
deserves the utmost respect. | strongly encourage
experimentation on the part of the reader, but only with
circuits powered by small batteries where there is no risk of
electric shock, fire, explosion, etc. High-power electric
circuits should be left to the care of trained professionals!
The Design Science License clearly states that neither | nor
any contributors to this book bear any liability for what is
done with its contents.
The use of SPICE
One of the best ways to learn how things work is to follow
the inductive approach: to observe specific instances of
things working and derive general conclusions from those
observations. In science education, labwork is the
traditionally accepted venue for this type of learning,
although in many cases labs are designed by educators to
reinforce principles previously learned through lecture or
textbook reading, rather than to allow the student to learn
on their own through a truly exploratory process.
Having taught myself most of the electronics that | know, |
appreciate the sense of frustration students may have in
teaching themselves from books. Although electronic
components are typically inexpensive, not everyone has the
means or opportunity to set up a laboratory in their own
homes, and when things go wrong there's no one to ask for
help. Most textbooks seem to approach the task of education
from a deductive perspective: tell the student how things
are supposed to work, then apply those principles to specific
instances that the student may or may not be able to
explore by themselves. The inductive approach, as useful as
it is, is hard to find in the pages of a book.
However, textbooks don't have to be this way. | discovered
this when | started to learn a computer program called
SPICE. It is a text-based piece of software intended to model
circuits and provide analyses of voltage, current, frequency,
etc. Although nothing is quite as good as building real
circuits to gain knowledge in electronics, computer
simulation is an excellent alternative. In learning how to use
this powerful tool, | made a discovery: SPICE could be used
within a textbook to present circuit simulations to allow
students to "observe" the phenomena for themselves. This
way, the readers could learn the concepts inductively (by
interpreting SPICE's output) as well as deductively (by
interpreting my explanations). Furthermore, in seeing SPICE
used over and over again, they should be able to
understand how to use it themselves, providing a perfectly
safe means of experimentation on their own computers with
circuit simulations of their own design.
Another advantage to including computer analyses in a
textbook is the empirical verification it adds to the concepts
presented. Without demonstrations, the reader is left to take
the author's statements on faith, trusting that what has
been written is indeed accurate. The problem with faith, of
course, is that it is only as good as the authority in which it
is placed and the accuracy of interpretation through which it
is understood. Authors, like all human beings, are liable to
err and/or communicate poorly. With demonstrations,
however, the reader can immediately see for themselves
that what the author describes is indeed true.
Demonstrations also serve to clarify the meaning of the text
with concrete examples.
SPICE is introduced in the book early on, and hopefully in a
gentle enough way that it doesn't create confusion. For
those wishing to learn more, a chapter in the Reference
volume (volume V) contains an overview of SPICE with many
example circuits. There may be more flashy (graphic) circuit
simulation programs in existence, but SPICE is free, a virtue
complementing the charitable philosophy of this book very
nicely.
Acknowledgements
First, | wish to thank my wife, whose patience during those
many and long evenings (and weekends!) of typing has
been extraordinary.
| also wish to thank those whose open-source software
development efforts have made this endeavor all the more
affordable and pleasurable. The following is a list of various
free computer software used to make this book, and the
respective programmers:
e GNU/Linux Operating System -- Linus Torvalds, Richard
Stallman, and a host of others too numerous to mention.
e Vim text editor -- Bram Moolenaar and others.
Xcircuit drafting program -- Tim Edwards.
SPICE circuit simulation program -- too many
contributors to mention.
e Nutmeg post-processor program for SPICE -- Wayne
Christopher.
e T-X text processing system -- Donald Knuth and others.
e Texinfo document formatting system -- Free Software
Foundation.
¢ LATEX document formatting system -- Leslie Lamport and
others.
e Gimp image manipulation program -- too many
contributors to mention.
Appreciation is also extended to Robert L. Boylestad, whose
first edition of Introductory Circuit Analysis taught me more
about electric circuits than any other book. Other important
texts in my electronics studies include the 1939 edition of
The “Radio” Handbook, Bernard Grob's second edition of
Introduction to Electronics I, and Forrest Mims' original
Engineer's Notebook.
Thanks to the staff of the Bellingham Antique Radio
Museum, who were generous enough to let me terrorize their
establishment with my camera and flash unit. Similar thanks
to the Fluke Corporation in Everett, Washington, who not
only let me photograph several pieces of equipment in their
primary standards laboratory, but proved their excellent
hosting skills to a large group of students and technical
professionals one evening in November of 2001.
| wish to specifically thank Jeffrey Elkner and all those at
Yorktown High School for being willing to host my book as
part of their Open Book Project, and to make the first effort
in contributing to its form and content. Thanks also to David
Sweet (website: [*]) and Ben Crowell (website: [*]) for
providing encouragement, constructive criticism, and a
wider audience for the online version of this book.
Thanks to Michael Stutz for drafting his Design Science
License, and to Richard Stallman for pioneering the concept
of copyleft.
Last but certainly not least, many thanks to my parents and
those teachers of mine who saw in me a desire to learn
about electricity, and who kindled that flame into a passion
for discovery and intellectual adventure. | honor you by
helping others as you have helped me.
Tony Kuphaldt, January 2002
"A candle loses nothing of its light when lighting
another"
Kahlil Gibran
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—|| 4] l_—
—| | +]
Appendix 2
CONTRIBUTOR LIST
How to contribute to this book
As a copylefted work, this book is open to revision and expansion by
any interested parties. The only "catch" is that credit must be given
where credit is due. This /s a copyrighted work: it is notin the public
domain!
If you wish to cite portions of this book in a work of your own, you
must follow the same guidelines as for any other copyrighted work.
Here is a Sample from the Design Science License:
The Work is copyright the Author. All rights to the Work are reserved
by the Author, except as specifically described below. This License
describes the terms and conditions under which the Author permits you
to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright Law.
Your right to operate, perform, read or otherwise interpret and/or
execute the Work is unrestricted; however, you do so at your own risk,
because the Work comes WITHOUT ANY WARRANTY -- see Section 7 ("NO
WARRANTY") below.
If you wish to modify this book in any way, you must document the
nature of those modifications in the "Credits" section along with your
name, and ideally, information concerning how you may be
contacted. Again, the Design Science License:
Permission is granted to modify or sample from a copy of the Work,
producing a derivative work, and to distribute the derivative work
under the terms described in the section for distribution above,
provided that the following terms are met:
(a) The new, derivative work is published under the terms of this
License.
(b) The derivative work is given a new name, so that its name or
title can not be confused with the Work, or with a version of
the Work, in any way.
(c) Appropriate authorship credit is given: for the differences
between the Work and the new derivative work, authorship is
attributed to you, while the material sampled or used from
the Work remains attributed to the original Author; appropriate
notice must be included with the new work indicating the nature
and the dates of any modifications of the Work made by you.
Given the complexities and security issues surrounding the
maintenance of files comprising this book, it is recommended that
you submit any revisions or expansions to the original author (Tony R.
Kuphaldt). You are, of course, welcome to modify this book directly by
editing your own personal copy, but we would all stand to benefit
from your contributions if your ideas were incorporated into the
online "master copy" where all the world can see it.
Credits
All entries arranged in alphabetical order of surname. Major
contributions are listed by individual name with some detail on the
nature of the contribution(s), date, contact info, etc. Minor
contributions (typo corrections, etc.) are listed by name only for
reasons of brevity. Please understand that when | classify a
contribution as "minor," it is in no way inferior to the effort or value of
a "major" contribution, just smaller in the sense of less text changed.
Any and all contributions are gratefully accepted. | am indebted to all
those who have given freely of their own knowledge, time, and
resources to make this a better book!
John Anhalt
« Date(s) of contribution(s): December 2008
e Nature of contribution: Updated lead-acid cell chemistry, Ch
11
¢ Contact at: jpa@anhalt.org
Benjamin Crowell, Ph.D.
« Date(s) of contribution(s): January 2001
e Nature of contribution: Suggestions on improving technical
accuracy of electric field and charge explanations in the first two
chapters.
¢ Contact at: crowell01@lightandmatter.com
Dennis Crunkilton
« Date(s) of contribution(s): January 2006 to present
e Nature of contribution: Mini table of contents, all chapters
except appedicies; html, latex, ps, pdf; See Devel/tutorial.Atm;
01/2006.
e DC network analysis ch, Mesh current section, Mesh current by
inspection, new material.i DC network analysis ch, Node voltage
method, new section.
e Ch3, Added AFCI paragraphs after GFCI, 10/09/2007.
e Contact at: liecibiblio(at) gmail.com
Tony R. Kuphaldt
« Date(s) of contribution(s): 1996 to present
¢ Nature of contribution: Original author.
e Contact at: liec0@lycos.com
Ron LaPlante
« Date(s) of contribution(s): October 1998
¢ Nature of contribution: Helped create the "table" concept for
use in analysis of series and parallel circuits.
Davy Van Nieuwenborgh
« Date(s) of contribution(s): October 2006
¢ Nature of contribution: DC network analysis ch, Mesh current
section, supplied solution to mesh problem, pointed out error in
text.
¢ Contact at: Theoretical Computer Science laboratory, Department
of Computer Science, Vrije Universiteit Brussel.
Ray A. Rayburn
« Date(s) of contribution(s): September 2009
¢ Nature of contribution: Nonapplicability of Maximum Power
Transfer Theorem to Hi-Fi audio amplifier.
e Contact at: http://forum.allaboutcircuits.com/member. php? u=54720
Jason Starck
« Date(s) of contribution(s): June 2000
¢ Nature of contribution: HTML formatting, some error
corrections.
¢ Contact at: jstarck@yhslug.tux.org
Warren Young
« Date(s) of contribution(s): August 2002
¢ Nature of contribution: Provided capacitor photographs for
chapter 13.
someonesdad@allaboutcircuits.com
Date(s) of contribution(s): November 2009
Nature of contribution: Chapter 8, troublehooting tip end of
Kelvin section.
Your name here
Date(s) of contribution(s): Month and year of contribution
Nature of contribution: Insert text here, describing how you
contributed to the book.
Contact at: my email@provider.net
Typo corrections and other "minor" contributions
The students of Bellingham Technical College's Instrumentation
program.
anonymous (July 2007) Ch 1, remove :registers. Ch 5, s/figures
something/figures is something/. Ch 6 s/The current/The current.
(September 2007) Ch 5, 8, 9,10, 11, 12, 13, 15. Numerous typos,
Clarifications.
Tony Armstrong (January 2003) Suggested diagram correction
in "Series and Parallel Combination Circuits" chapter.
James Boorn (January 2001) Clarification on SPICE simulation.
Dejan Budimir (January 2003) Clarification of Mesh Current
method explanation.
Sridhar Chitta, Assoc. Professor, Dept. of Instrumentation and
Control Engg., Vignan Institute of Technology and Science,
Deshmukhi Village, Pochampally Mandal, Nalgonda Distt, Andhra
Pradesh, India (December 2005) Chapter 13: CAPACITORS,
Clarification: s/note the direction of current/note the direction of
electron current/, 2-places
Colin Creitz (May 2007) Chapters: several, s/it's/its.
Larry Cramblett (September 2004) Typographical error
correction in "Nonlinear conduction" section.
Brad Drum (May 2006) Error correction in "Superconductivity"
section, Chapter 12: PHYSICS OF CONDUCTORS AND
INSULATORS. Degrees are not used as a modifier with kelvin(s), 3
changes.
¢ Jeff DeFreitas (March 2006)Improve appearance: replace “/" and
”"/" Chapters: Al, A2. Type errors Chapter 3: /am injurious
Spark/an injurious spark/, /in the even/inthe event/
Sean Donner (December 2004) Typographical error correction in
"Voltage and current" section, Chapter 1: BASIC CONCEPTS OF
ELECTRICITY,(by a the/ by the) (current of current/ of current).
(January 2005), Typographical error correction in "Fuses" section,
Chapter 12: THE PHYSICS OF CONDUCTORS AND INSULATORS
(Neither fuses nor circuit breakers were not designed to open /
Neither fuses nor circuit breakers were designed to open).
(January 2005), Typographical error correction in "Factors
Affecting Capacitance" section, Chapter 13: CAPACITORS,
(greater plate area gives greater capacitance; less plate area
gives less capacitance / greater plate area gives greater
Capacitance; less plate area gives less capacitance); "Factors
Affecting Capacitance" section, (thin layer if insulation/thin layer
of insulation).
(January 2005), Typographical error correction in "Practical
Considerations" section, Chapter 15: INDUCTORS, (there is not
such thing / there is no such thing).
(January 2005), Typographical error correction in "Voltage and
current calculations" section, Chapter 16: RC AND L/R TIME
CONSTANTS (voltage in current / voltage and current).
Manuel Duarte (August 2006): Ch: DC Metering Circuits
ammeter images: 00163.eps, 00164.eps; Ch: RC and L/R Time
Constants, simplified In() equation images 10263.eps, 10264.eps,
10266.eps, 10276.eps.
Aaron Forster (February 2003) Typographical error correction in
"Physics of Conductors and Insulators" chapter.
Bill Heath (September-December 2002) Correction on
illustration of atomic structure, and corrections of several
typographical errors.
Stefan Kluehspies (June 2003): Corrected spelling error in
Andrew Tannenbaum's name.
David M. St. Pierre (November 2007): Corrected spelling error
in Andrew Tanenbaum's name (from the title page of his book).
Geoffrey Lessel, Thompsons Station, TN (June 2005): Corrected
typo error in Ch 1 "If this charge (static electricity) is stationary,
and you won't realize-remove If; Ch 2 "Ohm's Law also make
intuitive sense if you apply if to the water-and-pipe analogy."
s/if/it; Chapter 2 "Ohm's Law is not very useful for analyzing the
behavior of components like these where resistance is varies with
voltage and current." remove "is"; Ch 3 "which halts fibrillation
and and gives the heart a chance to recover." double "and"; Ch 3
"To be safest, you should follow this procedure is checking, using,
and then checking your meter.... S/iS/of.
LouTheBlueGuru, allaboutcircuits.com, July 2005 Typographical
errors, in Ch 6 "the current through R1 is half:" s/half/twice;
“current through R1 is still exactly twice that of R2" s/R3/R2
Norm Meyrowitz , nkm, allaboutcircuits.com, July 2005
Typographical errors, in Ch 2.3 "where we don't know both
voltage and resistance:" s/resistance/current
Don Stalkowski (June 2002) Technical help with PostScript-to-
PDF file format conversion.
Joseph Teichman (June 2002) Suggestion and technical help
regarding use of PNG images instead of JPEG.
Derek Terveer (June 2006) Typographical errors, several in Ch
2 Se
Geoffrey Lessel (June 2005) Typographical error, s/It
discovered/It was discovered/ in Ch 1.
Austin@allaboutcircuits.com (July 2007) Ch 2, units of mass,
pound vs kilogram, near "units of pound" s/pound/kilogram/.
CATV@allaboutcircuits.com (April 2007) Telephone ring
voltage error, Ch 3.
line@allaboutcircuits.com (June 2005) Typographical error
correction in Volumes 1,2,3,5, various chapters ,(:s/visa-versa/vice
versa/).
rob843 @allaboutcircuits.com (April 2007) Telephone ring
voltage error, Ch 3.
bigtwenty@allaboutcircuits.com (July 2007) Ch 4 near
“different metric prefix”, s/right to left/left to right/.
jut@allaboutcircuits.com (September 2007) Ch 13 near S/if
were we to/if we were to/, S/a capacitors/a capacitor.
rxtxau@allaboutcircuits.com (October 2007) Ch 3, suggested,
GFCI terminology, non-US usage.
Stacy Mckenna Seip (November 2007) Ch 3 s/on hand/one
hand, Ch 4 s/weight/weigh, Ch 8 s/weight/weigh, s/left their/left
there, Ch 9 s/cannot spare/cannot afford/, Chl Clarification, static
electricity.
Cory Benjamin (November 2007) Ch 3 s/on hand/one hand.
Larry Weber (Feb 2008) Ch 3 s/on hand/one hand.
trunks14@allaboutrcircuits.com (Feb 2008) Ch 15 s/of of/of .
Greg Herrington (Feb 2008) Ch 1, Clarification: no neutron in
hydrogen atom.
mark44 (Feb 2008) Ch 1, s/naturaly/naturally/
Unregistered@allaboutcircuits.com (February 2008) Ch 1,
s/smokelsee/smokeless , s/ecconomic/economic/ .
Timothy Unregistered@allaboutcircuits.com (Feb 2008)
Changed default roman font to newcent.
Imranullah Syed (Feb 2008) Suggested centering of
uncaptioned schematics.
davidr@insyst_Itd.com (april 2008) Ch 5, s/results/result 2plcs.
Professor Thom@allaboutcircuits.com (Oct 2008) Ch 6, s/g/c
near Ecd and near 00435.png, 2plcs.
John Schwab (Dec 2008) Ch 1, Static Electricity, near Charles
Coulomb: rearrangement of text segments.
Olivier Derewonko (Dec 2008) Ch 4 s/orientation a
voltage/orientation of a/. Ch2 s/flow though/flow through/. Ch
Safe meter usage, REVIEW, s/,/./ . Ch5, s/is it/it is/.
dor@allaboutcircuits.com (June 2009) Ch 1,
s/nusiance/nuisance.
rspuzio@allaboutcircuits.com (September 2009) Ch 8,
s/logarithmic/nonlinear , 6-plcs.
David Lewis@allaboutcircuits.com (September 2009) Ch 1,
hide paragraph: Physical dimension also impacts conductivity. . .
etc.
Walter Odington@allaboutcircuits.com (January 2010) Ch 3,
s/hydration another/hydration is another/ .
tone_b@allaboutcircuits.com (January 2010) Ch 6, s/must
were/were/ .
Unregistered Guest@allaboutcircuits.com (July 2010) Chl,
S/is is/it iS/.
Unregistered Guest@allaboutcircuits.com (July 2010) Ch5,
added |2 to image 00090.png .
Unregistered Guest@allaboutcircuits.com (August 2010) Ch
1 , s/was one the/was one of the/.
e D. Crunkilton (June 2011) hi.latex, header file; updated link to
openbookproject.net .
Bob Arthur (Jan 2012) images: 00046.eps, 00047 .eps,00048.eps
00362.eps, graph line visibility fixed.
¢ vspriyan@allaboutcircuits.com (Jan 2013) Ch 10, Near:
voltages divided by their s/currents/resistances/ .
« Eugene Smirnoff (Jan 2013) Chl, s/an hypothetical/a
hypothetical/ . Ch 2 s/An historic/A historic/ .
Gulliveig@allaboutcircuits.com (Jan 2014) Ch4, s/significant
digits/mantissa, s/1000/999/ .
« slidercrank@allaboutcircuits.com (Feb 2014) Ch6, s/both
positive/both be positive/ .
¢ Skfir@allaboutcircuits.com (August 2015) Ch10,
S/suppling/supplying/ .
¢ John Wang (Sept 2017) Ch2, s/points 1 and 4/points 1 and 6/,
s/points 2 and 3/3 and 4/.
e Stewart Todd Morgan (Feb 2020) Ch3,
st+http://web.mit.edu/safety+https://ehs.mit.edu/workplace-
safety-program/electr
e DC (Sept 2017) Ch12, Chi3; Reformated various tables to
html/latex.
ical-safety/+ .
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt,
under the terms and conditions of the Design Science License.
—|/]|+4|l\—
—/ | 4]
Appendix 3
DESIGN SCIENCE LICENSE
Copyright © 1999-2000 Michael Stutz stutz@dsl.org
Verbatim copying of this document is permitted, in any
medium.
0. Preamble
Copyright law gives certain exclusive rights to the author of
a work, including the rights to copy, modify and distribute
the work (the "reproductive," "adaptative," and
"distribution" rights).
The idea of "copyleft" is to willfully revoke the exclusivity of
those rights under certain terms and conditions, so that
anyone can copy and distribute the work or properly
attributed derivative works, while all copies remain under
the same terms and conditions as the original.
The intent of this license is to be a general "copyleft" that
can be applied to any kind of work that has protection under
copyright. This license states those certain conditions under
which a work published under its terms may be copied,
distributed, and modified.
Whereas "design science" is a strategy for the development
of artifacts as a way to reform the environment (not people)
and subsequently improve the universal standard of living,
this Design Science License was written and deployed as a
strategy for promoting the progress of science and art
through reform of the environment.
1. Definitions
"License" shall mean this Design Science License. The
License applies to any work which contains a notice placed
by the work's copyright holder stating that it is published
under the terms of this Design Science License.
"Work" shall mean such an aforementioned work. The
License also applies to the output of the Work, only if said
output constitutes a "derivative work" of the licensed Work
as defined by copyright law.
“Object Form" shall mean an executable or performable form
of the Work, being an embodiment of the Work in some
tangible medium.
"Source Data" shall mean the origin of the Object Form,
being the entire, machine-readable, preferred form of the
Work for copying and for human modification (usually the
language, encoding or format in which composed or
recorded by the Author); plus any accompanying files,
scripts or other data necessary for installation, configuration
or compilation of the Work.
(Examples of "Source Data" include, but are not limited to,
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below. This License describes the terms and conditions
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modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright law.
Your right to operate, perform, read or otherwise interpret
and/or execute the Work is unrestricted; however, you do so
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WARRANTY -- see Section 7 ("NO WARRANTY") below.
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END OF TERMS AND CONDITIONS
[ $Id: dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]
— + —
aa
ch
: ns In Electric Cire
<< Volume II - AC
Copyright (C) 2000-2020, Tony R.
Kuphaldt
See the Design Science License (Appendix 3)
for details regarding copying and distribution
Revised July 25, 2007
Master Index
Chapter 1: BASIC AC THEORY
Chapter 2: COMPLEX NUMBERS
Chapter 3: REACTANCE AND IMPEDANCE -- INDUCTIVE
Chapter 4: REACTANCE AND IMPEDANCE -- CAPACITIVE
Chapter 5: REACTANCE AND IMPEDANCE -- R,_L, AND C
Chapter 6: RESONANCE
Chapter 7: MIXED-FREQUENCY AC SIGNALS
Chapter 8: FILTERS
Chapter 9: TRANSFORMERS
Chapter 10: POLYPHASE AC CIRCUITS
Chapter 11: POWER FACTOR
Chapter 12: AC METERING CIRCUITS
Chapter 13: AC MOTORS
Chapter 14: TRANSMISSION LINES
Appendix 1: ABOUT THIS BOOK
Appendix 2: CONTRIBUTOR LIST
Appendix 3: DESIGN SCIENCE LICENSE
Download printable versions of this
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"How do! view and/or print PostScript documents," you ask?
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There you'll find GSview and Ghostscript, two progams
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computer system, you can get by with a little skill and a
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or word processor, and contains all the instructions
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Sed (stands for Stream EDitor), a common UNIX utility
for performing search-and-replace commands on text
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LaTeX2e, a document formatting system designed as an
extension to TeX, Donald Knuth's outstanding text
processing system. You can also get by with just plain
TeX, but your printed output won't look quite as nice and
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If you opt for the smaller of the two files (ACtiny.tar.gz), you'll
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—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 1
BASIC AC THEORY
What is alternating current (AC)?
AC waveforms
Measurements of AC magnitude
Simple AC circuit calculations
AC phase
Principles of radio
Contributors
What is alternating current (AC)?
Most students of electricity begin their study with what is
known as direct current (DC), which is electricity flowing ina
constant direction, and/or possessing a voltage with
constant polarity. DC is the kind of electricity made by a
battery (with definite positive and negative terminals), or
the kind of charge generated by rubbing certain types of
materials against each other.
As useful and as easy to understand as DC is, it is not the
only “kind” of electricity in use. Certain sources of electricity
(most notably, rotary electro-mechanical generators)
naturally produce voltages alternating in polarity, reversing
positive and negative over time. Either as a voltage
switching polarity or as a current switching direction back
and forth, this “kind” of electricity is known as Alternating
Current (AC): Figure below
DIRECT CURRENT ALTERNATING CURRENT
(DC) (AC)
<— ] <—— ]--->
1-_~ +--- | —_>
Direct vs alternating current
Whereas the familiar battery symbol is used as a generic
symbol for any DC voltage source, the circle with the wavy
line inside is the generic symbol for any AC voltage source.
One might wonder why anyone would bother with such a
thing as AC. It is true that in some cases AC holds no
practical advantage over DC. In applications where
electricity is used to dissipate energy in the form of heat, the
polarity or direction of current is irrelevant, so long as there
iS enough voltage and current to the load to produce the
desired heat (power dissipation). However, with AC it is
possible to build electric generators, motors and power
distribution systems that are far more efficient than DC, and
so we find AC used predominately across the world in high
power applications. To explain the details of why this is so, a
bit of background knowledge about AC is necessary.
If a machine is constructed to rotate a magnetic field around
a set of stationary wire coils with the turning of a shaft, AC
voltage will be produced across the wire coils as that shaft is
rotated, in accordance with Faraday's Law of
electromagnetic induction. This is the basic operating
principle of an AC generator, also known as an a/ternator.
Figure below
Step #1 Step #2
HD
no current!
Load
Step #3 Step #4
/o™
no (| ) Car
- +
no ai | |
WV —.
het Load
Alternator operation
Notice how the polarity of the voltage across the wire coils
reverses as the opposite poles of the rotating magnet pass
by. Connected to a load, this reversing voltage polarity will
create a reversing current direction in the circuit. The faster
the alternator's shaft is turned, the faster the magnet will
Spin, resulting in an alternating voltage and current that
switches directions more often in a given amount of time.
While DC generators work on the same general principle of
electromagnetic induction, their construction is not as
simple as their AC counterparts. With a DC generator, the
coil of wire is mounted in the shaft where the magnet is on
the AC alternator, and electrical connections are made to
this spinning coil via stationary carbon “brushes” contacting
copper strips on the rotating shaft. All this is necessary to
switch the coil's changing output polarity to the external
circuit so the external circuit sees a constant polarity: Figure
below
Step #1 Step #2
Load
DC generator operation
The generator shown above will produce two pulses of
voltage per revolution of the shaft, both pulses in the same
direction (polarity). In order for a DC generator to produce
constant voltage, rather than brief pulses of voltage once
every 1/2 revolution, there are multiple sets of coils making
intermittent contact with the brushes. The diagram shown
above is a bit more simplified than what you would see in
real life.
The problems involved with making and breaking electrical
contact with a moving coil should be obvious (sparking and
heat), especially if the shaft of the generator is revolving at
high speed. If the atmosphere surrounding the machine
contains flammable or explosive vapors, the practical
problems of spark-producing brush contacts are even
greater. An AC generator (alternator) does not require
brushes and commutators to work, and so is immune to
these problems experienced by DC generators.
The benefits of AC over DC with regard to generator design
is also reflected in electric motors. While DC motors require
the use of brushes to make electrical contact with moving
coils of wire, AC motors do not. In fact, AC and DC motor
designs are very similar to their generator counterparts
(identical for the sake of this tutorial), the AC motor being
dependent upon the reversing magnetic field produced by
alternating current through its stationary coils of wire to
rotate the rotating magnet around on its shaft, and the DC
motor being dependent on the brush contacts making and
breaking connections to reverse current through the rotating
coil every 1/2 rotation (180 degrees).
So we know that AC generators and AC motors tend to be
simpler than DC generators and DC motors. This relative
simplicity translates into greater reliability and lower cost of
manufacture. But what else is AC good for? Surely there
must be more to it than design details of generators and
motors! Indeed there is. There is an effect of
electromagnetism known as mutual induction, whereby two
or more coils of wire placed so that the changing magnetic
field created by one induces a voltage in the other. If we
have two mutually inductive coils and we energize one coil
with AC, we will create an AC voltage in the other coil. When
used as such, this device is known as a transformer. Figure
below
Transformer
Induced AC
voltage
Transformer “transforms” AC voltage and current.
The fundamental significance of a transformer is its ability to
step voltage up or down from the powered coil to the
unpowered coil. The AC voltage induced in the unpowered
(“secondary”) coil is equal to the AC voltage across the
powered (“primary”) coil multiplied by the ratio of secondary
coil turns to primary coil turns. If the secondary coil is
powering a load, the current through the secondary coil is
just the opposite: primary coil current multiplied by the ratio
of primary to secondary turns. This relationship has a very
close mechanical analogy, using torque and speed to
represent voltage and current, respectively: Figure below
Speed multiplication geartrain
"Step-down" transformer
Large gear
(many teeth)
Small gear
(few teeth)
low voltage
> few turns Load
high current
Noh oeeee low current
high torque
low speed
Speed multiplication gear train steps torque down and
speed up. Step-down transformer steps voltage down and
current up.
If the winding ratio is reversed so that the primary coil has
less turns than the secondary coil, the transformer “steps
up” the voltage from the source level to a higher level at the
load: Figure below
"Step-up" transformer
Speed reduction geartrain
Large gear ;
(many teeth) high voltage
Small gear
(few teeth) low voltage
AC many turns $ eee
voltage
sou
low torque high torque low current
high speed low speed
Speed reduction gear train steps torque up and speed down.
Step-up transformer steps voltage up and current down.
The transformer's ability to step AC voltage up or down with
ease gives AC an advantage unmatched by DC in the realm
of power distribution in figure below. When transmitting
electrical power over long distances, it is far more efficient to
do so with stepped-up voltages and stepped-down currents
(smaller-diameter wire with less resistive power losses), then
step the voltage back down and the current back up for
industry, business, or consumer use.
high voltage
Power Plant \ Seas
Step-up <f ou.f uf) wA OO
... to other customers
low voltage
Step-down
Home or
Business low voltage
Transformers enable efficient long distance high voltage
transmission of electric energy.
Transformer technology has made long-range electric power
distribution practical. Without the ability to efficiently step
voltage up and down, it would be cost-prohibitive to
construct power systems for anything but close-range
(within a few miles at most) use.
As useful as transformers are, they only work with AC, not
DC. Because the phenomenon of mutual inductance relies
on changing magnetic fields, and direct current (DC) can
only produce steady magnetic fields, transformers simply
will not work with direct current. Of course, direct current
may be interrupted (pulsed) through the primary winding of
a transformer to create a changing magnetic field (as is
done in automotive ignition systems to produce high-
voltage spark plug power from a low-voltage DC battery),
but pulsed DC is not that different from AC. Perhaps more
than any other reason, this is why AC finds such widespread
application in power systems.
e REVIEW:
e DC stands for “Direct Current,” meaning voltage or
current that maintains constant polarity or direction,
respectively, over time.
e AC stands for “Alternating Current,” meaning voltage or
current that changes polarity or direction, respectively,
over time.
AC electromechanical generators, known as alternators,
are of simpler construction than DC electromechanical
generators.
AC and DC motor design follows respective generator
design principles very closely.
A transformer is a pair of mutually-inductive coils used
to convey AC power from one coil to the other. Often, the
number of turns in each coil is set to create a voltage
increase or decrease from the powered (primary) coil to
the unpowered (secondary) coil.
e Secondary voltage = Primary voltage (Secondary turns /
primary turns)
e Secondary current = Primary current (primary turns /
secondary turns)
AC waveforms
When an alternator produces AC voltage, the voltage
switches polarity over time, but does so in a very particular
manner. When graphed over time, the “wave” traced by this
voltage of alternating polarity from an alternator takes on a
distinct shape, known as a sine wave: Figure below
(the sine wave)
Time —>
Graph of AC voltage over time (the sine wave).
In the voltage plot from an electromechanical alternator, the
change from one polarity to the other is a smooth one, the
voltage level changing most rapidly at the zero
(“crossover”) point and most slowly at its peak. If we were to
graph the trigonometric function of “sine” over a horizontal
range of 0 to 360 degrees, we would find the exact same
pattern as in Table below.
Trigonometric “sine” function.
nn
0.5000 + 210 [0.5000 -
O7o71 #225 _~fo7o71-
eo ——aeeo + b40 Fosse.
0.9659 + 255 —~(0.9659-
25
rpeak|270——}1.0000 [peak
0.9659 + 285. 0.9659}
120 .8660_—/+ 300 /o.a660 |
+ _pis___fovo71_f _|
150” 0.5000 33005000
nes 0.2588 + 345 —-0.2588 |_|
180 0.0000 zero 360 0.0000 _zero_
The reason why an electromechanical alternator outputs
sine-wave AC is due to the physics of its operation. The
voltage produced by the stationary coils by the motion of
the rotating magnet is proportional to the rate at which the
magnetic flux is changing perpendicular to the coils
(Faraday's Law of Electromagnetic Induction). That rate is
greatest when the magnet poles are closest to the coils, and
least when the magnet poles are furthest away from the
coils. Mathematically, the rate of magnetic flux change due
to a rotating magnet follows that of a sine function, so the
voltage produced by the coils follows that same function.
If we were to follow the changing voltage produced by a coil
in an alternator from any point on the sine wave graph to
that point when the wave shape begins to repeat itself, we
would have marked exactly one cycle of that wave. This is
most easily shown by spanning the distance between
identical peaks, but may be measured between any
corresponding points on the graph. The degree marks on the
horizontal axis of the graph represent the domain of the
trigonometric sine function, and also the angular position of
our simple two-pole alternator shaft as it rotates: Figure
below
I~—- one wave cycle —>|
I~—- one wave cycle —+>|
Alternator shaft ——>
position (degrees)
Alternator voltage as function of shaft position (time).
Since the horizontal axis of this graph can mark the passage
of time as well as shaft position in degrees, the dimension
marked for one cycle is often measured in a unit of time,
most often seconds or fractions of a second. When expressed
as a measurement, this is often called the period of a wave.
The period of a wave in degrees is a/ways 360, but the
amount of time one period occupies depends on the rate
voltage oscillates back and forth.
A more popular measure for describing the alternating rate
of an AC voltage or current wave than period is the rate of
that back-and-forth oscillation. This is called frequency. The
modern unit for frequency is the Hertz (abbreviated Hz),
which represents the number of wave cycles completed
during one second of time. In the United States of America,
the standard power-line frequency is 60 Hz, meaning that
the AC voltage oscillates at a rate of 60 complete back-and-
forth cycles every second. In Europe, where the power
system frequency is 50 Hz, the AC voltage only completes
50 cycles every second. A radio station transmitter
broadcasting at a frequency of 100 MHz generates an AC
voltage oscillating at a rate of 100 million cycles every
second.
Prior to the canonization of the Hertz unit, frequency was
simply expressed as “cycles per second.” Older meters and
electronic equipment often bore frequency units of “CPS”
(Cycles Per Second) instead of Hz. Many people believe the
change from self-explanatory units like CPS to Hertz
constitutes a step backward in clarity. A similar change
occurred when the unit of “Celsius” replaced that of
“Centigrade” for metric temperature measurement. The
name Centigrade was based on a 100-count (“Centi-”) scale
(“-grade”) representing the melting and boiling points of
HO, respectively. The name Celsius, on the other hand,
gives no hint as to the unit's origin or meaning.
Period and frequency are mathematical reciprocals of one
another. That is to say, if a wave has a period of 10 seconds,
its frequency will be 0.1 Hz, or 1/10 of a cycle per second:
1
Frequency in Hertz = ——__________
Period in seconds
An instrument called an oscilloscope, Figure below, is used
to display a changing voltage over time on a graphical
screen. You may be familiar with the appearance of an ECG
or EKG (electrocardiograph) machine, used by physicians to
graph the oscillations of a patient's heart over time. The
ECG is a special-purpose oscilloscope expressly designed for
medical use. General-purpose oscilloscopes have the ability
to display voltage from virtually any voltage source, plotted
as a graph with time as the independent variable. The
relationship between period and frequency is very useful to
know when displaying an AC voltage or current waveform on
an oscilloscope screen. By measuring the period of the wave
on the horizontal axis of the oscilloscope screen and
reciprocating that time value (in seconds), you can
determine the frequency in Hertz.
OSCILLOSCOPE
vertical
trigger
Time period of sinewave is shown on oscilloscope.
Voltage and current are by no means the only physical
variables subject to variation over time. Much more common
to our everyday experience is sound, which is nothing more
than the alternating compression and decompression
(pressure waves) of air molecules, interpreted by our ears as
a physical sensation. Because alternating current is a wave
phenomenon, it shares many of the properties of other wave
phenomena, like sound. For this reason, sound (especially
structured music) provides an excellent analogy for relating
AC concepts.
In musical terms, frequency is equivalent to pitch. Low-pitch
notes such as those produced by a tuba or bassoon consist
of air molecule vibrations that are relatively slow (low
frequency). High-pitch notes such as those produced by a
flute or whistle consist of the same type of vibrations in the
air, only vibrating at a much faster rate (higher frequency).
Figure below is a table showing the actual frequencies for a
range of common musical notes.
Note Musical designation Frequency (in hertz)
A A; 220.00
A sharp (or B flat) A* or BY 233.08
B B, 246.94
C (middle) C 261.63
C sharp (or D flat) orb" 277.18
D D 293.66
D sharp (or E flat) D* or E° 311.13
E E 329.63
F r 349.23
F sharp (or G flat) F*or@ 369.99
G G 392.00
G sharp (or A flat) G* or A? 415.30
A A 440.00
A sharp (or B flat) A* or B® 466.16
B B 493.88
cS Q’ 523.25
The frequency in Hertz (Hz) is shown for various musical
notes.
Astute observers will notice that all notes on the table
bearing the same letter designation are related by a
frequency ratio of 2:1. For example, the first frequency
shown (designated with the letter “A”) is 220 Hz. The next
highest “A” note has a frequency of 440 Hz -- exactly twice
as many sound wave cycles per second. The same 2:1 ratio
holds true for the first A sharp (233.08 Hz) and the next A
sharp (466.16 Hz), and for all note pairs found in the table.
Audibly, two notes whose frequencies are exactly double
each other sound remarkably similar. This similarity in sound
is musically recognized, the shortest span on a musical scale
separating such note pairs being called an octave. Following
this rule, the next highest “A” note (one octave above 440
Hz) will be 880 Hz, the next lowest “A” (one octave below
220 Hz) will be 110 Hz. A view of a piano keyboard helps to
put this scale into perspective: Figure below
C* D* — Gat tol Bid eG at eal 8 eo ar
Be pe GP AP at be pe be pe Ge AP at
HT
DEFGABCDEFGABCDEFGA
J one octave —+|
An octave is shown on a musical keyboard.
As you Can see, one octave is equal to seven white keys'
worth of distance on a piano keyboard. The familiar musical
mnemonic (doe-ray-mee-fah-so-lah-tee) -- yes, the same
pattern immortalized in the whimsical Rodgers and
Hammerstein song sung in The Sound of Music -- covers one
octave from C to C.
While electromechanical alternators and many other
physical phenomena naturally produce sine waves, this is
not the only kind of alternating wave in existence. Other
“waveforms” of AC are commonly produced within electronic
circuitry. Here are but a few sample waveforms and their
common designations in figure below
Square wave Triangle wave
M— onewave cycle —*+| — onewave cycle —~>
Sawtooth wave
Some common waveshapes (waveforms).
These waveforms are by no means the only kinds of
waveforms in existence. They're simply a few that are
common enough to have been given distinct names. Even in
circuits that are supposed to manifest “pure” sine, square,
triangle, or sawtooth voltage/current waveforms, the real-life
result is often a distorted version of the intended
waveshape. Some waveforms are so complex that they defy
classification as a particular “type” (including waveforms
associated with many kinds of musical instruments).
Generally speaking, any waveshape bearing close
resemblance to a perfect sine wave is termed sinusoidal,
anything different being labeled as non-sinusoidal. Being
that the waveform of an AC voltage or current is crucial to its
impact in a circuit, we need to be aware of the fact that AC
waves come in a variety of shapes.
e REVIEW:
e AC produced by an electromechanical alternator follows
the graphical shape of a sine wave.
e One cycle of a wave is one complete evolution of its
Shape until the point that it is ready to repeat itself.
e The period of a wave is the amount of time it takes to
complete one cycle.
e Frequency is the number of complete cycles that a wave
completes in a given amount of time. Usually measured
in Hertz (Hz), 1 Hz being equal to one complete wave
cycle per second.
e Frequency = 1/(period in seconds)
Measurements of AC magnitude
So far we know that AC voltage alternates in polarity and AC
current alternates in direction. We also know that AC can
alternate in a variety of different ways, and by tracing the
alternation over time we can plot it as a “waveform.” We can
measure the rate of alternation by measuring the time it
takes for a wave to evolve before it repeats itself (the
“period”), and express this as cycles per unit time, or
“frequency.” In music, frequency is the same as pitch, which
is the essential property distinguishing one note from
another.
However, we encounter a measurement problem if we try to
express how large or small an AC quantity is. With DC, where
quantities of voltage and current are generally stable, we
have little trouble expressing how much voltage or current
we have in any part of a circuit. But how do you grant a
single measurement of magnitude to something that is
constantly changing?
One way to express the intensity, or magnitude (also called
the amplitude), of an AC quantity is to measure its peak
height on a waveform graph. This is known as the peak or
crest value of an AC waveform: Figure below
Peak
bd
Time —>
Peak voltage of a waveform.
Another way is to measure the total height between
opposite peaks. This is known as the peak-to-peak (P-P)
value of an AC waveform: Figure below
Peak-to-Peak
[a
Time —>
Peak-to-peak voltage of a waveform.
Unfortunately, either one of these expressions of waveform
amplitude can be misleading when comparing two different
types of waves. For example, a square wave peaking at 10
volts is obviously a greater amount of voltage for a greater
amount of time than a triangle wave peaking at 10 volts.
The effects of these two AC voltages powering a load would
be quite different: Figure below
(same load resistance)
4
10 V W) i 10 V
(peak) (peak
more heat energy less heat energy
dissipated dissipated
(N) a
)
A square wave produces a greater heating effect than the
same peak voltage triangle wave.
One way of expressing the amplitude of different
waveshapes in a more equivalent fashion is to
mathematically average the values of all the points ona
waveform's graph to a single, aggregate number. This
amplitude measure is known simply as the average value of
the waveform. If we average all the points on the waveform
algebraically (that is, to consider their s/gn, either positive
or negative), the average value for most waveforms is
technically zero, because all the positive points cancel out
all the negative points over a full cycle: Figure below
True average value of all points
(considering their signs) is zero!
The average value of a sinewave Is zero.
This, of course, will be true for any waveform having equal-
area portions above and below the “zero” line of a plot.
However, as a practical measure of a waveform's aggregate
value, “average” is usually defined as the mathematical
mean of all the points' abso/ute values over a cycle. In other
words, we calculate the practical average value of the
waveform by considering all points on the wave as positive
quantities, as if the waveform looked like this: Figure below
Practical average of points, all
values assumed to be positive.
Waveform seen by AC “average responding” meter.
Polarity-insensitive mechanical meter movements (meters
designed to respond equally to the positive and negative
half-cycles of an alternating voltage or current) register in
proportion to the waveform's (practical) average value,
because the inertia of the pointer against the tension of the
spring naturally averages the force produced by the varying
voltage/current values over time. Conversely, polarity-
sensitive meter movements vibrate uselessly if exposed to
AC voltage or current, their needles oscillating rapidly about
the zero mark, indicating the true (algebraic) average value
of zero for a symmetrical waveform. When the “average”
value of a waveform is referenced in this text, it will be
assumed that the “practical” definition of average is
intended unless otherwise specified.
Another method of deriving an aggregate value for
waveform amplitude is based on the waveform's ability to do
useful work when applied to a load resistance.
Unfortunately, an AC measurement based on work
performed by a waveform is not the same as that waveform's
“average” value, because the power dissipated by a given
load (work performed per unit time) is not directly
proportional to the magnitude of either the voltage or
Current impressed upon it. Rather, power is proportional to
the square of the voltage or current applied to a resistance
(P = E2/R, and P = [?R). Although the mathematics of such
an amplitude measurement might not be straightforward,
the utility of it is.
Consider a bandsaw and a jigsaw, two pieces of modern
woodworking equipment. Both types of saws cut with a thin,
toothed, motor-powered metal blade to cut wood. But while
the bandsaw uses a continuous motion of the blade to cut,
the jigsaw uses a back-and-forth motion. The comparison of
alternating current (AC) to direct current (DC) may be
likened to the comparison of these two saw types: Figure
below
Bandsaw
eS Jigsaw
blade
motion!
> blade
motion
(analogous to DC) (analogous to AC)
Bandsaw-jigsaw analogy of DC vs AC.
The problem of trying to describe the changing quantities of
AC voltage or current in a single, aggregate measurement is
also present in this saw analogy: how might we express the
speed of a jigsaw blade? A bandsaw blade moves with a
constant speed, similar to the way DC voltage pushes or DC
current moves with a constant magnitude. A jigsaw blade,
on the other hand, moves back and forth, its blade speed
constantly changing. What is more, the back-and-forth
motion of any two jigsaws may not be of the same type,
depending on the mechanical design of the saws. One
jigsaw might move its blade with a sine-wave motion, while
another with a triangle-wave motion. To rate a jigsaw based
on its peak blade speed would be quite misleading when
comparing one jigsaw to another (or a jigsaw with a
bandsaw!). Despite the fact that these different saws move
their blades in different manners, they are equal in one
respect: they all cut wood, and a quantitative comparison of
this common function can serve as a common basis for
which to rate blade speed.
Picture a jigsaw and bandsaw side-by-side, equipped with
identical blades (Same tooth pitch, angle, etc.), equally
capable of cutting the same thickness of the same type of
wood at the same rate. We might say that the two saws were
equivalent or equal in their cutting capacity. Might this
comparison be used to assign a “bandsaw equivalent” blade
speed to the jigsaw's back-and-forth blade motion; to relate
the wood-cutting effectiveness of one to the other? This is
the general idea used to assign a “DC equivalent”
measurement to any AC voltage or current: whatever
magnitude of DC voltage or current would produce the same
amount of heat energy dissipation through an equal
resistance:Figure below
~«~—5A RMS ---+ ~«~— iA
LOV 22 Za lov — 22 Za
RMS @ 393 ~ ~
~-- 5A RMS —> 50 W 5A— > 50W
power power
dissipated dissipated
Equal power dissipated through
equal resistance loads
An RMS voltage produces the same heating effect as a the
same DC voltage
In the two circuits above, we have the same amount of load
resistance (2 QO) dissipating the same amount of power in the
form of heat (50 watts), one powered by AC and the other by
DC. Because the AC voltage source pictured above is
equivalent (in terms of power delivered to a load) toa 10
volt DC battery, we would call this a “10 volt” AC source.
More specifically, we would denote its voltage value as
being 10 volts RMS. The qualifier “RMS” stands for Root
Mean Square, the algorithm used to obtain the DC
equivalent value from points on a graph (essentially, the
procedure consists of squaring all the positive and negative
points on a waveform graph, averaging those squared
values, then taking the square root of that average to obtain
the final answer). Sometimes the alternative terms
equivalent or DC equivalent are used instead of “RMS,” but
the quantity and principle are both the same.
RMS amplitude measurement is the best way to relate AC
quantities to DC quantities, or other AC quantities of
differing waveform shapes, when dealing with
measurements of electric power. For other considerations,
peak or peak-to-peak measurements may be the best to
employ. For instance, when determining the proper size of
wire (ampacity) to conduct electric power from a source toa
load, RMS current measurement is the best to use, because
the principal concern with current is overheating of the wire,
which is a function of power dissipation caused by current
through the resistance of the wire. However, when rating
insulators for service in high-voltage AC applications, peak
voltage measurements are the most appropriate, because
the principal concern here is insulator “flashover” caused by
brief spikes of voltage, irrespective of time.
Peak and peak-to-peak measurements are best performed
with an oscilloscope, which can capture the crests of the
waveform with a high degree of accuracy due to the fast
action of the cathode-ray-tube in response to changes in
voltage. For RMS measurements, analog meter movements
(D'Arsonval, Weston, iron vane, electrodynamometer) will
work so long as they have been calibrated in RMS figures.
Because the mechanical inertia and dampening effects of an
electromechanical meter movement makes the deflection of
the needle naturally proportional to the average value of the
AC, not the true RMS value, analog meters must be
specifically calibrated (or mis-calibrated, depending on how
you look at it) to indicate voltage or current in RMS units.
The accuracy of this calibration depends on an assumed
waveshape, usually a sine wave.
Electronic meters specifically designed for RMS
measurement are best for the task. Some instrument
manufacturers have designed ingenious methods for
determining the RMS value of any waveform. One such
manufacturer produces “True-RMS” meters with a tiny
resistive heating element powered by a voltage proportional
to that being measured. The heating effect of that resistance
element is measured thermally to give a true RMS value with
no mathematical calculations whatsoever, just the laws of
physics in action in fulfillment of the definition of RMS. The
accuracy of this type of RMS measurement is independent of
waveshape.
For “pure” waveforms, simple conversion coefficients exist
for equating Peak, Peak-to-Peak, Average (practical, not
algebraic), and RMS measurements to one another: Figure
below
RMS = 0.707 (Peak)
AVG = 0.637 (Peak)
P-P = 2 (Peak)
RMS = Peak
AVG = Peak
P-P = 2 (Peak)
RMS = 0.577 (Peak)
AVG = 0.5 (Peak)
P-P = 2 (Peak)
Conversion factors for common waveforms.
In addition to RMS, average, peak (crest), and peak-to-peak
measures of an AC waveform, there are ratios expressing the
proportionality between some of these fundamental
measurements. The crest factor of an AC waveform, for
instance, is the ratio of its peak (crest) value divided by its
RMS value. The form factor of an AC waveform is the ratio of
its RMS value divided by its average value. Square-shaped
waveforms always have crest and form factors equal to 1,
since the peak is the same as the RMS and average values.
Sinusoidal waveforms have an RMS value of 0.707 (the
reciprocal of the square root of 2) and a form factor of 1.11
(0.707/0.636). Triangle- and sawtooth-shaped waveforms
have RMS values of 0.577 (the reciprocal of square root of 3)
and form factors of 1.15 (0.577/0.5).
Bear in mind that the conversion constants shown here for
peak, RMS, and average amplitudes of sine waves, square
waves, and triangle waves hold true only for pure forms of
these waveshapes. The RMS and average values of distorted
waveshapes are not related by the same ratios: Figure below
RMS = ???
AVG = ???
P-P = 2 (Peak)
Arbitrary waveforms have no simple conversions.
This is a very important concept to understand when using
an analog D'Arsonval meter movement to measure AC
voltage or current. An analog D'Arsonval movement,
calibrated to indicate sine-wave RMS amplitude, will only be
accurate when measuring pure sine waves. If the waveform
of the voltage or current being measured is anything but a
pure sine wave, the indication given by the meter will not be
the true RMS value of the waveform, because the degree of
needle deflection in an analog D'Arsonval meter movement
iS proportional to the average value of the waveform, not the
RMS. RMS meter calibration is obtained by “skewing” the
span of the meter so that it displays a small multiple of the
average value, which will be equal to be the RMS value fora
particular waveshape and a particular waveshape only.
Since the sine-wave shape is most common in electrical
measurements, it is the waveshape assumed for analog
meter calibration, and the small multiple used in the
calibration of the meter is 1.1107 (the form factor:
0.707/0.636: the ratio of RMS divided by average for a
sinusoidal waveform). Any waveshape other than a pure sine
wave will have a different ratio of RMS and average values,
and thus a meter calibrated for sine-wave voltage or current
will not indicate true RMS when reading a non-sinusoidal
wave. Bear in mind that this limitation applies only to
simple, analog AC meters not employing “True-RMS”
technology.
e REVIEW:
e The amplitude of an AC waveform is its height as
depicted on a graph over time. An amplitude
measurement can take the form of peak, peak-to-peak,
average, or RMS quantity.
Peak amplitude is the height of an AC waveform as
measured from the zero mark to the highest positive or
lowest negative point on a graph. Also known as the
crest amplitude of a wave.
Peak-to-peak amplitude is the total height of an AC
waveform as measured from maximum positive to
maximum negative peaks on a graph. Often abbreviated
as “P-P”,
Average amplitude is the mathematical “mean” of all a
waveform's points over the period of one cycle.
Technically, the average amplitude of any waveform
with equal-area portions above and below the “zero” line
on a graph is zero. However, as a practical measure of
amplitude, a waveform's average value is often
calculated as the mathematical mean of all the points’
absolute values (taking all the negative values and
considering them as positive). For a sine wave, the
average value so calculated is approximately 0.637 of
its peak value.
e “RMS” stands for Root Mean Square, and is a way of
expressing an AC quantity of voltage or current in terms
functionally equivalent to DC. For example, 10 volts AC
RMS is the amount of voltage that would produce the
Same amount of heat dissipation across a resistor of
given value as a 10 volt DC power supply. Also Known as
the “equivalent” or “DC equivalent” value of an AC
voltage or current. For a sine wave, the RMS value is
approximately 0.707 of its peak value.
e The crest factor of an AC waveform is the ratio of its
peak (crest) to its RMS value.
e The form factor of an AC waveform is the ratio of its RMS
value to its average value.
e Analog, electromechanical meter movements respond
proportionally to the average value of an AC voltage or
current. When RMS indication is desired, the meter's
calibration must be “skewed” accordingly. This means
that the accuracy of an electromechanical meter's RMS
indication is dependent on the purity of the waveform:
whether it is the exact same waveshape as the
waveform used in calibrating.
Simple AC circuit calculations
Over the course of the next few chapters, you will learn that
AC circuit measurements and calculations can get very
complicated due to the complex nature of alternating
Current in circuits with inductance and capacitance.
However, with simple circuits (figure below) involving
nothing more than an AC power source and resistance, the
Same laws and rules of DC apply simply and directly.
AC circuit calculations for resistive circuits are the same as
for DC.
Rita = R, + R, + R,
Rootal = 1 kQ
Erotal 10 V
Liotal = = Lota = LOmA
Reotal = 1kQ
| R.
Eri = loraRi Epo = Lota Rs R3 = hols
Ep, =1V Er, =5 V E,3;=4V
Series resistances still add, parallel resistances still diminish,
and the Laws of Kirchhoff and Ohm still hold true. Actually,
as we will discover later on, these rules and laws a/ways hold
true, its just that we have to express the quantities of
voltage, current, and opposition to current in more advanced
mathematical forms. With purely resistive circuits, however,
these complexities of AC are of no practical consequence,
and so we can treat the numbers as though we were dealing
with simple DC quantities.
Because all these mathematical relationships still hold true,
we can make use of our familiar “table” method of
organizing circuit values just as with DC:
Total
R,
5 Volts
R, R;
re ee ee
Amps
R|_ too | soo | 400 | tk | Ohms
One major caveat needs to be given here: all measurements
of AC voltage and current must be expressed in the same
terms (peak, peak-to-peak, average, or RMS). If the source
voltage is given in peak AC volts, then all currents and
voltages subsequently calculated are cast in terms of peak
units. If the source voltage is given in AC RMS volts, then all
calculated currents and voltages are cast in AC RMS units as
well. This holds true for any calculation based on Ohm's
Laws, Kirchhoff's Laws, etc. Unless otherwise stated, all
values of voltage and current in AC circuits are generally
assumed to be RMS rather than peak, average, or peak-to-
peak. In some areas of electronics, peak measurements are
assumed, but in most applications (especially industrial
electronics) the assumption is RMS.
e REVIEW:
e All the old rules and laws of DC (Kirchhoff's Voltage and
Current Laws, Ohm's Law) still hold true for AC. However,
with more complex circuits, we may need to represent
the AC quantities in more complex form. More on this
later, | promise!
e The “table” method of organizing circuit values is still a
valid analysis tool for AC circuits.
AC phase
Things start to get complicated when we need to relate two
or more AC voltages or currents that are out of step with
each other. By “out of step,” | mean that the two waveforms
are not synchronized: that their peaks and zero points do
not match up at the same points in time. The graph in figure
below illustrates an example of this.
AB AB
Out of phase waveforms
The two waves shown above (A versus B) are of the same
amplitude and frequency, but they are out of step with each
other. In technical terms, this is called a phase shift. Earlier
we saw how we could plot a “sine wave” by calculating the
trigonometric sine function for angles ranging from 0 to 360
degrees, a full circle. The starting point of a sine wave was
zero amplitude at zero degrees, progressing to full positive
amplitude at 90 degrees, zero at 180 degrees, full negative
at 270 degrees, and back to the starting point of zero at 360
degrees. We can use this angle scale along the horizontal
axis of our waveform plot to express just how far out of step
one wave is with another: Figure below
degrees
(0) (0)
A 0 90 180 270 360 90 180 270 360
BO 90 180 270 360 9% 180 270 360
(0) (O)
degrees
Wave A leads wave B by 45°
The shift between these two waveforms is about 45 degrees,
the “A” wave being ahead of the “B” wave. A sampling of
different phase shifts is given in the following graphs to
better illustrate this concept: Figure below
Phase shift = 90 degrees
Ais ahead of B
(A "leads” B)
Phase shift = 90 degrees
Bis ahead of A
(B "leads” A)
Phase shift = 180 degrees
A and B waveforms are
mirror-images of each other
Phase shift = 0 degrees
AB A and B waveforms are
in perfect step with each other
Examples of phase shifts.
Because the waveforms in the above examples are at the
same frequency, they will be out of step by the same
angular amount at every point in time. For this reason, we
can express phase shift for two or more waveforms of the
same frequency as a constant quantity for the entire wave,
and not just an expression of shift between any two
particular points along the waves. That is, it is safe to say
something like, “voltage 'A' is 45 degrees out of phase with
voltage 'B'.” Whichever waveform is ahead in its evolution is
said to be /eading and the one behind is said to be /agging.
Phase shift, like voltage, is always a measurement relative
between two things. There's really no such thing asa
waveform with an abso/ute phase measurement because
there's no known universal reference for phase. Typically in
the analysis of AC circuits, the voltage waveform of the
power supply is used as a reference for phase, that voltage
stated as “xxx volts at 0 degrees.” Any other AC voltage or
Current in that circuit will have its phase shift expressed in
terms relative to that source voltage.
This is what makes AC circuit calculations more complicated
than DC. When applying Ohm's Law and Kirchhoff's Laws,
quantities of AC voltage and current must reflect phase shift
as well as amplitude. Mathematical operations of addition,
subtraction, multiplication, and division must operate on
these quantities of phase shift as well as amplitude.
Fortunately, there is a mathematical system of quantities
called complex numbers ideally suited for this task of
representing amplitude and phase.
Because the subject of complex numbers is so essential to
the understanding of AC circuits, the next chapter will be
devoted to that subject alone.
e REVIEW:
e Phase shift is where two or more waveforms are out of
step with each other.
e The amount of phase shift between two waves can be
expressed in terms of degrees, as defined by the degree
units on the horizontal axis of the waveform graph used
in plotting the trigonometric sine function.
e A leading waveform is defined as one waveform that is
ahead of another in its evolution. A /Jagging waveform is
one that is behind another. Example:
Phase shift = 90 degrees
Aleads B; B lagsA
e Calculations for AC circuit analysis must take into
consideration both amplitude and phase shift of voltage
and current waveforms to be completely accurate. This
requires the use of a mathematical system called
complex numbers.
Principles of radio
One of the more fascinating applications of electricity is in
the generation of invisible ripples of energy called radio
waves. The limited scope of this lesson on alternating
current does not permit full exploration of the concept, some
of the basic principles will be covered.
With Oersted's accidental discovery of electromagnetism, it
was realized that electricity and magnetism were related to
each other. When an electric current was passed through a
conductor, a magnetic field was generated perpendicular to
the axis of flow. Likewise, if a conductor was exposed to a
change in magnetic flux perpendicular to the conductor, a
voltage was produced along the length of that conductor. So
far, scientists knew that electricity and magnetism always
seemed to affect each other at right angles. However, a
major discovery lay hidden just beneath this seemingly
simple concept of related perpendicularity, and its unveiling
was one of the pivotal moments in modern science.
This breakthrough in physics is hard to overstate. The man
responsible for this conceptual revolution was the Scottish
physicist James Clerk Maxwell (1831-1879), who “unified”
the study of electricity and magnetism in four relatively tidy
equations. In essence, what he discovered was that electric
and magnetic fie/ds were intrinsically related to one another,
with or without the presence of a conductive path for
electrons to flow. Stated more formally, Maxwell's discovery
was this:
A changing electric field produces a perpendicular
magnetic field, and
A changing magnetic field produces a perpendicular
electric field.
All of this can take place in open space, the alternating
electric and magnetic fields supporting each other as they
travel through space at the speed of light. This dynamic
structure of electric and magnetic fields propagating
through space is better known as an e/ectromagnetic wave.
There are many kinds of natural radiative energy composed
of electromagnetic waves. Even light is electromagnetic in
nature. So are X-rays and “gamma” ray radiation. The only
difference between these kinds of electromagnetic radiation
is the frequency of their oscillation (alternation of the
electric and magnetic fields back and forth in polarity). By
using a source of AC voltage and a special device called an
antenna, we can create electromagnetic waves (of a much
lower frequency than that of light) with ease.
An antenna is nothing more than a device built to produce a
dispersing electric or magnetic field. Two fundamental types
of antennae are the dipole and the /oop: Figure below
Basic antenna designs
DIPOLE LOOP
—_()}-___—.
Dipole and loop antennae
While the dipole looks like nothing more than an open
circuit, and the loop a short circuit, these pieces of wire are
effective radiators of electromagnetic fields when connected
to AC sources of the proper frequency. The two open wires of
the dipole act as a sort of capacitor (two conductors
separated by a dielectric), with the electric field open to
dispersal instead of being concentrated between two
closely-spaced plates. The closed wire path of the loop
antenna acts like an inductor with a large air core, again
providing ample opportunity for the field to disperse away
from the antenna instead of being concentrated and
contained as in a normal inductor.
As the powered dipole radiates its changing electric field
into space, a changing magnetic field is produced at right
angles, thus sustaining the electric field further into space,
and so on as the wave propagates at the speed of light. As
the powered loop antenna radiates its changing magnetic
field into space, a changing electric field is produced at right
angles, with the same end-result of a continuous
electromagnetic wave sent away from the antenna. Either
antenna achieves the same basic task: the controlled
production of an electromagnetic field.
When attached to a source of high-frequency AC power, an
antenna acts as a transmitting device, converting AC
voltage and current into electromagnetic wave energy.
Antennas also have the ability to intercept electromagnetic
waves and convert their energy into AC voltage and current.
In this mode, an antenna acts as a receiving device: Figure
below
AC voltage Radio receivers
produced _
i \ AC current
produced
electromagnetic radiation electromagnetic radiation
TTT TTT
Radio transmitters
Basic radio transmitter and receiver
While there is much more that may be said about antenna
technology, this brief introduction is enough to give you the
general idea of what's going on (and perhaps enough
information to provoke a few experiments).
e REVIEW:
e James Maxwell discovered that changing electric fields
produce perpendicular magnetic fields, and vice versa,
even in empty space.
e A twin set of electric and magnetic fields, oscillating at
right angles to each other and traveling at the speed of
light, constitutes an electromagnetic wave.
e An antenna is a device made of wire, designed to radiate
a changing electric field or changing magnetic field
when powered by a high-frequency AC source, or
intercept an electromagnetic field and convert it to an
AC voltage or current.
e The dipole antenna consists of two pieces of wire (not
touching), primarily generating an electric field when
energized, and secondarily producing a magnetic field
In space.
e The /oop antenna consists of a loop of wire, primarily
generating a magnetic field when energized, and
secondarily producing an electric field in space.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Harvey Lew (February 7, 2004): Corrected typographical
error: “circuit” should have been “circle”.
Duane Damiano (February 25, 2003): Pointed out
magnetic polarity error in DC generator illustration.
Mark D. Zarella (April 28, 2002): Suggestion for improving
explanation of “average” waveform amplitude.
John Symonds (March 28, 2002): Suggestion for improving
explanation of the unit “Hertz.”
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—|/|+4]l\—
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 2
COMPLEX NUMBERS
Introduction
Vectors and AC waveforms
e Simple vector addition
e Complex vector addition
e Polar and rectangular notation
e Complex number arithmetic
e More on AC "polarity"
e Some examples with AC circuits
Contributors
Introduction
If | needed to describe the distance between two cities, |
could provide an answer consisting of a single number in
miles, kilometers, or some other unit of linear measurement.
However, if | were to describe how to travel from one city to
another, | would have to provide more information than just
the distance between those two cities; | would also have to
provide information about the direction to travel, as well.
The kind of information that expresses a single dimension,
such as linear distance, is called a sca/ar quantity in
mathematics. Scalar numbers are the kind of numbers
you've used in most all of your mathematical applications so
far. The voltage produced by a battery, for example, is a
scalar quantity. So is the resistance of a piece of wire (ohms),
or the current through it (amps).
However, when we begin to analyze alternating current
circuits, we find that quantities of voltage, current, and even
resistance (called /mpedance in AC) are not the familiar one-
dimensional quantities we're used to measuring in DC
circuits. Rather, these quantities, because they're dynamic
(alternating in direction and amplitude), possess other
dimensions that must be taken into account. Frequency and
phase shift are two of these dimensions that come into play.
Even with relatively simple AC circuits, where we're only
dealing with a single frequency, we still have the dimension
of phase shift to contend with in addition to the amplitude.
In order to successfully analyze AC circuits, we need to work
with mathematical objects and techniques capable of
representing these multi-dimensional quantities. Here is
where we need to abandon scalar numbers for something
better suited: complex numbers. Just like the example of
giving directions from one city to another, AC quantities in a
single-frequency circuit have both amplitude (analogy:
distance) and phase shift (analogy: direction). A complex
number is a single mathematical quantity able to express
these two dimensions of amplitude and phase shift at once.
Complex numbers are easier to grasp when they're
represented graphically. If | draw a line with a certain length
(magnitude) and angle (direction), | have a graphic
representation of a complex number which is commonly
known in physics as a vector. (Figure below)
—_—_—_—_—_—_—_ ————————-
length = 7 length = 10
angle = 0 degrees angle = 180 degrees
length = 5 length = 4
angle = 90 degrees angle = 270 degrees
(-90 degrees)
length = 9.43
length = 5.66 angle = 302.01 degrees
angle = 45 degrees (-57.99 degrees)
A vector has both magnitude and direction.
Like distances and directions on a map, there must be some
common frame of reference for angle figures to have any
meaning. In this case, directly right is considered to be 0°,
and angles are counted in a positive direction going counter-
clockwise: (Figure below)
The vector "compass"
90°
270° (-90°)
The vector compass
The idea of representing a number in graphical form is
nothing new. We all learned this in grade school with the
“number line:” (Figure below)
0 1 = 3 4 5 6 7 8 9 10
Number lIine.
We even learned how addition and subtraction works by
seeing how lengths (magnitudes) stacked up to give a final
answer: (Figure below)
5+3=8
J+ 5 —______—- +
[$< § —_$___»}— 3 ——}
Addition on a “number line”.
Later, we learned that there were ways to designate the
values between the whole numbers marked on the line.
These were fractional or decimal quantities: (Figure below)
3-1/2 or 3.5
0 1 2 3 4 5 6 Fi 8 9 10
Locating a fraction on the “number line”
Later yet we learned that the number line could extend to
the left of zero as well: (Figure below)
5 4 3 2 +1 01 2 3 #4 «5
“Number line” shows both positive and negative numbers.
These fields of numbers (whole, integer, rational, irrational,
real, etc.) learned in grade school share a common trait:
they're all one-dimensional. The straightness of the number
line illustrates this graphically. You can move up or down the
number line, but all “motion” along that line is restricted to
a single axis (horizontal). One-dimensional, scalar numbers
are perfectly adequate for counting beads, representing
weight, or measuring DC battery voltage, but they fall short
of being able to represent something more complex like the
distance and direction between two cities, or the amplitude
and phase of an AC waveform. To represent these kinds of
quantities, we need multidimensional representations. In
other words, we need a number line that can point in
different directions, and that's exactly what a vector is.
REVIEW:
e A scalarnumber is the type of mathematical object that
people are used to using in everyday life: a one-
dimensional quantity like temperature, length, weight,
etc.
e A complex number is a mathematical quantity
representing two dimensions of magnitude and
direction.
e A vector is a graphical representation of a complex
number. It looks like an arrow, with a starting point, a
tip, a definite length, and a definite direction.
Sometimes the word phasor is used in electrical
applications where the angle of the vector represents
phase shift between waveforms.
Vectors and AC waveforms
OK, so how exactly can we represent AC quantities of
voltage or current in the form of a vector? The length of the
vector represents the magnitude (or amplitude) of the
waveform, like this: (Figure below)
Waveform Vector representation
eee —
Amplitude
| ~—— Length ne
Ce a
Vector length represents AC voltage magnitude.
The greater the amplitude of the waveform, the greater the
length of its corresponding vector. The angle of the vector,
however, represents the phase shift in degrees between the
waveform in question and another waveform acting as a
“reference” in time. Usually, when the phase of a waveform
in a circuit is expressed, it is referenced to the power supply
voltage waveform (arbitrarily stated to be “at” 0°).
Remember that phase is always a re/ative measurement
between two waveforms rather than an absolute property.
(Figure below) (Figure below)
Waveforms Phase relations Vector representatians
(of "A" waveform with
reference to "B" waveform)
Phase shift = 0 degrees
AB A and B waveforms are — AB
in perfect step with each other
A
Phase shift = 90 degrees
A is ahead of B 90 degrees
(A "leads" B) =
Phase shift = 90 degrees >B
B is ahead of A -90 degrees
(B "leads" A)
A
Phase shift = 180 degrees 180 degrees
A and B waveforms are A ~B
mirror-images of each other
—>| k—
phase shift
Phase shift between waves and vector phase angle
The greater the phase shift in degrees between two
waveforms, the greater the angle difference between the
corresponding vectors. Being a relative measurement, like
voltage, phase shift (vector angle) only has meaning in
reference to some standard waveform. Generally this
“reference” waveform is the main AC power supply voltage
in the circuit. If there is more than one AC voltage source,
then one of those sources is arbitrarily chosen to be the
phase reference for all other measurements in the circuit.
This concept of a reference point is not unlike that of the
“ground” point in a circuit for the benefit of voltage
reference. With a clearly defined point in the circuit declared
to be “ground,” it becomes possible to talk about voltage
“on” or “at” single points in a circuit, being understood that
those voltages (always relative between two points) are
referenced to “ground.” Correspondingly, with a clearly
defined point of reference for phase it becomes possible to
speak of voltages and currents in an AC circuit having
definite phase angles. For example, if the current in an AC
circuit is described as “24.3 milliamps at -64 degrees,” it
means that the current waveform has an amplitude of 24.3
mA, and it lags 64° behind the reference waveform, usually
assumed to be the main source voltage waveform.
e REVIEW:
e When used to describe an AC quantity, the length of a
vector represents the amplitude of the wave while the
angle of a vector represents the phase angle of the wave
relative to some other (reference) waveform.
Simple vector addition
Remember that vectors are mathematical objects just like
numbers on a number line: they can be added, subtracted,
multiplied, and divided. Addition is perhaps the easiest
vector operation to visualize, so we'll begin with that. If
vectors with common angles are added, their magnitudes
(lengths) add up just like regular scalar quantities: (Figure
below)
lengh=6 = length=8 total length =6+8=14
> 2 ae
angle =Odegrees angle =Odegrees angle = O degrees
Vector magnitudes add like scalars for a common angle.
Similarly, if AC voltage sources with the same phase angle
are connected together in series, their voltages add just as
you might expect with DC batteries: (Figure below)
~f4 V]=
“In phase” AC voltages add like DC battery voltages.
Please note the (+) and (-) polarity marks next to the leads
of the two AC sources. Even though we know AC doesn't
have “polarity” in the same sense that DC does, these marks
are essential to knowing how to reference the given phase
angles of the voltages. This will become more apparent in
the next example.
If vectors directly opposing each other (180° out of phase)
are added together, their magnitudes (lengths) subtract just
like positive and negative scalar quantities subtract when
added: (Figure below)
length = 6 angle = 0 degrees
es
length =8 angle = 180 degrees
total length = 6 - 8 = -2 at Odegrees
<— or 2at 180 degrees
Directly opposing vector magnitudes subtract.
Similarly, if opposing AC voltage sources are connected in
series, their voltages subtract as you might expect with DC
batteries connected in an opposing fashion: (Figure below)
~~ f°
6V 8V
Odeg 180 deg 6V 8V
- + - + -
Opposing AC voltages subtract like opposing battery
voltages.
Determining whether or not these voltage sources are
opposing each other requires an examination of their
polarity markings and their phase angles. Notice how the
polarity markings in the above diagram seem to indicate
additive voltages (from left to right, we see - and + on the 6
volt source, - and + on the 8 volt source). Even though these
polarity markings would normally indicate an additive effect
in a DC circuit (the two voltages working together to
produce a greater total voltage), in this AC circuit they're
actually pushing in opposite directions because one of those
voltages has a phase angle of 0° and the other a phase
angle of 180°. The result, of course, is a total voltage of 2
volts.
We could have just as well shown the opposing voltages
subtracting in series like this: (Figure below)
Opposing voltages in spite of equal phase angles.
Note how the polarities appear to be opposed to each other
now, due to the reversal of wire connections on the 8 volt
source. Since both sources are described as having equal
phase angles (0°), they truly are opposed to one another,
and the overall effect is the same as the former scenario
with “additive” polarities and differing phase angles: a total
voltage of only 2 volts. (Figure below)
Nw SJ
6V 8V
0 deg 0 deg
- + + -
Just as there are two ways to express the phase of the
sources, there are two ways to express the resultant their
sum.
The resultant voltage can be expressed in two different
ways: 2 volts at 180° with the (-) symbol on the left and the
(+) symbol on the right, or 2 volts at 0° with the (+) symbol
on the left and the (-) symbol on the right. A reversal of
wires from an AC voltage source is the same as phase-
shifting that source by 180°. (Figure below)
8V 8
180 deg These voltage sources 0 deg
+4 =
= (\) are equivalent! ~~)
Example of equivalent voltage sources.
Complex vector addition
If vectors with uncommon angles are added, their
magnitudes (lengths) add up quite differently than that of
scalar magnitudes: (Figure below)
Vector addition
length = 10
angle = 53.13
sea lis A
i
length =6
angle = 0 degrees
6 at 0 degrees
length =8
+ 8at90 degrees
angle = 90 degrees eee ener
10 at 53.13 degrees
Vector magnitudes do not directly add for unequal angles.
If two AC voltages -- 90° out of phase -- are added together
by being connected in series, their voltage magnitudes do
not directly add or subtract as with scalar voltages in DC.
Instead, these voltage quantities are complex quantities,
and just like the above vectors, which add up ina
trigonometric fashion, a 6 volt source at 0° added to an 8
volt source at 90° results in 10 volts at a phase angle of
53.13°: (Figure below)
Ne ONG
6V 8V
0 deg 90 deg
- + - +
10 V
53.13 deg
The 6V and 8V sources add to 10V with the help of
trigonometry.
Compared to DC circuit analysis, this is very strange indeed.
Note that it is possible to obtain voltmeter indications of 6
and 8 volts, respectively, across the two AC voltage sources,
yet only read 10 volts for a total voltage!
There is no suitable DC analogy for what we're seeing here
with two AC voltages slightly out of phase. DC voltages can
only directly aid or directly oppose, with nothing in between.
With AC, two voltages can be aiding or opposing one
another to any degree between fully-aiding and fully-
opposing, inclusive. Without the use of vector (complex
number) notation to describe AC quantities, it would be very
difficult to perform mathematical calculations for AC circuit
analysis.
In the next section, we'll learn how to represent vector
quantities in symbolic rather than graphical form. Vector
and triangle diagrams suffice to illustrate the general
concept, but more precise methods of symbolism must be
used if any serious calculations are to be performed on these
quantities.
e REVIEW:
e DC voltages can only either directly aid or directly
oppose each other when connected in series. AC
voltages may aid or oppose to any degree depending on
the phase shift between them.
Polar and rectangular notation
In order to work with these complex numbers without
drawing vectors, we first need some kind of standard
mathematical notation. There are two basic forms of
complex number notation: po/arand rectangular.
Polar form is where a complex number is denoted by the
length (otherwise known as the magnitude, absolute value,
or modulus) and the angle of its vector (usually denoted by
an angle symbol that looks like this: Z). To use the map
analogy, polar notation for the vector from New York City to
San Diego would be something like “2400 miles, southwest.”
Here are two examples of vectors and their polar notations:
(Figure below)
8.06 Z -29,74°
oe Z 330.26")
8.49 745°
Note: the de cole fie aya for designating a vector’s angle
is this symbol: Z
cea 158.2 7.81 2 230.19°
(7.81 Z -129.81°)
Vectors with polar notations.
Standard orientation for vector angles in AC circuit
calculations defines 0° as being to the right (horizontal),
making 90° straight up, 180° to the left, and 270° straight
down. Please note that vectors angled “down” can have
angles represented in polar form as positive numbers in
excess of 180, or negative numbers less than 180. For
example, a vector angled Z 270° (straight down) can also be
said to have an angle of -90°. (Figure below) The above
vector on the right (7.81 Z 230.19°) can also be denoted as
7.81 Z -129.81°.
The vector "compass"
90°
180° 0°
270° (-90°)
The vector compass
Rectangular form, on the other hand, is where a complex
number is denoted by its respective horizontal and vertical
components. In essence, the angled vector is taken to be the
hypotenuse of a right triangle, described by the lengths of
the adjacent and opposite sides. Rather than describing a
vector's length and direction by denoting magnitude and
angle, it is described in terms of “how far left/right” and
“how far up/down.”
These two dimensional figures (horizontal and vertical) are
symbolized by two numerical figures. In order to distinguish
the horizontal and vertical dimensions from each other, the
vertical is prefixed with a lower-case “i” (in pure
mathematics) or “j” (in electronics). These lower-case letters
do not represent a physical variable (such as instantaneous
current, also symbolized by a lower-case letter “i”), but
rather are mathematical operators used to distinguish the
vector's vertical component from its horizontal component.
As a complete complex number, the horizontal and vertical
quantities are written as a sum: (Figure below)
Vl -_ \
4+ )4 ace -44j4
"4 right and 4 up" "4 right and 0 up/down" "4 left and 4 up"
4 -j4 “4+ j0 -4-j4
"4 right and 4 down" "4 left and 0 up/down" "4 left and 4 down"
In “rectangular” form the vector's length and direction are
denoted in terms of its horizontal and vertical span, the first
number representing the the horizontal (“real”) and the
second number (with the “j” prefix) representing the vertical
(“imaginary”) dimensions.
The horizontal component is referred to as the rea/
component, since that dimension is compatible with normal,
scalar (“real”) numbers. The vertical component is referred
to as the imaginary component, since that dimension lies in
a different direction, totally alien to the scale of the real
numbers. (Figure below)
+ "imaginary"
+)
- "real" + "real"
‘J
- "imaginary"
Vector compass showing real and imaginary axes
The “real” axis of the graph corresponds to the familiar
number line we saw earlier: the one with both positive and
negative values on it. The “imaginary” axis of the graph
corresponds to another number line situated at 90° to the
“real” one. Vectors being two-dimensional things, we must
have a two-dimensional “map” upon which to express them,
thus the two number lines perpendicular to each other:
(Figure below)
ILs
"imaginary"
number line 2
Vector compass with real and imaginary (“j”) number lines.
Either method of notation is valid for complex numbers. The
primary reason for having two methods of notation is for
ease of longhand calculation, rectangular form lending itself
to addition and subtraction, and polar form lending itself to
multiplication and division.
Conversion between the two notational forms involves
simple trigonometry. To convert from polar to rectangular,
find the real component by multiplying the polar magnitude
by the cosine of the angle, and the imaginary component by
multiplying the polar magnitude by the sine of the angle.
This may be understood more readily by drawing the
quantities as sides of a right triangle, the hypotenuse of the
triangle representing the vector itself (its length and angle
with respect to the horizontal constituting the polar form),
the horizontal and vertical sides representing the “real” and
“imaginary” rectangular components, respectively: (Figure
below)
length = 5
+]3
angle =
36.87°
+4
Magnitude vector in terms of real (4) and imaginary (j3)
components.
5 Z 36.87° (polar form)
(5)(cos 36.87°)=4 (real component)
(5)(sin 36.87°)=3 (imaginary component)
44 j3 (rectangular form)
To convert from rectangular to polar, find the polar
magnitude through the use of the Pythagorean Theorem
(the polar magnitude is the hypotenuse of a right triangle,
and the real and imaginary components are the adjacent
and opposite sides, respectively), and the angle by taking
the arctangent of the imaginary component divided by the
real component:
4+j3 (rectangular form)
c=Vatbh (pythagorean theorem)
polar magnitude = 4° + 3°
polar magnitude = 5
3
polar angle = arctan re
polar angle = 36.87°
5 2 36.87° = (polar form)
REVIEW:
Polar notation denotes a complex number in terms of its
vector's length and angular direction from the starting
point. Example: fly 45 miles Z 203° (West by
Southwest).
Rectangular notation denotes a complex number in
terms of its horizontal and vertical dimensions. Example:
drive 41 miles West, then turn and drive 18 miles South.
In rectangular notation, the first quantity is the “real”
component (horizontal dimension of vector) and the
second quantity is the “imaginary” component (vertical
dimension of vector). The imaginary component is
preceded by a lower-case “j,” sometimes called the /
operator.
Both polar and rectangular forms of notation for a
complex number can be related graphically in the form
of a right triangle, with the hypotenuse representing the
vector itself (polar form: hypotenuse length =
magnitude; angle with respect to horizontal side =
angle), the horizontal side representing the rectangular
“real” component, and the vertical side representing the
rectangular “imaginary” component.
Complex number arithmetic
Since complex numbers are legitimate mathematical
entities, just like scalar numbers, they can be added,
subtracted, multiplied, divided, squared, inverted, and such,
just like any other kind of number. Some scientific
calculators are programmed to directly perform these
operations on two or more complex numbers, but these
operations can also be done “by hand.” This section will
show you how the basic operations are performed. It is
highly recommended that you equip yourself with a
scientific calculator capable of performing arithmetic
functions easily on complex numbers. It will make your
study of AC circuit much more pleasant than if you're forced
to do all calculations the longer way.
Addition and subtraction with complex numbers in
rectangular form is easy. For addition, simply add up the real
components of the complex numbers to determine the real
component of the sum, and add up the imaginary
components of the complex numbers to determine the
imaginary component of the sum:
2+j5 175 - j34 -36 + j10
+ 4-3 + 80 -jl5 + 20 + j82
6+ j2 255 - j49 -16 + j92
When subtracting complex numbers in rectangular form,
simply subtract the real component of the second complex
number from the real component of the first to arrive at the
real component of the difference, and subtract the
imaginary component of the second complex number from
the imaginary component of the first to arrive the imaginary
component of the difference:
2 +55 175 - j34 -36+j10
= (4-33) - (80 - j15) - (20 +j82)
2+ j8 95 - j19 -56 - j72
For longhand multiplication and division, polar is the
favored notation to work with. When multiplying complex
numbers in polar form, simply multiply the polar magnitudes
of the complex numbers to determine the polar magnitude
of the product, and add the angles of the complex numbers
to determine the angle of the product:
(35 Z 65°10 Z -12°) = 350 2 53°
(124 Z 250°) 11 Z 100°) = 1364 2 -10°
or
1364 7 350°
(3 Z30°)(5 Z -30°)=15 70°
Division of polar-form complex numbers is also easy: simply
divide the polar magnitude of the first complex number by
the polar magnitude of the second complex number to arrive
at the polar magnitude of the quotient, and subtract the
angle of the second complex number from the angle of the
first complex number to arrive at the angle of the quotient:
7 65
Bact so
10 Z -12
24 225
Boe = 11.273 7 150°
11 Z 100
320 06 260
5 Z -30°
To obtain the reciprocal, or “invert” (1/x), a complex number,
simply divide the number (in polar form) into a scalar value
of 1, which is nothing more than a complex number with no
imaginary component (angle = 0):
l 1Z0
$< = ——_—_ = 002857 2-45
35Z65° +35 265°
_ |. 140 ga
107-122 102-12
to EO 85 5-10”
0.0032 Z 10° 0.0032 Z 10°
These are the basic operations you will need to know in
order to manipulate complex numbers in the analysis of AC
circuits. Operations with complex numbers are by no means
limited just to addition, subtraction, multiplication, division,
and inversion, however. Virtually any arithmetic operation
that can be done with scalar numbers can be done with
complex numbers, including powers, roots, solving
simultaneous equations with complex coefficients, and even
trigonometric functions (although this involves a whole new
perspective in trigonometry called hyperbolic functions
which is well beyond the scope of this discussion). Be sure
that you're familiar with the basic arithmetic operations of
addition, subtraction, multiplication, division, and inversion,
and you'll have little trouble with AC circuit analysis.
e REVIEW:
e To add complex numbers in rectangular form, add the
real components and add the imaginary components.
Subtraction is similar.
e To multiply complex numbers in polar form, multiply the
magnitudes and add the angles. To divide, divide the
magnitudes and subtract one angle from the other.
More on AC "polarity"
Complex numbers are useful for AC circuit analysis because
they provide a convenient method of symbolically denoting
phase shift between AC quantities like voltage and current.
However, for most people the equivalence between abstract
vectors and real circuit quantities is not an easy one to
grasp. Earlier in this chapter we saw how AC voltage sources
are given voltage figures in complex form (magnitude and
phase angle), as well as polarity markings. Being that
alternating current has no set “polarity” as direct current
does, these polarity markings and their relationship to phase
angle tends to be confusing. This section is written in the
attempt to clarify some of these issues.
Voltage is an inherently re/ative quantity. When we measure
a voltage, we have a choice in how we connect a voltmeter
or other voltage-measuring instrument to the source of
voltage, as there are two points between which the voltage
exists, and two test leads on the instrument with which to
make connection. In DC circuits, we denote the polarity of
voltage sources and voltage drops explicitly, using “+” and
“-" symbols, and use color-coded meter test leads (red and
black). If a digital voltmeter indicates a negative DC voltage,
we know that its test leads are connected “backward” to the
voltage (red lead connected to the “-” and black lead to the
wr).
Batteries have their polarity designated by way of intrinsic
symbology: the short-line side of a battery is always the
negative (-) side and the long-line side always the positive
(+): (Figure below)
HO
ey
7;
Conventional battery polarity.
Although it would be mathematically correct to represent a
battery's voltage as a negative figure with reversed polarity
markings, it would be decidedly unconventional: (Figure
below)
a
ay =
+]
Decidedly unconventional polarity marking.
Interpreting such notation might be easier if the “+” and “-”
polarity markings were viewed as reference points for
voltmeter test leads, the “+” meaning “red” and the “-”
meaning “black.” A voltmeter connected to the above
battery with red lead to the bottom terminal and black lead
to the top terminal would indeed indicate a negative voltage
(-6 volts). Actually, this form of notation and interpretation is
not as unusual as you might think: it is commonly
encountered in problems of DC network analysis where “+”
and “-” polarity marks are initially drawn according to
educated guess, and later interpreted as correct or
“backward” according to the mathematical sign of the figure
calculated.
In AC circuits, though, we don't deal with “negative”
quantities of voltage. Instead, we describe to what degree
one voltage aids or opposes another by phase: the time-shift
between two waveforms. We never describe an AC voltage
as being negative in sign, because the facility of polar
notation allows for vectors pointing in an opposite direction.
If one AC voltage directly opposes another AC voltage, we
simply say that one is 180° out of phase with the other.
Still, voltage is relative between two points, and we havea
choice in how we might connect a voltage-measuring
instrument between those two points. The mathematical
sign of a DC voltmeter's reading has meaning only in the
context of its test lead connections: which terminal the red
lead is touching, and which terminal the black lead is
touching. Likewise, the phase angle of an AC voltage has
meaning only in the context of knowing which of the two
points is considered the “reference” point. Because of this
fact, “+” and “-” polarity marks are often placed by the
terminals of an AC voltage in schematic diagrams to give the
stated phase angle a frame of reference.
Let's review these principles with some graphical aids. First,
the principle of relating test lead connections to the
mathematical sign of a DC voltmeter indication: (Figure
below)
+60 cal)
rT
6V
Test lead colors provide a frame of reference for interpreting
the sign (+ or -) of the meter's indication.
The mathematical sign of a digital DC voltmeter's display
has meaning only in the context of its test lead connections.
Consider the use of a DC voltmeter in determining whether
or not two DC voltage sources are aiding or opposing each
other, assuming that both sources are unlabeled as to their
polarities. Using the voltmeter to measure across the first
source: (Figure below)
The meter tells us +24 volts
a sal
4
va
cou Source 1 So ar
Total voltage?
(+) Reading indicates black Is (-), red is (+).
This first measurement of +24 across the left-hand voltage
source tells us that the black lead of the meter really is
touching the negative side of voltage source #1, and the red
lead of the meter really is touching the positive. Thus, we
know source #1 is a battery facing in this orientation:
(Figure below)
24 V
—|I
Source 1 Source 2
Total voltage?
24V source Is polarized (-) to (+).
Measuring the other unknown voltage source: (Figure below)
The meter tells us -17 volts
-1 10
4
va
cou Source 1 wed
Total voltage?
(-) Reading indicates black is (+), red is (-).
This second voltmeter reading, however, is a negative (-) 17
volts, which tells us that the black test lead is actually
touching the positive side of voltage source #2, while the
red test lead is actually touching the negative. Thus, we
know that source #2 is a battery facing in the opposite
direction: (Figure below)
24 V 17V
— 4} | |;
Source 1 Source 2
— Total voltage = 7 V =
17V source Is polarized (+) to (-)
It should be obvious to any experienced student of DC
electricity that these two batteries are opposing one
another. By definition, opposing voltages subtract from one
another, so we subtract 17 volts from 24 volts to obtain the
total voltage across the two: 7 volts.
We could, however, draw the two sources as nondescript
boxes, labeled with the exact voltage figures obtained by
the voltmeter, the polarity marks indicating voltmeter test
lead placement: (Figure below)
24V -17V
Source 1 Source 2
Voltmeter readings as read from meters.
According to this diagram, the polarity marks (which
indicate meter test lead placement) indicate the sources
aiding each other. By definition, aiding voltage sources add
with one another to form the total voltage, so we add 24
volts to -17 volts to obtain 7 volts: still the correct answer. If
we let the polarity markings guide our decision to either add
or subtract voltage figures -- whether those polarity
markings represent the true polarity or just the meter test
lead orientation -- and include the mathematical signs of
those voltage figures in our calculations, the result will
always be correct. Again, the polarity markings serve as
frames of reference to place the voltage figures'
mathematical signs in proper context.
The same is true for AC voltages, except that phase angle
substitutes for mathematical sign. In order to relate multiple
AC voltages at different phase angles to each other, we need
polarity markings to provide frames of reference for those
voltages' phase angles. (Figure below)
Take for example the following circuit:
10V 20° 6V 245°
- + - +
14.861 V 2 16.59°
Phase angle substitutes for + sign.
The polarity markings show these two voltage sources aiding
each other, so to determine the total voltage across the
resistor we must add the voltage figures of 10 V Z 0° and 6
V Z 45° together to obtain 14.861 V Z 16.59°. However, it
would be perfectly acceptable to represent the 6 volt source
asS6V Z 225°, with a reversed set of polarity markings, and
still arrive at the same total voltage: (Figure below)
10V 20° 6 V 2 225°
= + + =
14.861 V Z 16.59°
Reversing the voltmeter leads on the 6V source changes the
phase angle by 180°.
6 V Z 45° with negative on the left and positive on the right
is exactly the same as 6 V Z 225° with positive on the left
and negative on the right: the reversal of polarity markings
perfectly complements the addition of 180° to the phase
angle designation: (Figure below)
6V 245°
—(~)—
.../S equivalent to...
6 V Z 225°
cn -
Reversing polarity adds 180°to phase angle
Unlike DC voltage sources, whose symbols intrinsically
define polarity by means of short and long lines, AC voltage
symbols have no intrinsic polarity marking. Therefore, any
polarity marks must be included as additional symbols on
the diagram, and there is no one “correct” way in which to
place them. They must, however, correlate with the given
phase angle to represent the true phase relationship of that
voltage with other voltages in the circuit.
e REVIEW:
e Polarity markings are sometimes given to AC voltages in
circuit schematics in order to provide a frame of
reference for their phase angles.
Some examples with AC circuits
Let's connect three AC voltage sources in series and use
complex numbers to determine additive voltages. All the
rules and laws learned in the study of DC circuits apply to
AC circuits as well (Ohm's Law, Kirchhoff's Laws, network
analysis methods), with the exception of power calculations
(Joule's Law). The only qualification is that all variables must
be expressed in complex form, taking into account phase as
well as magnitude, and all voltages and currents must be of
the same frequency (in order that their phase relationships
remain constant). (Figure below)
22 V 2 -64° (\) E:
‘
I2V 235° (\)) E:
IS V 20° (\) E,
load
KVL allows addition of complex voltages.
The polarity marks for all three voltage sources are oriented
in such a way that their stated voltages should add to make
the total voltage across the load resistor. Notice that
although magnitude and phase angle is given for each AC
voltage source, no frequency value is specified. If this is the
case, it is assumed that all frequencies are equal, thus
meeting our qualifications for applying DC rules to an AC
circuit (all figures given in complex form, all of the same
frequency). The setup of our equation to find total voltage
appears as such:
Frotat = B, + E, + E;
Etat = (22 V Z -64°) + (12 V Z 35°) + (15 V 20°)
Graphically, the vectors add up as shown in Figure below.
Zan
Graphic addition of vector voltages.
The sum of these vectors will be a resultant vector
Originating at the starting point for the 22 volt vector (dot at
upper-left of diagram) and terminating at the ending point
for the 15 volt vector (arrow tip at the middle-right of the
diagram): (Figure below)
resultant vector
22 Z -64°
Resultant is equivalent to the vector sum of the three
original voltages.
In order to determine what the resultant vector's magnitude
and angle are without resorting to graphic images, we can
convert each one of these polar-form complex numbers into
rectangular form and add. Remember, we're adding these
figures together because the polarity marks for the three
voltage sources are oriented in an additive manner:
ISV Z0°=15+j0V
12
<
Z 35° = 9.8298 + j6.8829 V
22 V Z -64° = 9.6442 - j19.7735 V
15 +jo0 Vv
9.8298 + j6.8829V
+ 9.6442 -j19.7735 V
34.4740 - {12.8906 V
In polar form, this equates to 36.8052 volts Z -20.5018°.
What this means in real terms is that the voltage measured
across these three voltage sources will be 36.8052 volts,
lagging the 15 volt (0° phase reference) by 20.5018°. A
voltmeter connected across these points in a real circuit
would only indicate the polar magnitude of the voltage
(36.8052 volts), not the angle. An oscilloscope could be
used to display two voltage waveforms and thus provide a
phase shift measurement, but not a voltmeter. The same
principle holds true for AC ammeters: they indicate the polar
magnitude of the current, not the phase angle.
This is extremely important in relating calculated figures of
voltage and current to real circuits. Although rectangular
notation is convenient for addition and subtraction, and was
indeed the final step in our sample problem here, it is not
very applicable to practical measurements. Rectangular
figures must be converted to polar figures (specifically polar
magnitude) before they can be related to actual circuit
measurements.
We can use SPICE to verify the accuracy of our results. In
this test circuit, the 10 kQ resistor value is quite arbitrary.
It's there so that SPICE does not declare an open-circuit error
and abort analysis. Also, the choice of frequencies for the
simulation (60 Hz) is quite arbitrary, because resistors
respond uniformly for all frequencies of AC voltage and
current. There are other components (notably capacitors and
inductors) which do not respond uniformly to different
frequencies, but that is another subject! (Figure below)
3
Spice circuit schematic.
ac voltage addition
v1 10 ac 15 O sin
v2 2 1 ac 12 35 sin
v3 3 2 ac 22 -64 sin
rl 3 0 10k
.ac lin 1 60 60 I'm using a frequency of 60 Hz
.print ac v(3,0) vp(3,0) as a default value
.end
freq v(3) vp (3)
6.000E+01 3.681E+01 -2.050E+01
Sure enough, we get a total voltage of 36.81 volts Z -20.5°
(with reference to the 15 volt source, whose phase angle was
arbitrarily stated at zero degrees so as to be the “reference”
waveform).
At first glance, this is counter-intuitive. How is it possible to
obtain a total voltage of just over 36 volts with 15 volt, 12
volt, and 22 volt supplies connected in series? With DC, this
would be impossible, as voltage figures will either directly
add or subtract, depending on polarity. But with AC, our
“polarity” (phase shift) can vary anywhere in between full-
aiding and full-opposing, and this allows for such
paradoxical summing.
What if we took the same circuit and reversed one of the
supply's connections? Its contribution to the total voltage
would then be the opposite of what it was before: (Figure
below)
22V Z-64°
Polanty reversed on 7
source E,!
12V 235° (\V) Es
+
load
Polarity of E> (12V) is reversed.
Note how the 12 volt supply's phase angle is still referred to
as 35°, even though the leads have been reversed.
Remember that the phase angle of any voltage drop is
stated in reference to its noted polarity. Even though the
angle is still written as 35°, the vector will be drawn 180°
opposite of what it was before: (Figure below)
L2 235° (reversed) = 12 2 215°
or
-12 235°
Sze
Direction of E> is reversed.
The resultant (Sum) vector should begin at the upper-left
point (origin of the 22 volt vector) and terminate at the right
arrow tip of the 15 volt vector: (Figure below)
22 2 -64°
resultant vector
12 235° (reversed) = 12 7 215°
or
-12 235°
15 20°
Resultant is vector sum of voltage sources.
The connection reversal on the 12 volt supply can be
represented in two different ways in polar form: by an
addition of 180° to its vector angle (making it 12 volts Z
215°), or a reversal of sign on the magnitude (making it -12
volts Z 35°). Either way, conversion to rectangular form
yields the same result:
12V 235° (reversed) = 12V 2215° = -98298 - j6.8829 V
or
-12V 235° = -98298 - j6.8829 V
The resulting addition of voltages in rectangular form, then:
15 +j0 V
-9.8298 - [6.8829 V
+ 9.6442 -j19.7735 V
14.8143 - j26.6564 V
In polar form, this equates to 30.4964 V Z -60.9368°. Once
again, we will use SPICE to verify the results of our
calculations:
ac voltage addition
v1 10 ac 15 © sin
v2 12 ac 12 35 sin Note the reversal of node numbers 2
and 1
v3 3 2 ac 22 -64 sin to simulate the swapping of
connections
rl 3 0 10k
.ac Lin 1 60 60
.print ac v(3,0) vp(3,0)
.end
freq v(3) vp (3)
6.000E+01 3.050E+01 -6.094E+01
e REVIEW:
e All the laws and rules of DC circuits apply to AC circuits,
with the exception of power calculations (Joule's Law), so
long as all values are expressed and manipulated in
complex form, and all voltages and currents are at the
same frequency.
e When reversing the direction of a vector (equivalent to
reversing the polarity of an AC voltage source in relation
to other voltage sources), it can be expressed in either
of two different ways: adding 180° to the angle, or
reversing the sign of the magnitude.
e Meter measurements in an AC circuit correspond to the
polar magnitudes of calculated values. Rectangular
expressions of complex quantities in an AC circuit have
no direct, empirical equivalent, although they are
convenient for performing addition and subtraction, as
Kirchhoff's Voltage and Current Laws require.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—| | +4] l—
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 3
REACTANCE AND
IMPEDANCE -- INDUCTIVE
AC resistor circuits
AC inductor circuits
Series resistor-inductor circuits
Parallel resistor-inductor circuits
Inductor quirks
More on the “skin effect”
Contributors
AC resistor circuits
FE, = Ep l=,
Pure resistive AC circuit: resistor voltage and current are in
phase.
If we were to plot the current and voltage for a very simple
AC circuit consisting of a source and a resistor (Figure
above), it would look something like this: (Figure below)
Time —->
Voltage and current “in phase” for resistive circuit.
Because the resistor simply and directly resists the flow of
electrons at all periods of time, the waveform for the voltage
drop across the resistor is exactly in phase with the
waveform for the current through it. We can look at any
point in time along the horizontal axis of the plot and
compare those values of current and voltage with each other
(any “snapshot” look at the values of a wave are referred to
as instantaneous values, meaning the values at that instant
in time). When the instantaneous value for current is zero,
the instantaneous voltage across the resistor is also zero.
Likewise, at the moment in time where the current through
the resistor is at its positive peak, the voltage across the
resistor is also at its positive peak, and so on. At any given
point in time along the waves, Ohm's Law holds true for the
instantaneous values of voltage and current.
We can also calculate the power dissipated by this resistor,
and plot those values on the same graph: (Figure below)
Instantaneous AC power in a pure resistive circuit is always
positive.
Note that the power is never a negative value. When the
Current is positive (above the line), the voltage is also
positive, resulting in a power (p=ie) of a positive value.
Conversely, when the current is negative (below the line),
the voltage is also negative, which results in a positive value
for power (a negative number multiplied by a negative
number equals a positive number). This consistent “polarity”
of power tells us that the resistor is always dissipating
power, taking it from the source and releasing it in the form
of heat energy. Whether the current is positive or negative, a
resistor still dissipates energy.
AC inductor circuits
Inductors do not behave the same as resistors. Whereas
resistors simply oppose the flow of electrons through them
(by dropping a voltage directly proportional to the current),
inductors oppose changes in current through them, by
dropping a voltage directly proportional to the rate of
change of current. In accordance with Lenz's Law, this
induced voltage is always of such a polarity as to try to
maintain current at its present value. That is, if current is
increasing in magnitude, the induced voltage will “push
against” the electron flow; if current is decreasing, the
polarity will reverse and “push with” the electron flow to
oppose the decrease. This opposition to current change is
called reactance, rather than resistance.
Expressed mathematically, the relationship between the
voltage dropped across the inductor and rate of current
change through the inductor is as such:
— _ di
oa
The expression di/dt is one from calculus, meaning the rate
of change of instantaneous current (i) over time, in amps per
second. The inductance (L) is in Henrys, and the
instantaneous voltage (e), of course, is in volts. Sometimes
you will find the rate of instantaneous voltage expressed as
“vy” instead of “e” (v = Ldi/dt), but it means the exact same
thing. To show what happens with alternating current, let's
analyze a simple inductor circuit: (Figure below)
FE, = BE, l=],
Pure inductive circuit: Inductor current lags inductor voltage
by 90°.
If we were to plot the current and voltage for this very
simple circuit, it would look something like this: (Figure
below)
Pure inductive circuit, waveforms.
Remember, the voltage dropped across an inductor is a
reaction against the change in current through it. Therefore,
the instantaneous voltage is zero whenever the
instantaneous current is at a peak (zero change, or level
slope, on the current sine wave), and the instantaneous
voltage is at a peak wherever the instantaneous current is at
maximum change (the points of steepest slope on the
Current wave, where it crosses the zero line). This results in a
voltage wave that is 90° out of phase with the current wave.
Looking at the graph, the voltage wave seems to have a
“head start” on the current wave; the voltage “leads” the
current, and the current “lags” behind the voltage. (Figure
below)
current slope = 0 current slope = max. (+)
voltage = 0 voltage = max. (+)
: Time —~
“current slope = 0
t voltage = 0
current slope = max. (-)
voltage = max. (-)
Current lags voltage by 90° in a pure inductive circuit.
Things get even more interesting when we plot the power for
this circuit: (Figure below)
In a pure inductive circuit, instantaneous power may be
positive or negative
Because instantaneous power is the product of the
instantaneous voltage and the instantaneous current (p=ie),
the power equals zero whenever the instantaneous current
or voltage is zero. Whenever the instantaneous current and
voltage are both positive (above the line), the power is
positive. As with the resistor example, the power is also
positive when the instantaneous current and voltage are
both negative (below the line). However, because the
current and voltage waves are 90° out of phase, there are
times when one is positive while the other is negative,
resulting in equally frequent occurrences of negative
instantaneous power.
But what does negative power mean? It means that the
inductor is releasing power back to the circuit, while a
positive power means that it is absorbing power from the
circuit. Since the positive and negative power cycles are
equal in magnitude and duration over time, the inductor
releases just as much power back to the circuit as it absorbs
over the span of a complete cycle. What this means ina
practical sense is that the reactance of an inductor
dissipates a net energy of zero, quite unlike the resistance of
a resistor, which dissipates energy in the form of heat. Mind
you, this is for perfect inductors only, which have no wire
resistance.
An inductor's opposition to change in current translates to
an opposition to alternating current in general, which is by
definition always changing in instantaneous magnitude and
direction. This opposition to alternating current is similar to
resistance, but different in that it always results in a phase
shift between current and voltage, and it dissipates zero
power. Because of the differences, it has a different name:
reactance. Reactance to AC is expressed in ohms, just like
resistance is, except that its mathematical symbol is X
instead of R. To be specific, reactance associated with an
inductor is usually symbolized by the capital letter X with a
letter Las a subscript, like this: X,.
Since inductors drop voltage in proportion to the rate of
current change, they will drop more voltage for faster-
changing currents, and less voltage for slower-changing
currents. What this means is that reactance in ohms for any
inductor is directly proportional to the frequency of the
alternating current. The exact formula for determining
reactance is as follows:
X, = 2nfL
If we expose a 10 MH inductor to frequencies of 60, 120, and
2500 Hz, it will manifest the reactances in Table Figure
below.
Reactance of a 10 MH inductor:
Frequency (Hertz)|/Reactance (Ohms)
3.7699
20 7.5398
157.0796
In the reactance equation, the term “2nf” (everything on the
right-hand side except the L) has a special meaning unto
itself. It is the number of radians per second that the
alternating current is “rotating” at, if you imagine one cycle
of AC to represent a full circle's rotation. A radian is a unit of
angular measurement: there are 2m radians in one full circle,
just as there are 360° in a full circle. If the alternator
producing the AC is a double-pole unit, it will produce one
cycle for every full turn of shaft rotation, which is every 2m
radians, or 360°. If this constant of 2m is multiplied by
frequency in Hertz (cycles per second), the result will be a
figure in radians per second, known as the angular velocity
of the AC system.
Angular velocity may be represented by the expression 2nf,
or it may be represented by its own symbol, the lower-case
Greek letter Omega, which appears similar to our Roman
lower-case “w”: W. Thus, the reactance formula X, = 2nfL
could also be written as X; = WL.
It must be understood that this “angular velocity” is an
expression of how rapidly the AC waveforms are cycling, a
full cycle being equal to 2m radians. It is not necessarily
representative of the actual shaft speed of the alternator
producing the AC. If the alternator has more than two poles,
the angular velocity will be a multiple of the shaft speed. For
this reason, W is sometimes expressed in units of e/ectrical
radians per second rather than (plain) radians per second,
so as to distinguish it from mechanical motion.
Any way we express the angular velocity of the system, it is
apparent that it is directly proportional to reactance in an
inductor. As the frequency (or alternator shaft speed) is
increased in an AC system, an inductor will offer greater
opposition to the passage of current, and vice versa.
Alternating current in a simple inductive circuit is equal to
the voltage (in volts) divided by the inductive reactance (in
ohms), just as either alternating or direct current in a simple
resistive circuit is equal to the voltage (in volts) divided by
the resistance (in ohms). An example circuit is shown here:
(Figure below)
10 mH
Inductive reactance
(inductive reactance of 10 MH inductor at 60 Hz)
X, = 3.7699 Q
1- =
x
— 10V
3.7699 Q
1= 2.6526 A
However, we need to keep in mind that voltage and current
are not in phase here. As was shown earlier, the voltage has
a phase shift of +90° with respect to the current. (Figure
below) If we represent these phase angles of voltage and
current mathematically in the form of complex numbers, we
find that an inductor's opposition to current has a phase
angle, too:
Opposition __Voltage_
Current
Opposition __ 10V 290"
2.6526 A Z0°
Opposition =3.7699 Q 2 90°
or
0 + 53.7699 Q
For an inductor:
90° 90°
E
- 0
l Opposition
(X,)
Current lags voltage by 90° in an inductor.
Mathematically, we say that the phase angle of an inductor's
opposition to current is 90°, meaning that an inductor's
opposition to current is a positive imaginary quantity. This
phase angle of reactive opposition to current becomes
critically important in circuit analysis, especially for complex
AC circuits where reactance and resistance interact. It will
prove beneficial to represent any component's opposition to
current in terms of complex numbers rather than scalar
quantities of resistance and reactance.
REVIEW:
Inductive reactance is the opposition that an inductor
offers to alternating current due to its phase-shifted
storage and release of energy in its magnetic field.
Reactance is symbolized by the capital letter “X” and is
measured in ohms just like resistance (R).
Inductive reactance can be calculated using this
formula: X, = 2nfL
The angular velocity of an AC circuit is another way of
expressing its frequency, in units of electrical radians
per second instead of cycles per second. It is symbolized
by the lower-case Greek letter “omega,” or W.
Inductive reactance increases with increasing frequency.
In other words, the higher the frequency, the more it
opposes the AC flow of electrons.
Series resistor-inductor circuits
In the previous section, we explored what would happen in
simple resistor-only and inductor-only AC circuits. Now we
will mix the two components together in series form and
investigate the effects.
Take this circuit as an example to work with: (Figure below)
E,
E; = E,t E,
Lei L
Series resistor inductor circuit: Current lags applied voltage
by 0° to 90°.
The resistor will offer 5 Q of resistance to AC current
regardless of frequency, while the inductor will offer 3.7699
Q of reactance to AC current at 60 Hz. Because the resistor's
resistance is a real number (5 O Z 0°, or 5 + j0 QO), and the
inductor's reactance is an imaginary number (3.7699 O Z
90°, or 0 + j3.7699 Q), the combined effect of the two
components will be an opposition to current equal to the
complex sum of the two numbers. This combined opposition
will be a vector combination of resistance and reactance. In
order to express this opposition succinctly, we need a more
comprehensive term for opposition to current than either
resistance or reactance alone. This term is called /mpedance,
its symbol is Z, and it is also expressed in the unit of ohms,
just like resistance and reactance. In the above example, the
total circuit impedance is:
Zrotal = (5 Q resistance) + (3.7699 Q inductive reactance)
Zrotat = Q(R) + 3.7699 Q(X)
yd = (5 52 Z.0°) + (3.7699 Q 790°)
total
or
(5 + jO Q) + (0 + 53.7699 Q)
Ztail = 2 + j3-7699 Q or 6.262 Q 2 37.016°
tota
Impedance is related to voltage and current just as you
might expect, in a manner similar to resistance in Ohm's
Law:
Ohm's Law for AC circuits:
E=I1Z t-te 7-
Z I
All quantities expressed in
complex, not scalar, form
In fact, this is a far more comprehensive form of Ohm's Law
than what was taught in DC electronics (E=IR), just as
impedance is a far more comprehensive expression of
opposition to the flow of electrons than resistance is. Any
resistance and any reactance, separately or in combination
(series/parallel), can be and should be represented as a
single impedance in an AC circuit.
To calculate current in the above circuit, we first need to
give a phase angle reference for the voltage source, which is
generally assumed to be zero. (The phase angles of resistive
and inductive impedance are a/ways 0° and +90°,
respectively, regardless of the given phase angles for
voltage or current).
1= —
Z
_ 10V Z0°
6.262 2 237.016
l= 1.597 A Z -37.016°
As with the purely inductive circuit, the current wave lags
behind the voltage wave (of the source), although this time
the lag is not as great: only 37.016° as opposed to a full 90°
as was the case in the purely inductive circuit. (Figure
below)
phase shift =
37.016°
Current lags voltage in a series L-R circuit.
For the resistor and the inductor, the phase relationships
between voltage and current haven't changed. Voltage
across the resistor is in phase (0° shift) with the current
through it; and the voltage across the inductor is +90° out
of phase with the current going through it. We can verify this
mathematically:
Bey
Ep = (1.597 A Z -37.016°)(5 Q Z 0°)
E, = 7.9847 V Z -37.016°
Notice that the phase angle of E, is equal to
the phase angle of the current.
The voltage across the resistor has the exact same phase
angle as the current through it, telling us that E and | are in
phase (for the resistor only).
E=1Z
E, =1,Z,
E, = (1.597 A Z -37.016°)(3.7699 Q 7 90°)
E, = 6.0203 V 2 52.984°
Notice that the phase angle of E, is exactly
90° more than the phase angle of the current.
The voltage across the inductor has a phase angle of
52.984°, while the current through the inductor has a phase
angle of -37.016°, a difference of exactly 90° between the
two. This tells us that E and | are still 90° out of phase (for
the inductor only).
We can also mathematically prove that these complex
values add together to make the total voltage, just as
Kirchhoff's Voltage Law would predict:
Eta - ER + E,
= (7.9847 V Z -37.016°) + (6.0203 V Z 52.984°)
otal
E otal =10VZ 0°
Let's check the validity of our calculations with SPICE:
(Figure below)
10 V
60 Hz
10 mH
0 0
Spice circuit: R-L.
ac r-l circuit
v1 10 ac 10 sin
rl1 125
11 2 0 10m
.ac Lin 1 60 60
.print ac v(1,2) v(2,0) i(v1)
.print ac vp(1,2) vp(2,0) ip(v1)
.end
freq v(1,2) v(2)
6.000E+01 7.985E+00 6.020E+00
freq vp(1,2) vp(2)
6.000E+01 -3.702E+01 5.298E+01
i(vl)
1.597E+00
ip(v1)
1.430E+02
Interpreted SPICE results
E, = 7.985 V Z -37.02°
E, = 6.020 V 2 52.98°
1= 1.597 A Z 143.0°
Note that just as with DC circuits, SPICE outputs current
figures as though they were negative (180° out of phase)
with the supply voltage. Instead of a phase angle of
-37.016°, we get a current phase angle of 143° (-37° +
180°). This is merely an idiosyncrasy of SPICE and does not
represent anything significant in the circuit simulation itself.
Note how both the resistor and inductor voltage phase
readings match our calculations (-37.02° and 52.98°,
respectively), just as we expected them to.
With all these figures to keep track of for even such a simple
circuit as this, it would be beneficial for us to use the “table”
method. Applying a table to this simple series resistor-
inductor circuit would proceed as such. First, draw up a table
for E/I/Z figures and insert all component values in these
terms (in other words, don't insert actual resistance or
inductance values in Ohms and Henrys, respectively, into
the table; rather, convert them into complex figures of
impedance and write those in):
Total
R Eb
10+ j0 Volt
10. 70° “_
5 +j0 0 + 53.7699
520 3.7699 7 90°
Ohms
Although it isn't necessary, | find it helpful to write both the
rectangular and polar forms of each quantity in the table. If
you are using a calculator that has the ability to perform
complex arithmetic without the need for conversion between
rectangular and polar forms, then this extra documentation
is completely unnecessary. However, if you are forced to
perform complex arithmetic “longhand” (addition and
subtraction in rectangular form, and multiplication and
division in polar form), writing each quantity in both forms
will be useful indeed.
Now that our “given” figures are inserted into their
respective locations in the table, we can proceed just as with
DC: determine the total impedance from the individual
impedances. Since this is a series circuit, we know that
opposition to electron flow (resistance or impedance) adds
to form the total opposition:
R L Total
: 10 + j0 a
10 70° aha
oO
j 13.7699 5 +73.
7 5 4+] 0 + {3.765 e 5 + j3.7699 Shine
: 3.7699 4 90 6.262 4 37.016
Rule of series
circuits
Zrotal = Zp a Zi
Now that we know total voltage and total impedance, we
can apply Ohm's Law (I=E/Z) to determine total current:
1.2751 - j0.9614
1.597 4-37.01
5+ j0 0 + j3.7699 5 + j3.7699 Ghia
520 3.7699 2 90° 6.262 4 37.016°
Ohm's
Law
[==
Z
Just as with DC, the total current in a series AC circuit is
shared equally by all components. This is still true because
in a series circuit there is only a single path for electrons to
flow, therefore the rate of their flow must uniform
throughout. Consequently, we can transfer the figures for
current into the columns for the resistor and inductor alike:
R L Total
1.2751 - j0.9614 1.2751 - j0.9614 L.2751 - j0.96L4
1.597 4-37.01 1.597 4-37.01 L.597 Z-37.016°
7 5 + 53.7699
6.262 2 37.016°
Rule of series
circuits:
Lott = tk =
Now all that's left to figure is the voltage drop across the
resistor and inductor, respectively. This is done through the
use of Ohm's Law (E=IZ), applied vertically in each column
of the table:
6.3756 - j4.8071 3.6244 + j4.8071
E
7.9847 2 -37.016° | 6.0203 7 52.984° Volts
| | 1.2751 - jo.9614 1.2751 - j0.9614 L.2751-j0.9614 | ang
L.597 2 -37.016° L.597 2 -37.016° L597 2 -37.016°
i3 99 i3 99
Z 0 + {3.7695 5 + j3.7699 Ohms
3.7699 4 90° 6.262 2 37.016
E=IZ E=IZ
And with that, our table is complete. The exact same rules
we applied in the analysis of DC circuits apply to AC circuits
as well, with the caveat that all quantities must be
represented and calculated in complex rather than scalar
form. So long as phase shift is properly represented in our
calculations, there is no fundamental difference in how we
approach basic AC circuit analysis versus DC.
Now is a good time to review the relationship between these
calculated figures and readings given by actual instrument
measurements of voltage and current. The figures here that
directly relate to real-life measurements are those in polar
notation, not rectangular! In other words, if you were to
connect a voltmeter across the resistor in this circuit, it
would indicate 7.9847 volts, not 6.3756 (real rectangular)
or 4.8071 (imaginary rectangular) volts. To describe this in
graphical terms, measurement instruments simply tell you
how long the vector is for that particular quantity (voltage or
Current).
Rectangular notation, while convenient for arithmetical
addition and subtraction, is a more abstract form of notation
than polar in relation to real-world measurements. As |
stated before, | will indicate both polar and rectangular
forms of each quantity in my AC circuit tables simply for
convenience of mathematical calculation. This is not
absolutely necessary, but may be helpful for those following
along without the benefit of an advanced calculator. If we
were to restrict ourselves to the use of only one form of
notation, the best choice would be polar, because it is the
only one that can be directly correlated to real
measurements.
Impedance (Z) of a series R-L circuit may be calculated,
given the resistance (R) and the inductive reactance (X,).
Since E=IR, E=IX,, and E=IZ, resistance, reactance, and
impedance are proportional to voltage, respectively. Thus,
the voltage phasor diagram can be replaced by a similar
impedance diagram. (Figure below)
E
i . Z,
fat be
R
R
Voltage Impedance
Series: R-L circuit Impedance phasor diagram.
Example:
Given: A 40 Q resistor in series with a 79.58 millihenry
inductor. Find the impedance at 60 hertz.
X, = 2nfLl
X_ = 2m 60: 79.58x10-3
X. = 30 Q
R + jX,
40 + j30
|Z| = sqrt(40* + 307) = 500
arctangent(30/40) = 36.87°
40 + j30 = 50436.87°
REVIEW:
Impedance is the total measure of opposition to electric
current and is the complex (vector) sum of (“real”)
resistance and (“imaginary”) reactance. It is symbolized
by the letter “Z” and measured in ohms, just like
resistance (R) and reactance (X).
Impedances (Z) are managed just like resistances (R) in
series circuit analysis: series impedances add to form
the total impedance. Just be sure to perform all
calculations in complex (not scalar) form! Z7 44) = Z1 +
Li era Ze
A purely resistive impedance will always have a phase
angle of exactly 0° (Zp = RQ Z 0°).
A purely inductive impedance will always have a phase
angle of exactly +90° (Z, = X, O Z 90°).
Ohm's Law for AC circuits: E=1Z;1 = E/Z; Z = E/l
When resistors and inductors are mixed together in
circuits, the total impedance will have a phase angle
somewhere between 0° and +90°. The circuit current
will have a phase angle somewhere between 0° and
-90°.
Series AC circuits exhibit the same fundamental
properties as series DC circuits: current is uniform
throughout the circuit, voltage drops add to form the
total voltage, and impedances add to form the total
impedance.
Parallel resistor-inductor circuits
Let's take the same components for our series example
circuit and connect them in parallel: (Figure below)
l =1,+1,
E = 5, = 5,
Parallel R-L circuit.
Because the power source has the same frequency as the
series example circuit, and the resistor and inductor both
have the same values of resistance and inductance,
respectively, they must also have the same values of
impedance. So, we can begin our analysis table with the
same “given” values:
The only difference in our analysis technique this time is
that we will apply the rules of parallel circuits instead of the
rules for series circuits. The approach is fundamentally the
same as for DC. We know that voltage is shared uniformly by
all components in a parallel circuit, so we can transfer the
figure of total voltage (10 volts Z 0°) to all components
columns:
R L Total
: 10 + j0 10 +j0 LO + j0 om
10 20° 10 20° 10 20° aia
; 5 +0 0 + 53.7699 aoe
520 3.7699 7 90
Rule of parallel
circuits:
Broint = Ep = Ey
Now we can apply Ohm's Law (I=E/Z) vertically to two
columns of the table, calculating current through the resistor
and current through the inductor:
R L Total
E LO + jO 10+ jO LO + j0 elk
10 20° lo 20° 10 20° a
2+ j0 0 - j2.6526
| Am
Z 5 +0 0+ j3 pel Ghs
520 3.7699 4 90
Ohm's Ohm's
Law Law
_E -E—
Z Z
Just as with DC circuits, branch currents in a parallel AC
circuit add to form the total current (Kirchhoff's Current Law
still holds true for AC as it did for DC):
0 - j2.6526 6526
2.6526 Z -90° 1 Z -52.984°
Rule of parallel
circuits:
Liotal = In + L
Finally, total impedance can be calculated by using Ohm's
Law (Z=E/I) vertically in the “Total” column. Incidentally,
parallel impedance can also be calculated by using a
reciprocal formula identical to that used in calculating
parallel resistances.
Zoarallel = -—_—_-
ae Tt 1 N
The only problem with using this formula is that it typically
involves a lot of calculator keystrokes to carry out. And if
you're determined to run through a formula like this
“longhand,” be prepared for a very large amount of work!
But, just as with DC circuits, we often have multiple options
in calculating the quantities in our analysis tables, and this
example is no different. No matter which way you calculate
total impedance (Ohm's Law or the reciprocal formula), you
will arrive at the same figure:
2 - j2.6526
3.322 Z -52.984°
0 + j3.7699 1.8122 + j2.4035
3.7699 2 90° 3.0102 4 52.984°
Ohm's Rule of parallel
Law OF circuits:
E L
Z=— Zrcta1 = ———
i Sige
Zp Zi
e REVIEW:
e Impedances (Z) are managed just like resistances (R) in
parallel circuit analysis: parallel impedances diminish to
form the total impedance, using the reciprocal formula.
Just be sure to perform all calculations in complex (not
scalar) form! Zyo¢a; = 1/(1/Z] + 1/Z> +... 1/Z,)
e Ohm's Law for AC circuits: E = 1Z; 1 = E/Z; Z = E/l
e When resistors and inductors are mixed together in
parallel circuits (just as in series circuits), the total
impedance will have a phase angle somewhere between
0° and +90°. The circuit current will have a phase angle
somewhere between 0° and -90°.
Parallel AC circuits exhibit the same fundamental
properties as parallel DC circuits: voltage is uniform
throughout the circuit, branch currents add to form the
total current, and impedances diminish (through the
reciprocal formula) to form the total impedance.
Inductor quirks
In an ideal case, an inductor acts as a purely reactive device.
That is, its opposition to AC current is strictly based on
inductive reaction to changes in current, and not electron
friction as is the case with resistive components. However,
inductors are not quite so pure in their reactive behavior. To
begin with, they're made of wire, and we know that all wire
possesses some measurable amount of resistance (unless its
superconducting wire). This built-in resistance acts as
though it were connected in series with the perfect
inductance of the coil, like this: (Figure below)
Equivalent circuit for a real inductor
Wire resistance
R
Ideal inductor
|
Inductor Equivalent circuit of a real inductor.
Consequently, the impedance of any real inductor will
always be a complex combination of resistance and
inductive reactance.
Compounding this problem is something called the skin
effect, which is AC's tendency to flow through the outer
areas of a conductor's cross-section rather than through the
middle. When electrons flow in a single direction (DC), they
use the entire cross-sectional area of the conductor to move.
Electrons switching directions of flow, on the other hand,
tend to avoid travel through the very middle of a conductor,
limiting the effective cross-sectional area available. The skin
effect becomes more pronounced as frequency increases.
Also, the alternating magnetic field of an inductor energized
with AC may radiate off into space as part of an
electromagnetic wave, especially if the AC is of high
frequency. This radiated energy does not return to the
inductor, and so it manifests itself as resistance (power
dissipation) in the circuit.
Added to the resistive losses of wire and radiation, there are
other effects at work in iron-core inductors which manifest
themselves as additional resistance between the leads.
When an inductor is energized with AC, the alternating
magnetic fields produced tend to induce circulating currents
within the iron core known as eddy currents. These electric
currents in the iron core have to overcome the electrical
resistance offered by the iron, which is not as good a
conductor as copper. Eddy current losses are primarily
counteracted by dividing the iron core up into many thin
sheets (laminations), each one separated from the other by
a thin layer of electrically insulating varnish. With the cross-
section of the core divided up into many electrically isolated
sections, current cannot circulate within that cross-sectional
area and there will be no (or very little) resistive losses from
that effect.
As you might have expected, eddy current losses in metallic
inductor cores manifest themselves in the form of heat. The
effect is more pronounced at higher frequencies, and can be
so extreme that it is sometimes exploited in manufacturing
processes to heat metal objects! In fact, this process of
“inductive heating” is often used in high-purity metal
foundry operations, where metallic elements and alloys must
be heated in a vacuum environment to avoid contamination
by air, and thus where standard combustion heating
technology would be useless. It is a “non-contact”
technology, the heated substance not having to touch the
coil(s) producing the magnetic field.
In high-frequency service, eddy currents can even develop
within the cross-section of the wire itself, contributing to
additional resistive effects. To counteract this tendency,
special wire made of very fine, individually insulated strands
called Litz wire (short for Litzendraht) can be used. The
insulation separating strands from each other prevent eddy
currents from circulating through the whole wire's cross-
sectional area.
Additionally, any magnetic hysteresis that needs to be
overcome with every reversal of the inductor's magnetic
field constitutes an expenditure of energy that manifests
itself as resistance in the circuit. Some core materials (such
as ferrite) are particularly notorious for their hysteretic
effect. Counteracting this effect is best done by means of
proper core material selection and limits on the peak
magnetic field intensity generated with each cycle.
Altogether, the stray resistive properties of a real inductor
(wire resistance, radiation losses, eddy currents, and
hysteresis losses) are expressed under the single term of
“effective resistance:” (Figure below)
Equivalent circuit for a real inductor
"Effective" resistance
R
Ideal inductor
L
Equivalent circuit of a real inductor with skin-effect,
radiation, eddy current, and hysteresis losses.
It is worthy to note that the skin effect and radiation losses
apply just as well to straight lengths of wire in an AC circuit
as they do a coiled wire. Usually their combined effect is too
small to notice, but at radio frequencies they can be quite
large. A radio transmitter antenna, for example, is designed
with the express purpose of dissipating the greatest amount
of energy in the form of electromagnetic radiation.
Effective resistance in an inductor can be a serious
consideration for the AC circuit designer. To help quantify
the relative amount of effective resistance in an inductor,
another value exists called the Q factor, or “quality factor”
which is calculated as follows:
xX
aes
The symbol “Q” has nothing to do with electric charge
(coulombs), which tends to be confusing. For some reason,
the Powers That Be decided to use the same letter of the
alphabet to denote a totally different quantity.
The higher the value for “Q,” the “purer” the inductor is.
Because its so easy to add additional resistance if needed, a
high-Q inductor is better than a low-Q inductor for design
purposes. An ideal inductor would have a Q of infinity, with
zero effective resistance.
Because inductive reactance (X) varies with frequency, so
will Q. However, since the resistive effects of inductors (wire
skin effect, radiation losses, eddy current, and hysteresis)
also vary with frequency, Q does not vary proportionally
with reactance. In order for a Q value to have precise
meaning, it must be specified at a particular test frequency.
Stray resistance isn't the only inductor quirk we need to be
aware of. Due to the fact that the multiple turns of wire
comprising inductors are separated from each other by an
insulating gap (air, varnish, or some other kind of electrical
insulation), we have the potential for capacitance to develop
between turns. AC capacitance will be explored in the next
chapter, but it suffices to say at this point that it behaves
very differently from AC inductance, and therefore further
“taints” the reactive purity of real inductors.
More on the “skin effect”
As previously mentioned, the skin effect is where alternating
current tends to avoid travel through the center of a solid
conductor, limiting itself to conduction near the surface.
This effectively limits the cross-sectional conductor area
available to carry alternating electron flow, increasing the
resistance of that conductor above what it would normally
be for direct current: (Figure below)
Cross-sectional area of a round
conductor available for conducting
DC current
"DC resistance"
Cross-sectional area of the same
conductor available for conducting
low-frequency AC
"AC resistance"
Cross-sectional area of the same
conductor available for conducting
high-frequency AC
"AC resistance”
Skin effect: skin depth decreases with increasing frequency.
The electrical resistance of the conductor with all its cross-
sectional area in use is known as the “DC resistance,” the
“AC resistance” of the same conductor referring to a higher
figure resulting from the skin effect. As you can see, at high
frequencies the AC current avoids travel through most of the
conductor's cross-sectional area. For the purpose of
conducting current, the wire might as well be hollow!
In some radio applications (antennas, most notably) this
effect is exploited. Since radio-frequency (“RF”) AC currents
wouldn't travel through the middle of a conductor anyway,
why not just use hollow metal rods instead of solid metal
wires and save both weight and cost? (Figure below) Most
antenna structures and RF power conductors are made of
hollow metal tubes for this reason.
In the following photograph you can see some large
inductors used in a 50 KW radio transmitting circuit. The
inductors are hollow copper tubes coated with silver, for
excellent conductivity at the “skin” of the tube:
High power inductors formed from hollow tubes.
The degree to which frequency affects the effective
resistance of a solid wire conductor is impacted by the
gauge of that wire. As a rule, large-gauge wires exhibit a
more pronounced skin effect (change in resistance from DC)
than small-gauge wires at any given frequency. The
equation for approximating skin effect at high frequencies
(greater than 1 MHZ) is as follows:
Rac= (Rocky f-
Where,
Rac = AC resistance at given frequency "f"
Roc = Resistance at zero frequency (DC)
k = Wire gage factor (see table below)
f= Frequency of AC in MHz (MegaHertz)
Table below gives approximate values of “k” factor for
various round wire sizes.
“k” factor for various AWG wire sizes.
245 8 (B48 |
2099.0 0 —+e7.6
7.6
1/0804 are
a 55 22 |e.ae
6 79 - -
For example, a length of number 10-gauge wire with a DC
end-to-end resistance of 25 Q would have an AC (effective)
resistance of 2.182 kQ at a frequency of 10 MHz:
Rac= (Rocky Y f
Rac = (25 2)(27.6) 10
Rao = 2.182 kQ
Please remember that this figure is not impedance, and it
does not consider any reactive effects, inductive or
Capacitive. This is simply an estimated figure of pure
resistance for the conductor (that opposition to the AC flow
of electrons which does dissipate power in the form of heat),
corrected for the skin effect. Reactance, and the combined
effects of reactance and resistance (impedance), are entirely
different matters.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jim Palmer (June 2001): Identified and offered correction
for typographical error in complex number calculation.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—|/]|4]l\—
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 4
REACTANCE AND
IMPEDANCE -- CAPACITIVE
AC resistor circuits
AC capacitor circuits
Series resistor-capacitor circuits
Parallel resistor-capacitor circuits
Capacitor quirks
Contributors
AC resistor circuits
E, = Ep l=,
Pure resistive AC circuit: voltage and current are in phase.
If we were to plot the current and voltage for a very simple
AC circuit consisting of a source and a resistor, (Figure
above) it would look something like this: (Figure below)
Time —->
Voltage and current “in phase” for resistive circuit.
Because the resistor allows an amount of current directly
proportional to the voltage across it at all periods of time,
the waveform for the current is exactly in phase with the
waveform for the voltage. We can look at any point in time
along the horizontal axis of the plot and compare those
values of current and voltage with each other (any
“snapshot” look at the values of a wave are referred to as
instantaneous values, meaning the values at that /nstant in
time). When the instantaneous value for voltage is zero, the
instantaneous current through the resistor is also zero.
Likewise, at the moment in time where the voltage across
the resistor is at its positive peak, the current through the
resistor is also at its positive peak, and so on. At any given
point in time along the waves, Ohm's Law holds true for the
instantaneous values of voltage and current.
We can also calculate the power dissipated by this resistor,
and plot those values on the same graph: (Figure below)
Instantaneous AC power in a resistive circuit is always
positive.
Note that the power is never a negative value. When the
Current is positive (above the line), the voltage is also
positive, resulting in a power (p=ie) of a positive value.
Conversely, when the current is negative (below the line),
the voltage is also negative, which results in a positive value
for power (a negative number multiplied by a negative
number equals a positive number). This consistent “polarity”
of power tells us that the resistor is always dissipating
power, taking it from the source and releasing it in the form
of heat energy. Whether the current is positive or negative, a
resistor still dissipates energy.
AC capacitor circuits
Capacitors do not behave the same as resistors. Whereas
resistors allow a flow of electrons through them directly
proportional to the voltage drop, capacitors oppose changes
in voltage by drawing or supplying current as they charge or
discharge to the new voltage level. The flow of electrons
“through” a capacitor is directly proportional to the rate of
change of voltage across the capacitor. This opposition to
voltage change is another form of reactance, but one that is
precisely opposite to the kind exhibited by inductors.
Expressed mathematically, the relationship between the
current “through” the capacitor and rate of voltage change
across the capacitor is as such:
i=C —
dt
The expression de/dt is one from calculus, meaning the rate
of change of instantaneous voltage (e) over time, in volts
per second. The capacitance (C) is in Farads, and the
instantaneous current (i), of course, is in amps. Sometimes
you will find the rate of instantaneous voltage change over
time expressed as dv/dt instead of de/dt: using the lower-
case letter “v” instead or “e” to represent voltage, but it
means the exact same thing. To show what happens with
alternating current, let's analyze a simple capacitor circuit:
(Figure below)
FE, = Er, l=1,
Pure capacitive circuit: capacitor voltage lags capacitor
current by 90°
If we were to plot the current and voltage for this very
simple circuit, it would look something like this: (Figure
below)
Pure capacitive circuit waveforms.
Remember, the current through a capacitor is a reaction
against the change in voltage across it. Therefore, the
instantaneous current is zero whenever the instantaneous
voltage is at a peak (zero change, or level slope, on the
voltage sine wave), and the instantaneous current is ata
peak wherever the instantaneous voltage is at maximum
change (the points of steepest slope on the voltage wave,
where it crosses the zero line). This results in a voltage wave
that is -90° out of phase with the current wave. Looking at
the graph, the current wave seems to have a “head start” on
the voltage wave; the current “leads” the voltage, and the
voltage “lags” behind the current. (Figure below)
voltage slope = 0 voltage slope = max. (+)
current = 0 current = max. (+)
\ voltage slope = 0
current = 0
voltage slope = max. (-)
current = max. (-)
Voltage lags current by 90° in a pure capacitive circuit.
As you might have guessed, the same unusual power wave
that we saw with the simple inductor circuit is present in the
simple capacitor circuit, too: (Figure below)
In a pure capacitive circuit, the instantaneous power may be
positive or negative.
As with the simple inductor circuit, the 90 degree phase
shift between voltage and current results in a power wave
that alternates equally between positive and negative. This
means that a capacitor does not dissipate power as it reacts
against changes in voltage; it merely absorbs and releases
power, alternately.
A capacitor's opposition to change in voltage translates to
an opposition to alternating voltage in general, which is by
definition always changing in instantaneous magnitude and
direction. For any given magnitude of AC voltage at a given
frequency, a capacitor of given size will “conduct” a certain
magnitude of AC current. Just as the current through a
resistor is a function of the voltage across the resistor and
the resistance offered by the resistor, the AC current through
a Capacitor is a function of the AC voltage across it, and the
reactance offered by the capacitor. As with inductors, the
reactance of a capacitor is expressed in ohms and
symbolized by the letter X (or X- to be more specific).
Since capacitors “conduct” current in proportion to the rate
of voltage change, they will pass more current for faster-
changing voltages (as they charge and discharge to the
same voltage peaks in less time), and less current for slower-
changing voltages. What this means is that reactance in
ohms for any capacitor is /nverse/ly proportional to the
frequency of the alternating current. (Table below)
1
aa 2nfC
Reactance of a 100 uF capacitor:
Frequency (Hertz)|/Reactance (Ohms)
26.5258
20 13.2629
0.6366
Please note that the relationship of capacitive reactance to
frequency is exactly opposite from that of inductive
reactance. Capacitive reactance (in ohms) decreases with
increasing AC frequency. Conversely, inductive reactance (in
ohms) increases with increasing AC frequency. Inductors
oppose faster changing currents by producing greater
voltage drops; capacitors oppose faster changing voltage
drops by allowing greater currents.
As with inductors, the reactance equation's 2nf term may be
replaced by the lower-case Greek letter Omega (W), which is
referred to as the angular velocity of the AC circuit. Thus, the
equation Xc = 1/(2mfC) could also be written as Xc = 1/(WC),
with w cast in units of radians per second.
Alternating current in a simple capacitive circuit is equal to
the voltage (in volts) divided by the capacitive reactance (in
ohms), just as either alternating or direct current in a simple
resistive circuit is equal to the voltage (in volts) divided by
the resistance (in ohms). The following circuit illustrates this
mathematical relationship by example: (Figure below)
LO V
60 Hz Cc 100 LF
Capacitive reactance.
— 10V
26.5258 Q
1=0.3770 A
However, we need to keep in mind that voltage and current
are not in phase here. As was shown earlier, the current has
a phase shift of +90° with respect to the voltage. If we
represent these phase angles of voltage and current
mathematically, we can calculate the phase angle of the
Capacitor's reactive opposition to current.
Opposition = SORA
Current
Opposition = ee
0.3770 A Z90°
Opposition =26.5258 Q Z -90°
For a capacitor:
90°
-90°
A
I
ae O° a
E Opposition
(X,)
Voltage lags current by 90° in a capacitor.
Mathematically, we say that the phase angle of a capacitor's
opposition to current is -90°, meaning that a capacitor's
opposition to current is a negative imaginary quantity.
(Figure above) This phase angle of reactive opposition to
current becomes critically important in circuit analysis,
especially for complex AC circuits where reactance and
resistance interact. It will prove beneficial to represent any
component's opposition to current in terms of complex
numbers, and not just scalar quantities of resistance and
reactance.
e REVIEW:
e Capacitive reactance is the opposition that a capacitor
offers to alternating current due to its phase-shifted
storage and release of energy in its electric field.
Reactance is symbolized by the capital letter “X” and is
measured in ohms just like resistance (R).
e Capacitive reactance can be calculated using this
formula: X¢ = 1/(2nfC)
e Capacitive reactance decreases with increasing
frequency. In other words, the higher the frequency, the
less it opposes (the more it “conducts”) the AC flow of
electrons.
Series resistor-capacitor circuits
In the last section, we learned what would happen in simple
resistor-only and capacitor-only AC circuits. Now we will
combine the two components together in series form and
investigate the effects. (Figure below)
E; = E,t Er
i=l =ie
Series capacitor circuit: voltage lags current by 0° to 90°.
The resistor will offer 5 O of resistance to AC current
regardless of frequency, while the capacitor will offer
26.5258 QO of reactance to AC current at 60 Hz. Because the
resistor's resistance is a real number (5 Q Z 0°, or 5 + j0 Q),
and the capacitor's reactance is an imaginary number
(26.5258 O Z -90°, or O - j26.5258 QO), the combined effect
of the two components will be an opposition to current equal
to the complex sum of the two numbers. The term for this
complex opposition to current is impedance, its symbol is Z,
and it is also expressed in the unit of ohms, just like
resistance and reactance. In the above example, the total
circuit impedance is:
Zrotal = (5 Q resistance) + (26.5258 Q capacitive reactance)
Zrotat = 3 2 (R) + 26.5258 Q (X_)
Zrotal = (3 Q Z O°) + (26.5258 Q Z -90°)
or
(5 + jO0 Q) + (0 - j26.5258 Q)
Ztail = 2 - j26.5258 Q or 26.993 Q Z -79.325°
tota
Impedance is related to voltage and current just as you
might expect, in a manner similar to resistance in Ohm's
Law:
Ohm's Law for AC circuits:
E=I1Z a 7S
Z I
All quantities expressed in
complex, not scalar, form
In fact, this is a far more comprehensive form of Ohm's Law
than what was taught in DC electronics (E=IR), just as
impedance is a far more comprehensive expression of
opposition to the flow of electrons than simple resistance is.
Any resistance and any reactance, separately or in
combination (series/parallel), can be and should be
represented as a single impedance.
To calculate current in the above circuit, we first need to
give a phase angle reference for the voltage source, which is
generally assumed to be zero. (The phase angles of resistive
and capacitive impedance are a/ways 0° and -90°,
respectively, regardless of the given phase angles for
voltage or current).
l= =
Z
- 10V 70°
26.933 QZ -79.325°
1= 370.5 mA Z 79.325°
As with the purely capacitive circuit, the current wave is
leading the voltage wave (of the source), although this time
the difference is 79.325° instead of a full 90°. (Figure below)
phase shift =
~=— 79.325 degrees
Voltage lags current (current leads voltage)in a series R-C
circuit.
As we learned in the AC inductance chapter, the “table”
method of organizing circuit quantities is a very useful tool
for AC analysis just as it is for DC analysis. Let's place out
known figures for this series circuit into a table and continue
the analysis using this tool:
R & Total
10 + j0
100° Volts
68.623 m + j364.06m
Amps
370.5m 4 79.325°
i _7)) 9 _7;D 9
z 5 +jo 0 - j26.5258 5 - j26.5258 ; Ohms
520 26.5258 4 -90° 26.993 4 -79.325
Current in a series circuit is shared equally by all
components, so the figures placed in the “Total” column for
current can be distributed to all other columns as well:
Total
R Cc
10 +j0 om
10 70° oe
68.623m + j364.06m |68.623m + j364.06m |68.623m + j364.06m
370.5m 4 79.325° 370.5m 2 79.325° 370.5m Z 79.325°
5 +j0 0 - j26.5258 5 -j26.5258
Z °o
520 26.5258 Z -90° 26.993 Z -79.32
Rule of series
circuits:
Lrotal = Ih = i.
Amps
Continuing with our analysis, we can apply Ohm's Law
(E=IR) vertically to determine voltage across the resistor
and capacitor:
R Cc Total
343.11m + j1.8203 9.6569 - j1.8203 10 + jo
E anet ° ° Volts
1.8523 4 79.325 9.8269 / -10.675 1040
68.623m + j364.06m |68.623m+ j364.06m |68.623m-+ j364.06m Amps
370.5m Z 79.325° 370.5m 4 79.325° 370.5m Z 79.325°
0 - j26.5258 5 - j26.5258
s Ohms
26.5258 4 -90° 26.993 Z -79.325°
Ohm's Ohm's
Law Law
E=I[Z E=1Z
Notice how the voltage across the resistor has the exact
Same phase angle as the current through it, telling us that E
and | are in phase (for the resistor only). The voltage across
the capacitor has a phase angle of -10.675°, exactly 90° /ess
than the phase angle of the circuit current. This tells us that
the capacitor's voltage and current are still 90° out of phase
with each other.
Let's check our calculations with SPICE: (Figure below)
Spice circuit: R-C.
ac r-c circuit
v1 10 ac 10 sin
ri. 1°25
cl 2 0 100u
.ac lin 1 60 60
.print ac v(1,2) v(2,0) i(vl)
.print ac vp(1,2) vp(2,0) ip(v1)
.end
freq v(1,2) v(2) i(vl1)
6.000E+01 1.852E+00 9.827E+00 3.705E-01
freq vp(1,2) vp(2) ip(v1)
6.000E+01 7.933E+01 -1.067E+01 -1.007E+02
Interpreted SPICE results
EB, = 1,852 ¥V 779,33"
E, = 9.827 V Z -10.67°
1= 370.5 mA Z -100.7°
Once again, SPICE confusingly prints the current phase
angle at a value equal to the real phase angle plus 180° (or
minus 180°). However, its a simple matter to correct this
figure and check to see if our work is correct. In this case,
the -100.7° output by SPICE for current phase angle equates
to a positive 79.3°, which does correspond to our previously
calculated figure of 79.325°.
Again, it must be emphasized that the calculated figures
corresponding to real-life voltage and current measurements
are those in po/ar form, not rectangular form! For example, if
we were to actually build this series resistor-capacitor circuit
and measure voltage across the resistor, our voltmeter
would indicate 1.8523 volts, not 343.11 millivolts (real
rectangular) or 1.8203 volts (imaginary rectangular). Real
instruments connected to real circuits provide indications
corresponding to the vector length (magnitude) of the
calculated figures. While the rectangular form of complex
number notation is useful for performing addition and
subtraction, it is a more abstract form of notation than polar,
which alone has direct correspondence to true
measurements.
Impedance (Z) of a series R-C circuit may be calculated,
given the resistance (R) and the capacitive reactance (Xc¢).
Since E=IR, E=IX-, and E=IZ, resistance, reactance, and
impedance are proportional to voltage, respectively. Thus,
the voltage phasor diagram can be replaced by a similar
impedance diagram. (Figure below)
1 ER 1 R
C PY ! » !
E, “AYE, Z Xe
Voltage Impedance
Series: R-C circuit Impedance phasor diagram.
Example:
Given: A 40 Q resistor in series with a 88.42 microfarad
capacitor. Find the impedance at 60 hertz.
Xc = 1/(2nfC)
Xc = 1/(2m 60: 88.42x10°°)
X = 30 0
Z=R- jXc
Z = 40 - j30
[Z| = sqrt(402 + (-30)7) = 500
4Z = arctangent(-30/40) = -36.87°
Z = 40 - j30 = 504-36.87°
REVIEW:
Impedance is the total measure of opposition to electric
current and is the complex (vector) sum of (“real”)
resistance and (“imaginary”) reactance.
Impedances (Z) are managed just like resistances (R) in
series circuit analysis: series impedances add to form
the total impedance. Just be sure to perform all
calculations in complex (not scalar) form! Z44, = Z1 +
Lo ee Ze
Please note that impedances always add in series,
regardless of what type of components comprise the
impedances. That is, resistive impedance, inductive
impedance, and capacitive impedance are to be treated
the same way mathematically.
A purely resistive impedance will always have a phase
angle of exactly 0° (Zp =RQZ O°).
A purely capacitive impedance will always have a phase
angle of exactly -90° (Z- = X-Q Z -90°).
Ohm's Law for AC circuits: E = 1Z;1 = E/Z; Z = E/l
When resistors and capacitors are mixed together in
circuits, the total impedance will have a phase angle
somewhere between 0° and -90°.
Series AC circuits exhibit the same fundamental
properties as series DC circuits: current is uniform
throughout the circuit, voltage drops add to form the
total voltage, and impedances add to form the total
impedance.
Parallel resistor-capacitor circuits
Using the same value components in our series example
circuit, we will connect them in parallel and see what
happens: (Figure below)
1 i
E Cc
oe Oo® lO V ' loo_|C
© ct YG) RESO
L0.7° 60 Hz
E le
1 =1,+1,
E =E,=E-
Parallel R-C circuit.
Because the power source has the same frequency as the
series example circuit, and the resistor and capacitor both
have the same values of resistance and capacitance,
respectively, they must also have the same values of
impedance. So, we can begin our analysis table with the
same “given” values:
0 - j26.5258
26.5258 Z -90°
This being a parallel circuit now, we know that voltage is
shared equally by all components, so we can place the
figure for total voltage (10 volts Z O°) in all the columns:
R Cc Total
10+ j0
Volts
10+ j0 L0 + j0
E ° ° °
1040 1040 1040
5 + j0 0 - j26.5258
Z . . Ohms
Rule of parallel
circuits:
Exotai = Ep =E>
Now we can apply Ohm's Law (I=E/Z) vertically to two
columns in the table, calculating current through the resistor
and current through the capacitor:
R Cc Total
E LO + jO 10+ jO LO + j0 Volt
10 20° 10 40° 10 20° li
2+ j0 0 + j376.99m
| Amps
anes
7 5 +j0 O - 26.5258 ae
520 26.5258 4 -90°
|
Ohm's Ohm's
Law Law
_E -E—
Z Z
Just as with DC circuits, branch currents in a parallel AC
circuit add up to form the total current (Kirchhoff's Current
Law again):
10 +j0 LO + j0 LO + j0 om
10 70° 10 70° 10 70° ae
2 +50 0 + j376.99m 2 + j376.99m
| i 3 Amps
220 376.99m 2 90° 2.0352 7 10.675
j _ ; 8
Zz 5 + j0 0 - j26.525 : lenis
520° 26.5258 2-90
R Cc Total
Rule of parallel
circuits:
Lott =p +e
Finally, total impedance can be calculated by using Ohm's
Law (Z=E/I) vertically in the “Total” column. As we saw in
the AC inductance chapter, parallel impedance can also be
calculated by using a reciprocal formula identical to that
used in calculating parallel resistances. It is noteworthy to
mention that this parallel impedance rule holds true
regardless of the kind of impedances placed in parallel. In
other words, it doesn't matter if we're calculating a circuit
composed of parallel resistors, parallel inductors, parallel
Capacitors, or some combination thereof: in the form of
impedances (Z), all the terms are common and can be
applied uniformly to the same formula. Once again, the
parallel impedance formula looks like this:
Zoarallel = aT Cale: Le, (a
The only drawback to using this equation is the significant
amount of work required to work it out, especially without
the assistance of a calculator capable of manipulating
complex quantities. Regardless of how we calculate total
impedance for our parallel circuit (either Ohm's Law or the
reciprocal formula), we will arrive at the same figure:
R Cc Total
0 + j376.99m 2 + j376.99m
376.99m 2 90° 2.0352 Z 10.675°
0 - j26.5258 4.8284 - j910.14m
26.5258 Z -90° 4.9135 2 -10.675°
Ohm's Rule of parallel
Law circuits:
E L
Z — — = A
I Zrotal “i si
Zr Zo
e REVIEW:
e Impedances (Z) are managed just like resistances (R) in
parallel circuit analysis: parallel impedances diminish to
form the total impedance, using the reciprocal formula.
Just be sure to perform all calculations in complex (not
scalar) form! Zyo¢a; = 1/(1/Z, + 1/Z> + ...1/Z,)
e Ohm's Law for AC circuits: E=1Z;1=E/Z; Z = E/|
e When resistors and capacitors are mixed together in
parallel circuits (just as in series circuits), the total
impedance will have a phase angle somewhere between
0° and -90°. The circuit current will have a phase angle
somewhere between 0° and +902.
e Parallel AC circuits exhibit the same fundamental
properties as parallel DC circuits: voltage is uniform
throughout the circuit, branch currents add to form the
total current, and impedances diminish (through the
reciprocal formula) to form the total impedance.
Capacitor quirks
As with inductors, the ideal capacitor is a purely reactive
device, containing absolutely zero resistive (power
dissipative) effects. In the real world, of course, nothing is so
perfect. However, capacitors have the virtue of generally
being purer reactive components than inductors. It is a lot
easier to design and construct a capacitor with low internal
series resistance than it is to do the same with an inductor.
The practical result of this is that real capacitors typically
have impedance phase angles more closely approaching 90°
(actually, -90°) than inductors. Consequently, they will tend
to dissipate less power than an equivalent inductor.
Capacitors also tend to be smaller and lighter weight than
their equivalent inductor counterparts, and since their
electric fields are almost totally contained between their
plates (unlike inductors, whose magnetic fields naturally
tend to extend beyond the dimensions of the core), they are
less prone to transmitting or receiving electromagnetic
“noise” to/from other components. For these reasons, circuit
designers tend to favor capacitors over inductors wherever a
design permits either alternative.
Capacitors with significant resistive effects are said to be
lossy, in reference to their tendency to dissipate (“lose”)
power like a resistor. The source of capacitor loss is usually
the dielectric material rather than any wire resistance, as
wire length in a capacitor is very minimal.
Dielectric materials tend to react to changing electric fields
by producing heat. This heating effect represents a loss in
power, and is equivalent to resistance in the circuit. The
effect is more pronounced at higher frequencies and in fact
can be so extreme that it is sometimes exploited in
manufacturing processes to heat insulating materials like
plastic! The plastic object to be heated is placed between
two metal plates, connected to a source of high-frequency
AC voltage. Temperature is controlled by varying the voltage
or frequency of the source, and the plates never have to
contact the object being heated.
This effect is undesirable for capacitors where we expect the
component to behave as a purely reactive circuit element.
One of the ways to mitigate the effect of dielectric “loss” is
to choose a dielectric material less susceptible to the effect.
Not all dielectric materials are equally “lossy.” A relative
scale of dielectric loss from least to greatest is given in Table
below.
Dielectric loss
Material
Vacuum
olystyrene
Mica
Glass
ow-K ceramic
lastic film (Mylar)
aper
igh-K ceramic
Aluminum oxide
Tantalum pentoxide
Dielectric resistivity manifests itself both as a series anda
parallel resistance with the pure capacitance: (Figure below)
Equivalent circuit for a real capacitor
R
series
Ideal oe
Capacitor ee
Real capacitor has both series and parallel resistance.
Fortunately, these stray resistances are usually of modest
impact (low series resistance and high parallel resistance),
much less significant than the stray resistances present in
an average inductor.
Electrolytic capacitors, known for their relatively high
Capacitance and low working voltage, are also Known for
their notorious lossiness, due to both the characteristics of
the microscopically thin dielectric film and the electrolyte
paste. Unless specially made for AC service, electrolytic
Capacitors should never be used with AC unless it is mixed
(biased) with a constant DC voltage preventing the
Capacitor from ever being subjected to reverse voltage. Even
then, their resistive characteristics may be too severe a
shortcoming for the application anyway.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+]l—
—||+]l—
Lessons In Electric Circuits
-- Volume Il
Chapter 5
REACTANCE AND
IMPEDANCE -- R, L, AND C
e Review of R, X, and Z
e Series R, L, and C
e Parallel R, L, and C
e Series-parallel R, L, and C
e Susceptance and Admittance
e Summary
e Contributors
Review of R, X, and Z
Before we begin to explore the effects of resistors, inductors,
and capacitors connected together in the same AC circuits,
let's briefly review some basic terms and facts.
Resistance is essentially friction against the motion of
electrons. It is present in all conductors to some extent
(except superconductors!), most notably in resistors. When
alternating current goes through a resistance, a voltage drop
iS produced that is in-phase with the current. Resistance is
mathematically symbolized by the letter “R” and is measured
in the unit of ohms (Q).
Reactance is essentially /nertia against the motion of
electrons. It is present anywhere electric or magnetic fields
are developed in proportion to applied voltage or current,
respectively; but most notably in capacitors and inductors.
When alternating current goes through a pure reactance, a
voltage drop is produced that is 90° out of phase with the
current. Reactance is mathematically symbolized by the letter
“X” and is measured in the unit of ohms (Q).
Impedance is a comprehensive expression of any and all
forms of opposition to electron flow, including both resistance
and reactance. It is present in all circuits, and in all
components. When alternating current goes through an
impedance, a voltage drop is produced that is somewhere
between 0° and 90° out of phase with the current. Impedance
is mathematically symbolized by the letter “Z” and is
measured in the unit of ohms (Q), in complex form.
Perfect resistors (Figure below) possess resistance, but not
reactance. Perfect inductors and perfect capacitors (Figure
below) possess reactance but no resistance. All components
possess impedance, and because of this universal quality, it
makes sense to translate all component values (resistance,
inductance, capacitance) into common terms of impedance as
the first step in analyzing an AC circuit.
Resistor |900 Inductor 100mH Capacitor lLOUF
L59.L5 Hz 159.15 Hz
: R= 1000 R=00 | R=00
X=00 X = L002 “| x=1000
Z=1002 20° Z= 1002 290° Z = 1002 2-90°
Perfect resistor, inductor, and capacitor.
The impedance phase angle for any component is the phase
shift between voltage across that component and current
through that component. For a perfect resistor, the voltage
drop and current are a/ways in phase with each other, and so
the impedance angle of a resistor is said to be 0°. For an
perfect inductor, voltage drop always leads current by 90°,
and so an inductor's impedance phase angle is said to be
+90°. For a perfect capacitor, voltage drop always lags
current by 90°, and so a capacitor's impedance phase angle is
said to be -90°.
Impedances in AC behave analogously to resistances in DC
circuits: they add in series, and they diminish in parallel. A
revised version of Ohm's Law, based on impedance rather
than resistance, looks like this:
Ohm's Law for AC circuits:
_E
Z
E-IZ I z= —
All quantities expressed in
complex, not scalar, form
Kirchhoff's Laws and all network analysis methods and
theorems are true for AC circuits as well, so long as quantities
are represented in complex rather than scalar form. While this
qualified equivalence may be arithmetically challenging, it is
conceptually simple and elegant. The only real difference
between DC and AC circuit calculations is in regard to power.
Because reactance doesn't dissipate power as resistance
does, the concept of power in AC circuits is radically different
from that of DC circuits. More on this subject in a later
chapter!
Series R, L, and C
Let's take the following example circuit and analyze it: (Figure
below)
650 mH
1.5 UF
Example series R, L, and C circuit.
The first step is to determine the reactances (in ohms) for the
inductor and the capacitor.
X, = 20fL
X, = (2)(1%)(60 Hz)(650 mH)
X, = 245.042
Xo = ae
(2)(11)(60 Hz)(1.5 LF)
X, = 1.7684 kQ
The next step is to express all resistances and reactances ina
mathematically common form: impedance. (Figure below)
Remember that an inductive reactance translates into a
positive imaginary impedance (or an impedance at +90°),
while a capacitive reactance translates into a negative
imaginary impedance (impedance at -90°). Resistance, of
course, is still regarded as a purely “real” impedance (polar
angle of 0°):
Zp=250+jOQ or 2502 20°
Z, =0+4j245.08.Q or 245.042 290°
Z-=0-j1.7684k Q or 1.7684 kQ Z -90°
120 V
: 245.04 Q 290°
60 Hz
1.7684 kQ Z -90°
Example series R, L, and C circuit with component values
replaced by impedances.
Now, with all quantities of opposition to electric current
expressed in a common, complex number format (as
impedances, and not as resistances or reactances), they can
be handled in the same way as plain resistances in a DC
circuit. This is an ideal time to draw up an analysis table for
this circuit and insert all the “given” figures (total voltage,
and the impedances of the resistor, inductor, and capacitor).
R L Cc Total
E 120+ j0
120 20°
Volts
Amps
7 250 + jo 0+j245.04 O - jL.7684k
250 2 OP 245.04 2 90° L7684k 2 -90°
Ohms
Unless otherwise specified, the source voltage will be our
reference for phase shift, and so will be written at an angle of
0°. Remember that there is no such thing as an “absolute”
angle of phase shift for a voltage or current, since its always a
quantity relative to another waveform. Phase angles for
impedance, however (like those of the resistor, inductor, and
Capacitor), are known absolutely, because the phase
relationships between voltage and current at each component
are absolutely defined.
Notice that I'm assuming a perfectly reactive inductor and
Capacitor, with impedance phase angles of exactly +90 and
-90°, respectively. Although real components won't be perfect
in this regard, they should be fairly close. For simplicity, I'll
assume perfectly reactive inductors and capacitors from now
on in my example calculations except where noted otherwise.
Since the above example circuit is a series circuit, we know
that the total circuit impedance is equal to the sum of the
individuals, so:
Zrotal =Zp+Z,_+Z,
Zrota| = (250 + jO Q) + (0 + j245.04 Q) + (0 -j1.7684k Q)
Zota = 250 - j1.5233k Q or 1.5437 kQ Z -80.680°
Inserting this figure for total impedance into our table:
R L | Total
120+ j0
E 120 2 0° a
| Amps
75 a 7 - 75H _ 57334
3 250+ jO 0 - j1.7684k 250 - j1.5233k ic
250 40° 1.768 4k 4 -90° 1.5437k 2 -80.680°
Rule of series
circuits:
Zecca) = Zp + Z_+Ze
We can now apply Ohm's Law (I=E/R) vertically in the “Total”
column to find total current for this series circuit:
R Total
120+ jo
E J
1202 0° vole
12.539m + 76.708m.
| Amps
77.734m 2 80.680°
Zz 250 + jO 0+ j245.04 O -j1.7684k 250 - j 1.5233k Ohms
250 40° 245.04 4 90° 1.768 4k 24 -90° 1.5437k 2 -80.680°
Being a series circuit, current must be equal through all
components. Thus, we can take the figure obtained for total
Current and distribute it to each of the other columns:
L Total
120+ j
E +j0
120 4 0°
| 12.589m + 76.708m | 12.589m + 76.708m | 12.589m+ 76.708m | 12.589m+ 76.708m Amps
77.74m 2 80.680° 77.734m 2 80.680° 77.74m 2 80.680° 77.734m Z 80.680°
2 250 + jo 0+ j245.04 0 - j1.7684k 250 - j 1.5233k Ohms
250 20° 245.04 4 90° 1.768 4k 2 -90° 1.5437k Z -80.680°
Rule of series
circuits:
Teat = 1p =1L =k
Volts
Now we're prepared to apply Ohm's Law (E=IZ) to each of the
individual component columns in the table, to determine
voltage drops:
R L Cc Total
E 3.1472 + j19.177 -18.797 + j3.0343 135.65 - J22.262 120+ jO Volt
19.434 280.680? | 19.0484 170.68° | 137.46 2-9.3199° 120 2 0° _—
| 12.589m + 76.708m 12.589m + 76.708m | 12.589m + 76.708m 12.589m + 76.708m Amps
77.734m 4 80.680° 77.734m 24 80.680° 77.734m 2 80.680° 77.734m 24 80.680°
5 745 . 7 - 950-11 5733k
7 250+ jo 0+ j245.04 0 -j1.7684k : 250 - j1.5233k Ohms
250 20 245.04 4 90° 1.768 4k 4 -90 1.5437k 2 -80.680°
Ohm’‘s Ohm's Ohm’‘s
Law Law Law
E=Iz E=i E=Z
Notice something strange here: although our supply voltage
is only 120 volts, the voltage across the capacitor is 137.46
volts! How can this be? The answer lies in the interaction
between the inductive and capacitive reactances. Expressed
as impedances, we can see that the inductor opposes current
in a manner precisely opposite that of the capacitor.
Expressed in rectangular form, the inductor's impedance has
a positive imaginary term and the capacitor has a negative
imaginary term. When these two contrary impedances are
added (in series), they tend to cancel each other out!
Although they're still added together to produce a sum, that
sum is actually /ess than either of the individual (capacitive
or inductive) impedances alone. It is analogous to adding
together a positive and a negative (scalar) number: the sum
is a quantity less than either one's individual absolute value.
If the total impedance in a series circuit with both inductive
and capacitive elements is less than the impedance of either
element separately, then the total current in that circuit must
be greater than what it would be with only the inductive or
only the capacitive elements there. With this abnormally high
current through each of the components, voltages greater
than the source voltage may be obtained across some of the
individual components! Further consequences of inductors'
and capacitors’ opposite reactances in the same circuit will be
explored in the next chapter.
Once you've mastered the technique of reducing all
component values to impedances (Z), analyzing any AC
circuit is only about as difficult as analyzing any DC circuit,
except that the quantities dealt with are vector instead of
scalar. With the exception of equations dealing with power
(P), equations in AC circuits are the same as those in DC
circuits, using impedances (Z) instead of resistances (R).
Ohm's Law (E=I!Z) still holds true, and so do Kirchhoff's
Voltage and Current Laws.
To demonstrate Kirchhoff's Voltage Law in an AC circuit, we
can look at the answers we derived for component voltage
drops in the last circuit. KVL tells us that the algebraic sum of
the voltage drops across the resistor, inductor, and capacitor
should equal the applied voltage from the source. Even
though this may not look like it is true at first sight, a bit of
complex number addition proves otherwise:
Ep +E, +E. should equal Eotal
3.1472 + jl9.177V Ep
-18.797 +j3.0848 VE,
+ 135.65-j22.262V E-
120+ j0 V Ea
Aside from a bit of rounding error, the sum of these voltage
drops does equal 120 volts. Performed on a calculator
(preserving all digits), the answer you will receive should be
exactly 120 + jO volts.
We can also use SPICE to verify our figures for this circuit:
(Figure below)
1.5 LF
Example series R, L, and C SPICE circuit.
-C Circuit
ac l
0 ac 120 sin
2
r
vl 1
rl 1
l1 2
cl 3 0 1.5u
.ac Lin 1 60 60
.print ac v(1,2) v(2,3) v(3,0) i(vl)
.print ac vp(1,2) vp(2,3) vp(3,0) ip(v1)
.end
freq v(1,2) v(2,3) v(3) i(vl)
6.000E+01 1.943E+01 1.905E+01 1.375E+02 7.773E-02
freq vp(1,2) vp (2,3) vp(3) ip(v1)
6.000E+01 8.068E+01 1.707E+02 -9.320E+00 -9.932E+01
Interpreted SPICE results
E, = 19.43 V Z 80.68°
E, = 19.05 V Z 170.7°
E, = 137.5 V Z -9.320°
1= 77.73 mA Z -99.32° (actual phase angle = 80.68")
The SPICE simulation shows our hand-calculated results to be
accurate.
As you can see, there is little difference between AC circuit
analysis and DC circuit analysis, except that all quantities of
voltage, current, and resistance (actually, impedance) must
be handled in complex rather than scalar form so as to
account for phase angle. This is good, since it means all
you've learned about DC electric circuits applies to what
you're learning here. The only exception to this consistency is
the calculation of power, which is so unique that it deserves a
chapter devoted to that subject alone.
e REVIEW:
¢ Impedances of any kind add in series: Zy 44, = Z1 + Zo +.
a
e Although impedances add in series, the total impedance
for a circuit containing both inductance and capacitance
may be less than one or more of the individual
impedances, because series inductive and capacitive
impedances tend to cancel each other out. This may lead
to voltage drops across components exceeding the supply
voltage!
e All rules and laws of DC circuits apply to AC circuits, so
long as values are expressed in complex form rather than
scalar. The only exception to this principle is the
calculation of power, which is very different for AC.
Parallel R, L, and C
We can take the same components from the series circuit and
rearrange them into a parallel configuration for an easy
example circuit: (Figure below)
120 V
60 Hz
Example R, L, and C parallel circuit.
The fact that these components are connected in parallel
instead of series now has absolutely no effect on their
individual impedances. So long as the power supply is the
same frequency as before, the inductive and capacitive
reactances will not have changed at all: (Figure below)
120 V
60 Hz
250Q 20° 1.7684 kQ Z -90°
245.04 Q 290°
Example R, L, and C parallel circuit with impedances
replacing component values.
With all component values expressed as impedances (Z), we
can set up an analysis table and proceed as in the last
example problem, except this time following the rules of
parallel circuits instead of series:
R L Cc Total
120 +jo
E Volt
120 20° _—
| Amps
> 2 . -
2 250 + j0 0+j245.04 0 - jL.7684k Ohms
250 4 0° 245.04 4 90° L.7684k 2 -90°
Knowing that voltage is shared equally by all components ina
parallel circuit, we can transfer the figure for total voltage to
all component columns in the table:
R L Cc Total
c 120 + jo 120 + jo 120+jo 120+j0 ma
120 20 120 2 0 12020 1202 0 a:
| Amps
25 245, - j1.7684k
Zz ako O+j 5.04 O - j1.7684k erie
250 20° 245.04 2 90° 1.7684k 2 -90°
Rule of parallel
circuits:
Fea! = Ep = B, = Ee
Now, we can apply Ohm's Law (I=E/Z) vertically in each
column to determine current through each component:
R L Total
c 120 + jo 120 + jO 120+ j0 120+j0 er
120 20° 120 20° 120 2 0° 1202 0° =
480m + jO
| Amps
480m 4 0° P
? 250 + jo 0+ j245.04 0 - j1.7684k Ohms
250 40° 245.04 4 90° 1.768 4k 2 -90°
Ohm’‘s Ohm's Ohm’‘s
Law Law Law
= = == t= =
Z Z Z
There are two strategies for calculating total current and total
impedance. First, we could calculate total impedance from all
the individual impedances in parallel (Zyo¢a; = 1/(1/Zp + 1/Z,
+ 1/Z-), and then calculate total current by dividing source
voltage by total impedance (I=E/Z). However, working
through the parallel impedance equation with complex
numbers is no easy task, with all the reciprocations (1/Z). This
is especially true if you're unfortunate enough not to have a
calculator that handles complex numbers and are forced to do
it all by hand (reciprocate the individual impedances in polar
form, then convert them all to rectangular form for addition,
then convert back to polar form for the final inversion, then
invert). The second way to calculate total current and total
impedance is to add up all the branch currents to arrive at
total current (total current in a parallel circuit -- AC or DC -- is
equal to the sum of the branch currents), then use Ohm's Law
to determine total impedance from total voltage and total
current (Z=E/l).
R
120 + jo 120+j0
E Volt
120 20° 12020 ai
480m + jO 0 - j489.71m 480m-}42.85m | a.
480 2 OP 489.71m Z -90° 639.03m / -41.311°
5 250+ j0 0 + j245.04 0 - j1.7684k 141.05 +J123.96 |
250 20° 245.04 2 90° 1.7684k 2 -90° 187.79 2 41.311°
Either method, performed properly, will provide the correct
answers. Let's try analyzing this circuit with SPICE and see
what happens: (Figure below)
Example parallel R, L, and C SPICE circuit. Battery symbols
are “dummy” voltage sources for SPICE to use as current
measurement points. All are set to O volts.
ac r-l-c circuit
vl 10 ac 120 sin
vi 12 ac 0
vir 2 3 ac 0
vil 2 4 ac 0
rbogus 4 5 le-12
vic 2 6 ac 0
rl 3 0 250
l1 5 0 650m
cl 6 0 1.5u
.ac Lin 1 60 60
.print ac i(vi) i(vir) i(vil) i(vic)
.print ac ip(vi) ip(vir) ip(vil) ip(vic)
.end
freq i(vi) i(vir) i(vil)
6.000E+01 6.390E-01 4.800E-01 4.897E-01
freq ip(v1) ip(vir) ip(vil)
6.000E+01 -4.131E+01 0.000E+00 -9.000E+01
Interpreted SPICE results
lm = 639.0 mA Z -41.31°
1, = 480 mA Z 0°
1, = 489.7 mA Z -90°
1. = 67.86 mA Z 90°
i(vic)
6.786E-02
ip(vic)
9.000E+01
It took a little bit of trickery to get SPICE working as we would
like on this circuit (installing “dummy” voltage sources in
each branch to obtain current figures and installing the
“dummy” resistor in the inductor branch to prevent a direct
inductor-to-voltage source loop, which SPICE cannot tolerate),
but we did get the proper readings. Even more than that, by
installing the dummy voltage sources (current meters) in the
proper directions, we were able to avoid that idiosyncrasy of
SPICE of printing current figures 180° out of phase. This way,
our current phase readings came out to exactly match our
hand calculations.
Series-parallel R, L, and C
Now that we've seen how series and parallel AC circuit
analysis is not fundamentally different than DC circuit
analysis, it should come as no surprise that series-parallel
analysis would be the same as well, just using complex
numbers instead of scalar to represent voltage, current, and
impedance.
Take this series-parallel circuit for example: (Figure below)
Example series-parallel R, L, and C circuit.
The first order of business, as usual, is to determine values of
impedance (Z) for all components based on the frequency of
the AC power source. To do this, we need to first determine
values of reactance (X) for all inductors and capacitors, then
convert reactance (X) and resistance (R) figures into proper
impedance (Z) form:
Reactances and Resistances:
XL = 2rtL
X, = (2)(7)(60 Hz)(650 mH)
Xe, = ————_—_—_____—_
(2)(7)(60 Hz)(4.7 UF)
Xe, = 564.3802 X, = 245.04 Q
Xoo = L.7684 kQ
Ze, = 0-j564.38Q2 or 564.382 2-90°
Z, =0+4j245.08Q or 245.042 290°
Ze. = 0-jl.7684k Q or 1.7684kQ Z-90°
Zp=410+j0OQ or 47022Z0°
Now we can set up the initial values in our table:
Cc L G R Total
120 + jo
120 20°
0 - j50433 0+ j245.04 O - jl. 7634b 470 + jO Ohms
564.33 4 90° 245.04 4 90° L.7684k 4 -90° 470 2 0°
Being a series-parallel combination circuit, we must reduce it
to a total impedance in more than one step. The first step is to
combine L and C; as a Series combination of impedances, by
adding their impedances together. Then, that impedance will
be combined in parallel with the impedance of the resistor, to
arrive at another combination of impedances. Finally, that
quantity will be added to the impedance of C, to arrive at the
total impedance.
In order that our table may follow all these steps, it will be
necessary to add additional columns to it so that each step
may be represented. Adding more columns horizontally to the
table shown above would be impractical for formatting
reasons, so | will place a new row of columns underneath,
each column designated by its respective component
combination:
Total
iC, R//(L~ C3) C,-([RW#(L—C,)]
E Volts
| Amps
Z Ohms
Calculating these new (combination) impedances will require
complex addition for series combinations, and the “reciprocal”
formula for complex impedances in parallel. This time, there
is no avoidance of the reciprocal formula: the required figures
can be arrived at no other way!
Total
Lt, Ri (L — C3) C,-IRW(L—C,]
c 120+j0
120 20°
0 - j1.5233k 429,15 -j132.41 429.15 - 696.79 | o.
1.5233k 2-90 449.11 2-17.147° | 818.34 2 -58.371°
Volts
Rule of series Rule of series
circuits: circuits:
Z.o2=2.+2e Zroal = 21 + Zpercr
Rue of parallel
circuits:
l
ZRL-C» = =Ay. 21-—
Tea
Ze 2h
Seeing as how our second table contains a column for “Total,”
we can Safely discard that column from the first table. This
gives us one table with four columns and another table with
three columns.
Now that we know the total impedance (818.34 Q Z -58.371°)
and the total voltage (120 volts Z 0°), we can apply Ohm's
Law (lI=E/Z) vertically in the “Total” column to arrive at a
figure for total current:
Total
L--C, R//(L—- C,) C,—-([R“#(L—C,)]
Volts
76.399m + j124.36m
146.64m 2 58.371°
O-j1.5233k 429.15 -j 152.41 429.15 - j696.79
1.5233k 2 -90° 449.11 2-17.147° 818.34 4 -58.371°
Amps
Ohms
At this point we ask ourselves the question: are there any
components or component combinations which share either
the total voltage or the total current? In this case, both C, and
the parallel combination R//(L--C,) share the same (total)
current, since the total impedance is composed of the two
sets of impedances in series. Thus, we can transfer the figure
for total current into both columns:
res L a R
E Volts
| | 76.899m + J124.36m A
<9 4710 mps
——» | 146.64m 2 58.371
0 - j564.38 0+j245.04 0 - {1.7684 470 + jO
Ohms
564.38 4 -90° 245.04 4 90° 1.7684k 4 -90° 470 20°
Rule of series
en cwouilts:
Trseal = Ley = Lp irri
Total
C,-—([RW/(L—-C,)
Volts
76.899m + j124.36m | 76.899m + j124.86m
Amps
146.64m 2 58.371° 146.64 4 58.371° P
Z O- j1.5233k 429.15 -j 152.41 429.15 - j696.79
Ohms
1.5233k 4 -90° 449.11 4-17.147° 818.34 4 -58.371°
Rule of series
circuits:
Trseal = Ley = Lp rer)
Now, we can calculate voltage drops across C, and the series-
parallel combination of R//(L--Cz) using Ohm's Law (E=!Z)
vertically in those table columns:
C; L ‘om R
E | 70.467 - 43.400 ce
82.760 2 -31.629° aa
76.899 +j 124.86m
| Amps
146.64: 4 58.371°
O- 564.38 0+ j245.04 O- j1.7684k 470 +j0
Z Ohms
564.38 2 -90° 245.04 290° 1.7684k 2 -90° 470 20°
Ohm's
Law
E=Z
Total
L-C, R//(L—-C,) C,-—(RW/(L-C,)
: 49.533 + j43.400 120+ jO dete
65.857 2 41.225° 12020 __
76.8991 + j124.86m | 76.899m + j124.86m Amps
146.64m 2 58.371° 146.64: 4 58.371°
QO - j1.5233k 429.15 -j 1352.41 429.15 - j696.79
1.5233k 2 -90° 449.11 4 -17.147° 818.34 2 -58.371°
E
Ohm's
Law
E=Z
Ohms
A quick double-check of our work at this point would be to
see whether or not the voltage drops across C, and the series-
parallel combination of R//(L--C3) indeed add up to the total.
According to Kirchhoff's Voltage Law, they should!
Exot Should be equal to E-, + Epi _c2)
70.467 - j43.400 V
+ 49.533 + j43.400 V
120+jOV ~—— /Indeed, it is!
That last step was merely a precaution. In a problem with as
many steps as this one has, there is much opportunity for
error. Occasional cross-checks like that one can save a person
a lot of work and unnecessary frustration by identifying
problems prior to the final step of the problem.
After having solved for voltage drops across C, and the
combination R//(L--C3), we again ask ourselves the question:
what other components share the same voltage or current? In
this case, the resistor (R) and the combination of the inductor
and the second capacitor (L--C,) share the same voltage,
because those sets of impedances are in parallel with each
other. Therefore, we can transfer the voltage figure just solved
for into the columns for R and L--C;:
re L om R
E 70.467 - j43.400 49.533 + j43.400 Vol
olts
82.760 2 -31.629° 65.857 241.225° | _
76.899m + j 124.86m
| Amps
146.64: 4 58.371°
? 0 - 564.38 O + j245.04 0 - j1.7684k 470 +j0 Ohms
564.38 2 -90° 245.04 4 90° 1.7684k 2 -90° 470 20°
Rule of parallel
circuits:
Ep wr-e2) = Ep =ELe
Total
L=<¢; Ri(L —C3) C,—[R# (L—C.))
E 49.533 + j43.400 49.533 + j43.400 Vol
5 | 65.857.241.225° | 65.857 241.225° on
| 76.899m + j124.86m | 76.899m + j124.86m Amps
146.64: 4 58.371° 146.641m 4 58.371°
r O-j1.5233k 429.15 - j132.41 429.15 - j696.79 Ohms
1.5233k 2 -90° 449.11 4 -17.147° 818.34 2 -58.371°
Rule of parallel
Se circuits:
Epyacz = Ep = ELicz
Now we're all set for calculating current through the resistor
and through the series combination L--C5. All we need to do is
apply Ohm's Law (I=E/Z) vertically in both of those columns:
c, L ol R
g | 70.467 - j43.400 49.533 + j43.400 ie
ons
82.760 2 -31.629° 65.857 2 41.225°
76.8991 + j 124.86m
146.64: 4 58.371°
Zz O-j 564.38 O+ j245.04 0 - j1.7684k 4+70+j0 Ohms
564.38 2 -90° 245.04 4 90° 1.7684k 2 -90° 470 20°
Ohm's
Law
l= =.
Zz
Total
L—C,; R//(L—C3) C,—(R/V/(L—C)]
49.533 + j43.400
65.857 4 41.225°
49.533 + j43.400 120+ jo
65.857 4 41.225° 120020
76.899m + j124.86m | 76.899m + j124.86m Airis
146.64m 4 58.371° | 146.64m 2 §8.371°
429.15 -j 132.41 429.15 - |696.79
Ohms
449.11 4 -17.147° 818.34 4 -58.371°
Volts
-28.490m + j32.516m
43.232m 2 131.22°
O- j1.5233k
1.5233k 2 -90°
Another quick double-check of our work at this point would
be to see if the current figures for L--C, and R add up to the
total current. According to Kirchhoff's Current Law, they
should:
Ipi_—c2) Should be equal to1,z + 1,_-
105.39m + j92.341m
+ -28.490m + j32.5 16m
76.899m + j124.86m —+— Indeed, it is!
Since the L and C, are connected in series, and since we know
the current through their series combination impedance, we
can distribute that current figure to the L and C, columns
following the rule of series circuits whereby series
components share the same current:
C, E Cc, R
g | 70467 -j43.400 49.533 + [43.400 oe
82.760 Z -31.629° 65.857 2 41.225° a
| | 76.899m +)124.86m |-28.490m + j32.516m |-28.490m + J32.516m | 105.39m +j92.341m |
146.64m 258.371° | 43.232m 2 131.22° | 43.232m 2 131.22° | 140.12m 2 41.225° P
7 0 - j564.38 O + j245.04 O- j1.7684k 470+j0
564.38 2 -90° 245.04 290° 1.7684k 2 -90° 470 20°
Rule of series
circuits:
Ile =Lal.
With one last step (actually, two calculations), we can
complete our analysis table for this circuit. With impedance
and current figures in place for L and C;, all we have to do is
apply Ohm's Law (E=IZ) vertically in those two columns to
calculate voltage drops.
fa L c R
g | 70.487 - j43.400 7.968 - j6.981 57.501 + 50.382 49.533 + j43.400 ate
82.760 2 -31.629° | 10.594 4 221.22° 76.451 2 41.225 65.857 2 41.225° ais
105.39m + j92.341m
76.8991 +j124.86m |-28.490m + j32.516m |-28.490m +j32.516m
146.64: 4 58.371° 43.232m 4 131.22° | 43.232m 2 131.22° | 140.12m 4 41.225°
O- j564.38
O+ j245.04 O-j1.7684k 4+70+j0 Ohms
564.38 2 -90° 245.04 290° 1.7684k 2 -90° 470 20°
Law Law
E=Iz E=IzZ
Amps
Now, let's turn to SPICE for a computer verification of our
work:
more "dummy" voltage sources to
act as current measurement points
in the SPICE analysis (all set to 0
volts).
120 V
60 Hz
Example series-parallel R, L, C SPICE circuit.
ac series-parallel r-l-c circuit
v1 10 ac 120 sin
vit 1 2 ac 0
vilc 3 4 ac 0
vir 3 6 ac 0
cl 2 3 4.7u
Ll 4 5 650m
c2 5 0 1.5u
r 6 0 470
.ac Lin 1 60 60
.print ac v(2,3) ) i(vit) ip(vit)
.print ac v(4,5) vp(4,5) i(vilc) ip(vilc)
.print ac v(5,0) ) i(vilc) ip(vilc)
.print ac v(6,0) ) i(vir) ip(vir)
.end
freq v(2,3) vp(2,3) i(vit)
Cl
6.000E+01 8.2/76E+01 -3.163E+01 1.466E-01
ip(vit)
5.837E+01
freq v(4,5) vp (4,5) i(vilc) ip(vilc)
L
6.Q00E+01 1.059E+01 -1.388E+02 4.323E-02 1.312E+02
freq v(5) vp (5) i(vilc) ip(vilc)
seers 7.645E+01 4.122E+01 4.323E-02 1.312E+02
freq v(6) vp (6) i(vir) ip(vir)
S aoeeei 6.586E+01 4.122E+01 1.401E-01 4.122E+01
Each line of the SPICE output listing gives the voltage,
voltage phase angle, current, and current phase angle for Cy,
L, C5, and R, in that order. As you can see, these figures do
concur with our hand-calculated figures in the circuit analysis
table.
As daunting a task as series-parallel AC circuit analysis may
appear, it must be emphasized that there is nothing really
new going on here besides the use of complex numbers.
Ohm's Law (in its new form of E=IZ) still holds true, as do the
voltage and current Laws of Kirchhoff. While there is more
potential for human error in carrying out the necessary
complex number calculations, the basic principles and
techniques of series-parallel circuit reduction are exactly the
same.
e REVIEW:
e Analysis of series-parallel AC circuits is much the same as
series-parallel DC circuits. The only substantive difference
is that all figures and calculations are in complex (not
scalar) form.
e It is important to remember that before series-parallel
reduction (simplification) can begin, you must determine
the impedance (Z) of every resistor, inductor, and
capacitor. That way, all component values will be
expressed in common terms (Z) instead of an
incompatible mix of resistance (R), inductance (L), and
Capacitance (C).
Susceptance and Admittance
In the study of DC circuits, the student of electricity comes
across a term meaning the opposite of resistance:
conductance. It is a useful term when exploring the
mathematical formula for parallel resistances: Roaratie! = 1 /
(1/R, + 1/R> +... 1/R,,). Unlike resistance, which diminishes
as more parallel components are included in the circuit,
conductance simply adds. Mathematically, conductance is the
reciprocal of resistance, and each 1/R term in the “parallel
resistance formula” is actually a conductance.
Whereas the term “resistance” denotes the amount of
opposition to flowing electrons in a circuit, “conductance”
represents the ease of which electrons may flow. Resistance is
the measure of how much a circuit resists current, while
conductance is the measure of how much a circuit conducts
current. Conductance used to be measured in the unit of
mhos, or “ohms” spelled backward. Now, the proper unit of
measurement is Siemens. When symbolized ina
mathematical formula, the proper letter to use for
conductance is “G”.
Reactive components such as inductors and capacitors
oppose the flow of electrons with respect to time, rather than
with a constant, unchanging friction as resistors do. We call
this time-based opposition, reactance, and like resistance we
also measure it in the unit of ohms.
As conductance is the complement of resistance, there is also
a complementary expression of reactance, called
susceptance. Mathematically, it is equal to 1/X, the reciprocal
of reactance. Like conductance, it used to be measured in the
unit of mhos, but now is measured in Siemens. Its
mathematical symbol is “B”, unfortunately the same symbol
used to represent magnetic flux density.
The terms “reactance” and “susceptance” have a certain
linguistic logic to them, just like resistance and conductance.
While reactance is the measure of how much a circuit reacts
against change in current over time, susceptance is the
measure of how much a circuit is susceptib/e to conducting a
changing current.
If one were tasked with determining the total effect of several
parallel-connected, pure reactances, one could convert each
reactance (X) to a susceptance (B), then add susceptances
rather than diminish reactances: Xparalie: = 1/(1/X1 + 1/X2 + ..
_ 1/X,). Like conductances (G), susceptances (B) add in
parallel and diminish in series. Also like conductance,
susceptance is a scalar quantity.
When resistive and reactive components are interconnected,
their combined effects can no longer be analyzed with scalar
quantities of resistance (R) and reactance (X). Likewise,
figures of conductance (G) and susceptance (B) are most
useful in circuits where the two types of opposition are not
mixed, i.e. either a purely resistive (conductive) circuit, ora
purely reactive (Susceptive) circuit. In order to express and
quantify the effects of mixed resistive and reactive
components, we had to have a new term: impedance,
measured in ohms and symbolized by the letter “Z”.
To be consistent, we need a complementary measure
representing the reciprocal of impedance. The name for this
measure is admittance. Admittance is measured in (guess
what?) the unit of Siemens, and its symbol is “Y”. Like
impedance, admittance is a complex quantity rather than
scalar. Again, we see a certain logic to the naming of this new
term: while impedance is a measure of how much alternating
current is /mpeded in a circuit, admittance is a measure of
how much current is admitted.
Given a scientific calculator capable of handling complex
number arithmetic in both polar and rectangular forms, you
may never have to work with figures of susceptance (B) or
admittance (Y). Be aware, though, of their existence and their
meanings.
Summary
With the notable exception of calculations for power (P), all
AC circuit calculations are based on the same general
principles as calculations for DC circuits. The only significant
difference is that fact that AC calculations use complex
quantities while DC calculations use scalar quantities. Ohm's
Law, Kirchhoff's Laws, and even the network theorems
learned in DC still hold true for AC when voltage, current, and
impedance are all expressed with complex numbers. The
same troubleshooting strategies applied toward DC circuits
also hold for AC, although AC can certainly be more difficult
to work with due to phase angles which aren't registered by a
handheld multimeter.
Power is another subject altogether, and will be covered in its
own chapter in this book. Because power in a reactive circuit
is both absorbed and released -- not just dissipated as it is
with resistors -- its mathematical handling requires a more
direct application of trigonometry to solve.
When faced with analyzing an AC circuit, the first step in
analysis is to convert all resistor, inductor, and capacitor
component values into impedances (Z), based on the
frequency of the power source. After that, proceed with the
same steps and strategies learned for analyzing DC circuits,
using the “new” form of Ohm's Law: E=1Z ; |I=E/Z ; and Z=E/|
Remember that only the calculated figures expressed in polar
form apply directly to empirical measurements of voltage and
current. Rectangular notation is merely a useful tool for us to
add and subtract complex quantities together. Polar notation,
where the magnitude (length of vector) directly relates to the
magnitude of the voltage or current measured, and the angle
directly relates to the phase shift in degrees, is the most
practical way to express complex quantities for circuit
analysis.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Jason Starck (June 2000): HTML document formatting, which
led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—| | 4/l—
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 6
RESONANCE
e An electric pendulum
e Simple series resonance
e Applications of resonance
e Resonance in series-parallel circuits
e Q and bandwidth of a resonant circuit
o Series resonant circuits
o Parallel resonant circuits
Contributors
An electric pendulum
Capacitors store energy in the form of an electric field, and
electrically manifest that stored energy as a potential: static
voltage. Inductors store energy in the form of a magnetic
field, and electrically manifest that stored energy asa
kinetic motion of electrons: current. Capacitors and
inductors are flip-sides of the same reactive coin, storing
and releasing energy in complementary modes. When these
two types of reactive components are directly connected
together, their complementary tendencies to store energy
will produce an unusual result.
If either the capacitor or inductor starts out in a charged
state, the two components will exchange energy between
them, back and forth, creating their own AC voltage and
current cycles. If we assume that both components are
subjected to a sudden application of voltage (say, from a
momentarily connected battery), the capacitor will very
quickly charge and the inductor will oppose change in
current, leaving the capacitor in the charged state and the
inductor in the discharged state: (Figure below)
Battery mom entarily
connected to start the cycle e=— e
=a id - a Time —>
L cL ;
capacitor charged: voltage at (+) peak
inductor discharged: Zero current
Capacitor charged: voltage at (+) peak, inductor
discharged: zero current.
The capacitor will begin to discharge, its voltage decreasing.
Meanwhile, the inductor will begin to build up a “charge” in
the form of a magnetic field as current increases in the
circuit: (Figure below)
¥ +
f Time —>
_
capacitor discharging: voltage decreasing
inductor charging: current increasing
Capacitor discharging: voltage decreasing, Inductor
charging: current increasing.
The inductor, still charging, will keep electrons flowing in the
circuit until the capacitor has been completely discharged,
leaving zero voltage across it: (Figure below)
e=-°""" Pa
is---- D
=—— rs
Pf Time —>
en
capacitor fully discharged: zero voltage
inductor fully charged: maximum current
Capacitor fully discharged: zero voltage, inductor fully
charged: maximum current.
The inductor will maintain current flow even with no voltage
applied. In fact, it will generate a voltage (like a battery) in
order to keep current in the same direction. The capacitor,
being the recipient of this current, will begin to accumulate
a charge in the opposite polarity as before: (Figure below)
Time —>
_
capacitor charging: voltage increasing (in opposite polarity)
inductor discharging: current decreasing
Capacitor charging: voltage increasing (in opposite polarity),
inductor discharging: current decreasing.
When the inductor is finally depleted of its energy reserve
and the electrons come to a halt, the capacitor will have
reached full (voltage) charge in the opposite polarity as
when it started: (Figure below)
Time —>
capacitor fully charged: voltage at (-) peak
inductor fully discharged: zero current
Capacitor fully charged: voltage at (-) peak, inductor fully
discharged: zero current.
Now we're at a condition very similar to where we started:
the capacitor at full charge and zero current in the circuit.
The capacitor, as before, will begin to discharge through the
inductor, causing an increase in current (in the opposite
direction as before) and a decrease in voltage as it depletes
its own energy reserve: (Figure below)
Time —>
capacitor discharging: voltage decreasing
inductor charging: current increasing
Capacitor discharging: voltage decreasing, inductor
Charging: current increasing.
Eventually the capacitor will discharge to zero volts, leaving
the inductor fully charged with full current through it:
(Figure below)
Time —>
capacitor fully discharged: zero voltage
inductor fully charged: current at (-) peak
Capacitor fully discharged: zero voltage, inductor fully
charged: current at (-) peak.
The inductor, desiring to maintain current in the same
direction, will act like a source again, generating a voltage
like a battery to continue the flow. In doing so, the capacitor
will begin to charge up and the current will decrease in
magnitude: (Figure below)
Time —>
capacitor charging: voltage increasing
inductor discharging: current decreasing
Capacitor charging: voltage increasing, inductor
discharging: current decreasing.
Eventually the capacitor will become fully charged again as
the inductor expends all of its energy reserves trying to
maintain current. The voltage will once again be at its
positive peak and the current at zero. This completes one
full cycle of the energy exchange between the capacitor and
inductor: (Figure below)
Time —>
capacitor fully charged: voltage at (+) peak
inductor fully discharged: zero current
Capacitor fully charged: voltage at (+) peak, inductor fully
discharged: zero current.
This oscillation will continue with steadily decreasing
amplitude due to power losses from stray resistances in the
circuit, until the process stops altogether. Overall, this
behavior is akin to that of a pendulum: as the pendulum
mass swings back and forth, there is a transformation of
energy taking place from kinetic (motion) to potential
(height), in a similar fashion to the way energy is transferred
in the capacitor/inductor circuit back and forth in the
alternating forms of current (kinetic motion of electrons) and
voltage (potential electric energy).
At the peak height of each swing of a pendulum, the mass
briefly stops and switches directions. It is at this point that
potential energy (height) is at a maximum and kinetic
energy (motion) is at zero. As the mass swings back the
other way, it passes quickly through a point where the string
iS pointed straight down. At this point, potential energy
(height) is at zero and kinetic energy (motion) is at
maximum. Like the circuit, a pendulum's back-and-forth
oscillation will continue with a steadily dampened
amplitude, the result of air friction (resistance) dissipating
energy. Also like the circuit, the pendulum's position and
velocity measurements trace two sine waves (90 degrees
out of phase) over time: (Figure below)
maximum potential energy,
zero kinetic energy
mass
—_->-
zero potential energy,
maximum kinetic energy
potential energy = ——
kinetic energy = ~----
Pendelum transfers energy between kinetic and potential
energy as it swings low to high.
In physics, this kind of natural sine-wave oscillation for a
mechanical system is called Simple Harmonic Motion (often
abbreviated as “SHM”). The same underlying principles
govern both the oscillation of a capacitor/inductor circuit
and the action of a pendulum, hence the similarity in effect.
It is an interesting property of any pendulum that its
periodic time is governed by the length of the string holding
the mass, and not the weight of the mass itself. That is why a
pendulum will keep swinging at the same frequency as the
oscillations decrease in amplitude. The oscillation rate is
independent of the amount of energy stored in it.
The same is true for the capacitor/inductor circuit. The rate
of oscillation is strictly dependent on the sizes of the
Capacitor and inductor, not on the amount of voltage (or
Current) at each respective peak in the waves. The ability for
such a circuit to store energy in the form of oscillating
voltage and current has earned it the name tank circuit. Its
property of maintaining a single, natural frequency
regardless of how much or little energy is actually being
stored in it gives it special significance in electric circuit
design.
However, this tendency to oscillate, or resonate, at a
particular frequency is not limited to circuits exclusively
designed for that purpose. In fact, nearly any AC circuit with
a combination of capacitance and inductance (commonly
called an “LC circuit”) will tend to manifest unusual effects
when the AC power source frequency approaches that
natural frequency. This is true regardless of the circuit's
intended purpose.
If the power supply frequency for a circuit exactly matches
the natural frequency of the circuit's LC combination, the
circuit is said to be in a state of resonance. The unusual
effects will reach maximum in this condition of resonance.
For this reason, we need to be able to predict what the
resonant frequency will be for various combinations of L and
C, and be aware of what the effects of resonance are.
e REVIEW:
e A capacitor and inductor directly connected together
form something called a tank circuit, which oscillates (or
resonates) at one particular frequency. At that
frequency, energy is alternately shuffled between the
capacitor and the inductor in the form of alternating
voltage and current 90 degrees out of phase with each
other.
e When the power supply frequency for an AC circuit
exactly matches that circuit's natural oscillation
frequency as set by the Land C components, a condition
of resonance will have been reached.
resonance
A condition of resonance will be experienced in a tank
circuit (Figure below) when the reactances of the capacitor
and inductor are equal to each other. Because inductive
reactance increases with increasing frequency and
Capacitive reactance decreases with increasing frequency,
there will only be one frequency where these two reactances
will be equal.
100 mH
Simple parallel resonant circuit (tank circuit).
In the above circuit, we have a 10 uF capacitor and a 100
MH inductor. Since we know the equations for determining
the reactance of each at a given frequency, and we're
looking for that point where the two reactances are equal to
each other, we can set the two reactance formulae equal to
each other and solve for frequency algebraically:
L
27tC
X_ = 2ntLh Xc=
. .. Setting the two equal to each other,
representing a condition of equal reactance
(resonance)...
L
27tC
2mtL =
Multiplying both sides by f eliminates the f
term in the denominator of the fraction .
L
27C
2nfL =
Dividing both sides by 2nL leaves f by itself
on the left-hand side of the equation . . .
2 L
272nLC
Taking the square root of both sides of the
equation leaves f by itself on the left side. . .
e-_ Vi
\/2n2nLC
... Simplifying. . .
So there we have it: a formula to tell us the resonant
frequency of a tank circuit, given the values of inductance
(L) in Henrys and capacitance (C) in Farads. Plugging in the
values of Land Cin our example circuit, we arrive ata
resonant frequency of 159.155 Hz.
What happens at resonance is quite interesting. With
Capacitive and inductive reactances equal to each other, the
total impedance increases to infinity, meaning that the tank
circuit draws no current from the AC power source! We can
calculate the individual impedances of the 10 uF capacitor
and the 100 mH inductor and work through the parallel
impedance formula to demonstrate this mathematically:
X, = 2nfL
X, = (2)(%)( 159.155 Hz)( 100 mH)
X, = 100 2
l
Xn=
© 2RfC
Xc= l
(2)(1)(159.155 Hz)(10 LF)
X,.= 1002
As you might have guessed, | chose these component values
to give resonance impedances that were easy to work with
(100 Q even). Now, we use the parallel impedance formula
to see what happens to total Z:
l
Z parallel = 1 1
Z. ‘ Ze
l
Zoarallel =
parallel i : i
100 Q 7 90° 100 Q Z -90°
z _ l
parallel — "pia AA Pan
0.01 27-90" + 0.01 290
Zraratea= —- Undefined!
0
We can't divide any number by zero and arrive at a
meaningful result, but we can say that the result approaches
a value of infinity as the two parallel impedances get closer
to each other. What this means in practical terms is that, the
total impedance of a tank circuit is infinite (behaving as an
open circuit) at resonance. We can plot the consequences of
this over a wide power supply frequency range with a short
SPICE simulation: (Figure below)
1
lOuF L,3100mH
Resonant circuit sutitable for SPICE simulation.
freq i(vl) 3.162E-04 1.000E-03 3.162E-03
1.0E-02
.OOOE+02 9.632E-03
.053E+02 8.506E-03 .
eS ee Ket
105E+02 7.455E-03 . ; *
1.158E+02 6.470E-03 . *
1.211E+02 5.542E-03 . 4
1.263E+02 4.663E-03 . _*
1.316E+02 3.828E-03 .
1.368E+02 3.033E-03
1.421E+02 2.271E-03
1.474E+02 1.540E-03
1\526E402- 8 7373E-04 -,
1.579E+02 1.590E-04 .
1.632E+02 4.969E-04 .
1.684E+02 1.132E-03
1.737E+02 1.749E-03
1.789E+02 2.350E-03
1.842E+02 2.934E-03
1.895E+02 3.505E-03
1.947E+02 4.063E-03
2.000E+02 4.609E-03
tank circuit frequency sweep
vl 10 ac 1 sin
cl 10 10u
* rbogus is necessary to eliminate a direct loop
* between vl and 11, which SPICE can't handle
rbogus 1 2 le-12
Ll 2 0 100m
.ac Lin 20 100 200
.plot ac i(vl)
.end
The 1 pico-ohm (1 pQ) resistor is placed in this SPICE
analysis to overcome a limitation of SPICE: namely, that it
cannot analyze a circuit containing a direct inductor-voltage
source loop. (Figure below) A very low resistance value was
chosen so as to have minimal effect on circuit behavior.
This SPICE simulation plots circuit current over a frequency
range of 100 to 200 Hz in twenty even steps (100 and 200
Hz inclusive). Current magnitude on the graph increases
from left to right, while frequency increases from top to
bottom. The current in this circuit takes a sharp dip around
the analysis point of 157.9 Hz, which is the closest analysis
point to our predicted resonance frequency of 159.155 Hz. It
is at this point that total current from the power source falls
to zero.
The plot above is produced from the above spice circuit file (
* cir), the command (.plot) in the last line producing the text
plot on any printer or terminal. A better looking plot is
produced by the “nutmeg” graphical post-processor, part of
the spice package. The above spice ( *.cir) does not require
the plot (.plot) command, though it does no harm. The
following commands produce the plot below: (Figure below)
Spice -b -r resonant.raw resonant.cir
( -b batch mode, -r raw file, input is resonant.cir)
nutmeg resonant. raw
From the nutmeg prompt:
>setplot acl (setplot {enter} for list of plots)
>display (for list of signals)
>plot mag(vl#branch)
(magnitude of complex current vector
vl#branch)
mA — mag(v1l#branch)
10,0 grrsesseeseenseessetsnestesen pinnae :
0,0° = =
100,0 1500 200.0
frequency Hz
Nutmeg produces plot of current I(v1) for parallel resonant
circuit.
Incidentally, the graph output produced by this SPICE
computer analysis is more generally known as a Bode plot.
Such graphs plot amplitude or phase shift on one axis and
frequency on the other. The steepness of a Bode plot curve
characterizes a circuit's “frequency response,” or how
sensitive it is to changes in frequency.
e REVIEW:
e Resonance occurs when capacitive and inductive
reactances are equal to each other.
e For a tank circuit with no resistance (R), resonant
frequency can be calculated with the following formula:
f l
resonant — —
77 /
e The total impedance of a parallel LC circuit approaches
infinity as the power supply frequency approaches
resonance.
e A Bode plotis a graph plotting waveform amplitude or
phase on one axis and frequency on the other.
Simple series resonance
A similar effect happens in series inductive/capacitive
circuits. (Figure below) When a state of resonance is reached
(capacitive and inductive reactances equal), the two
impedances cancel each other out and the total impedance
drops to zero!
10 LF
100 mH
Simple series resonant circuit.
At 159.155 Hz:
Z, =0+jl00Q Z-=0-jlo0Q
Zseries = ZL + Zc
Zeeries = (0 + {100 22) + (0 - j100 2)
Z 0
With the total series impedance equal to 0 Q at the resonant
frequency of 159.155 Hz, the result is a short circuit across
the AC power source at resonance. In the circuit drawn
above, this would not be good. I'll add a small resistor
(Figure below) in series along with the capacitor and the
inductor to keep the maximum circuit current somewhat
limited, and perform another SPICE analysis over the same
range of frequencies: (Figure below)
10 LF
100 mH
Series resonant circuit suitable for SPICE.
series lc circuit
vl 10 ac 1 sin
rl 121
cl 2 3 10u
11 3 0 100m
.ac Lin 20 100 200
.plot ac i(vl)
.end
mA — mag(vi#branch)
frequency Hz
Series resonant circuit plot of current I(v1).
As before, circuit current amplitude increases from bottom to
top, while frequency increases from left to right. (Figure
above) The peak is still seen to be at the plotted frequency
point of 157.9 Hz, the closest analyzed point to our
predicted resonance point of 159.155 Hz. This would
suggest that our resonant frequency formula holds as true
for simple series LC circuits as it does for simple parallel LC
circuits, which is the case:
f l
resonant — — =
2m \V LC
A word of caution is in order with series LC resonant circuits:
because of the high currents which may be present in a
series LC circuit at resonance, it is possible to produce
dangerously high voltage drops across the capacitor and the
inductor, as each component possesses significant
impedance. We can edit the SPICE netlist in the above
example to include a plot of voltage across the capacitor
and inductor to demonstrate what happens: (Figure below)
series lc circuit
vl 10 ac 1 sin
rl 121
cl 2 3 10u
l1 3 0 100m
.ac Lin 20 100 200
.plot ac i(vl) v(2,3) v(3)
.end
Units — vm(3) —vm(2,3)
— 100*mag(v1#branch) 4
100,0 150,0 200,,0
frequency Hz
Plot of Vc=V(2,3) 70 V peak, V,=v(3) 70 V peak,
/=1(V1# branch) 0.532 A peak
According to SPICE, voltage across the capacitor and
inductor reach a peak somewhere around 70 volts! This is
quite impressive for a power supply that only generates 1
volt. Needless to say, caution is in order when
experimenting with circuits such as this. This SPICE voltage
is lower than the expected value due to the small (20)
number of steps in the AC analysis statement (.ac lin 20 100
200). What is the expected value?
Given: f, = 159.155 Hz, L = 100mH, R = 1
X, = 2nflL = 2m(159.155) (100mH)=j 1000
Xo = 1/(2nfC) = 1/(2m(159.155) (10UF)) = -j1000
Z = 1 +j100 -j100 = 190
I=V/Z = (1 V)/(1 9) = 1A
, = IZ = (1 A)(j100) = j100 V
Vc = IZ = (1 A)(-j100) = -j100 V
Ve = IR = (1 A)(1)= 1 V
Vtotal = Vi + Vc + Vp
Vtotal = J100 -j3100 +1 =1V
The expected values for capacitor and inductor voltage are
100 V. This voltage will stress these components to that
level and they must be rated accordingly. However, these
voltages are out of phase and cancel yielding a total voltage
across all three components of only 1 V, the applied voltage.
The ratio of the capacitor (or inductor) voltage to the
applied voltage is the “Q” factor.
Q = VL/Vpa = Vc/Vp
e REVIEW:
e The total impedance of a series LC circuit approaches
zero as the power supply frequency approaches
resonance.
e The same formula for determining resonant frequency in
a simple tank circuit applies to simple series circuits as
well.
Extremely high voltages can be formed across the
individual components of series LC circuits at resonance,
due to high current flows and substantial individual
component impedances.
Applications of resonance
So far, the phenomenon of resonance appears to be a
useless curiosity, or at most a nuisance to be avoided
(especially if series resonance makes for a short-circuit
across our AC voltage source!). However, this is not the case.
Resonance is a very valuable property of reactive AC
circuits, employed in a variety of applications.
One use for resonance is to establish a condition of stable
frequency in circuits designed to produce AC signals.
Usually, a parallel (tank) circuit is used for this purpose, with
the capacitor and inductor directly connected together,
exchanging energy between each other. Just as a pendulum
can be used to stabilize the frequency of a clock
mechanism's oscillations, so can a tank circuit be used to
stabilize the electrical frequency of an AC oscillator circuit.
As was noted before, the frequency set by the tank circuit is
solely dependent upon the values of L and C, and not on the
magnitudes of voltage or current present in the oscillations:
(Figure below)
the natural frequency
of the "tank circuit”
helps to stabilize
oscillations
... tothe rest of
the "oscillator"
circuit
Resonant circuit serves as stable frequency source.
Another use for resonance is in applications where the
effects of greatly increased or decreased impedance at a
particular frequency is desired. A resonant circuit can be
used to “block” (present high impedance toward) a
frequency or range of frequencies, thus acting as a sort of
frequency “filter” to strain certain frequencies out of a mix
of others. In fact, these particular circuits are called filters,
and their design constitutes a discipline of study all by itself:
(Figure below)
Tank circuit presents a
high impedance to a narrow
range of frequencies, blocking
them from getting to the load
(v) AC source of.
mixed frequencies
load
Resonant circuit serves as filter.
In essence, this is how analog radio receiver tuner circuits
work to filter, or select, one station frequency out of the mix
of different radio station frequency signals intercepted by
the antenna.
e REVIEW:
e Resonance can be employed to maintain AC circuit
oscillations at a constant frequency, just as a pendulum
can be used to maintain constant oscillation speed ina
timekeeping mechanism.
e Resonance can be exploited for its impedance
properties: either dramatically increasing or decreasing
impedance for certain frequencies. Circuits designed to
screen certain frequencies out of a mix of different
frequencies are called fi/ters.
Resonance in series-parallel circuits
In simple reactive circuits with little or no resistance, the
effects of radically altered impedance will manifest at the
resonance frequency predicted by the equation given
earlier. In a parallel (tank) LC circuit, this means infinite
impedance at resonance. In a series LC circuit, it means zero
impedance at resonance:
f l
resonant — —
2% VE LE
However, as soon as significant levels of resistance are
introduced into most LC circuits, this simple calculation for
resonance becomes invalid. We'll take a look at several LC
circuits with added resistance, using the same values for
Capacitance and inductance as before: 10 uF and 100 mH,
respectively. According to our simple equation, the resonant
frequency should be 159.155 Hz. Watch, though, where
current reaches maximum or minimum in the following
SPICE analyses:
Parallel LC with resistance in series with L
lOWF =L. 100 mH
Parallel LC circuit with resistance in series with L.
resonant circuit
vl 10 ac 1 sin
cl 10 10u
rl 12 100
l1 2 0 100m
.ac Lin 20 100 200
.plot ac i(vl)
end
mA — mag(vl#branch)
9,0: a a al nnn ene ene :
8 ba SUN OU ROR E ORR O OPER HERR AHHH HNN H HON EES <
os = =
frequency Hz
Resistance in series with L produces minimum current at
136.8 Hz instead of calculated 159.2 Hz
Minimum current at 136.8 Hz instead of 159.2 Hz!
Parallel LC with resistance in series with C
Parallel LC with resistance in serieis with C.
Here, an extra resistor (Rpogus) (Figure below)is necessary to
prevent SPICE from encountering trouble in analysis. SPICE
can't handle an inductor connected directly in parallel with
any voltage source or any other inductor, so the addition of
a series resistor is necessary to “break up” the voltage
source/inductor loop that would otherwise be formed. This
resistor is chosen to be a very low value for minimum impact
on the circuit's behavior.
resonant circuit
vl 10 ac 1 sin
rl 1 2 100
cl 2 0 10u
rbogus 1 3 le-12
11 3 0 100m
.ac Lin 20 100 400
.plot ac i(vl)
end
Minimum current at roughly 180 Hz instead of 159.2 Hz!
mA — mag(vl#branch)
100,0 200,0 300,0 400,0
frequency Hz
Resistance in series with C shifts minimum current from
calculated 159.2 Hz to roughly 180 Hz.
Switching our attention to series LC circuits, (Figure below)
we experiment with placing significant resistances in
parallel with either L or C. In the following series circuit
examples, a 1 Q resistor (Rj) is placed in series with the
inductor and capacitor to limit total current at resonance.
The “extra” resistance inserted to influence resonant
frequency effects is the 100 Q resistor, Ro. The results are
shown in (Figure below).
Series LC with resistance in parallel with L
Series LC resonant circuit with resistance in parallel with L.
resonant circuit
vl 10 ac 1 sin
rl 121
cl 2 3 10u
11 3 0 100m
r2 3 0 100
.ac Lin 20 100 400
.plot ac i(vl)
.end
Maximum current at roughly 178.9 Hz instead of 159.2 Hz!
100,0 200,0 300,0 400,0
frequency Hz
Series resonant circuit with resistance in parallel with L
shifts maximum current from 159.2 Hz to roughly 180 Hz.
And finally, a series LC circuit with the significant resistance
in parallel with the capacitor. (Figure below) The shifted
resonance is shown in (Figure below)
Series LC with resistance in parallel with C
Series LC resonant circuit with rsistance in parallel with C.
resonant circuit
vl 10 ac 1 sin
rl 121
cl 2 3 10u
r2 2 3 100
11 3 0 100m
.ac Lin 20 100 200
.plot ac i(vl)
.end
Maximum current at 136.8 Hz instead of 159.2 Hz!
mA — mag(vl#branch)
frequency Hz
Resistance in parallel with C in series resonant circuit shifts
curreent maximum from calculated 159.2 Hz to about 136.8
Hz.
The tendency for added resistance to skew the point at
which impedance reaches a maximum or minimum in an LC
circuit is called antiresonance. The astute observer will
notice a pattern between the four SPICE examples given
above, in terms of how resistance affects the resonant peak
of a circuit:
e Parallel (“tank”) LC circuit:
e Rin series with L: resonant frequency shifted down
e Rin series with C: resonant frequency shifted up
e Series LC circuit:
e Rin parallel with L: resonant frequency shifted up
e Rin parallel with C: resonant frequency shifted down
Again, this illustrates the complementary nature of
Capacitors and inductors: how resistance in series with one
creates an antiresonance effect equivalent to resistance in
parallel with the other. If you look even closer to the four
SPICE examples given, you'll see that the frequencies are
shifted by the same amount, and that the shape of the
complementary graphs are mirror-images of each other!
Antiresonance is an effect that resonant circuit designers
must be aware of. The equations for determining
antiresonance “shift” are complex, and will not be covered in
this brief lesson. It should suffice the beginning student of
electronics to understand that the effect exists, and what its
general tendencies are.
Added resistance in an LC circuit is no academic matter.
While it is possible to manufacture capacitors with negligible
unwanted resistances, inductors are typically plagued with
substantial amounts of resistance due to the long lengths of
wire used in their construction. What is more, the resistance
of wire tends to increase as frequency goes up, due toa
strange phenomenon known as the skin effect where AC
current tends to be excluded from travel through the very
center of a wire, thereby reducing the wire's effective cross-
sectional area. Thus, inductors not only have resistance, but
changing, frequency-dependent resistance at that.
As if the resistance of an inductor's wire weren't enough to
cause problems, we also have to contend with the “core
losses” of iron-core inductors, which manifest themselves as
added resistance in the circuit. Since iron is a conductor of
electricity as well as a conductor of magnetic flux, changing
flux produced by alternating current through the coil will
tend to induce electric currents in the core itself (eddy
currents). This effect can be thought of as though the iron
core of the transformer were a sort of secondary transformer
coil powering a resistive load: the less-than-perfect
conductivity of the iron metal. This effects can be minimized
with laminated cores, good core design and high-grade
materials, but never completely eliminated.
One notable exception to the rule of circuit resistance
causing a resonant frequency shift is the case of series
resistor-inductor-capacitor (“RLC”) circuits. So long as a//
components are connected in series with each other, the
resonant frequency of the circuit will be unaffected by the
resistance. (Figure below) The resulting plot is shown in
(Figure below).
Series LC with resistance in series
0
Series LC with resistance in series.
series rlc circuit
vl 10 ac 1 sin
rl 1 2 100
cl 2 3 10u
11 3 0 100m
.ac Lin 20 100 200
.plot ac i(vl)
end
Maximum current at 159.2 Hz once again!
frequency Hz
Resistance in series resonant circuit leaves current
maximum at calculated 159.2 Hz, broadening the curve.
Note that the peak of the current graph (Figure below) has
not changed from the earlier series LC circuit (the one with
the 1 O token resistance in it), even though the resistance is
now 100 times greater. The only thing that has changed is
the “sharpness” of the curve. Obviously, this circuit does not
resonate as strongly as one with less series resistance (it is
said to be “less selective”), but at least it has the same
natural frequency!
It is noteworthy that antiresonance has the effect of
dampening the oscillations of free-running LC circuits such
as tank circuits. In the beginning of this chapter we saw how
a Capacitor and inductor connected directly together would
act something like a pendulum, exchanging voltage and
current peaks just like a pendulum exchanges kinetic and
potential energy. In a perfect tank circuit (no resistance),
this oscillation would continue forever, just as a frictionless
pendulum would continue to swing at its resonant frequency
forever. But frictionless machines are difficult to find in the
real world, and so are lossless tank circuits. Energy lost
through resistance (or inductor core losses or radiated
electromagnetic waves or...) in a tank circuit will cause the
oscillations to decay in amplitude until they are no more. If
enough energy losses are present in a tank circuit, it will fail
to resonate at all.
Antiresonance's dampening effect is more than just a
curiosity: it can be used quite effectively to eliminate
unwanted oscillations in circuits containing stray
inductances and/or capacitances, as almost all circuits do.
Take note of the following L/R time delay circuit: (Figure
below)
switch
L/R time delay circuit
The idea of this circuit is simple: to “charge” the inductor
when the switch is closed. The rate of inductor charging will
be set by the ratio L/R, which is the time constant of the
circuit in seconds. However, if you were to build such a
circuit, you might find unexpected oscillations (AC) of
voltage across the inductor when the switch is closed.
(Figure below) Why is this? There's no capacitor in the
circuit, so how can we have resonant oscillation with just an
inductor, resistor, and battery?
ideal L/R voltage curve = ------
actual L/R voltage curve =
Inductor ringing due to resonance with stray capacitance.
All inductors contain a certain amount of stray capacitance
due to turn-to-turn and turn-to-core insulation gaps. Also,
the placement of circuit conductors may create stray
Capacitance. While clean circuit layout is important in
eliminating much of this stray capacitance, there will always
be some that you cannot eliminate. If this causes resonant
problems (unwanted AC oscillations), added resistance may
be a way to combat it. If resistor R is large enough, it will
cause a condition of antiresonance, dissipating enough
energy to prohibit the inductance and stray capacitance
from sustaining oscillations for very long.
Interestingly enough, the principle of employing resistance
to eliminate unwanted resonance is one frequently used in
the design of mechanical systems, where any moving object
with mass is a potential resonator. A very common
application of this is the use of shock absorbers in
automobiles. Without shock absorbers, cars would bounce
wildly at their resonant frequency after hitting any bump in
the road. The shock absorber's job is to introduce a strong
antiresonant effect by dissipating energy hydraulically (in
the same way that a resistor dissipates energy electrically).
REVIEW:
Added resistance to an LC circuit can cause a condition
known as antiresonance, where the peak impedance
effects happen at frequencies other than that which
gives equal capacitive and inductive reactances.
e Resistance inherent in real-world inductors can
contribute greatly to conditions of antiresonance. One
source of such resistance is the skin effect, caused by
the exclusion of AC current from the center of
conductors. Another source is that of core losses in iron-
core inductors.
e In a simple series LC circuit containing resistance (an
“RLC” circuit), resistance does not produce
antiresonance. Resonance still occurs when capacitive
and inductive reactances are equal.
Q_and bandwidth of a resonant circuit
The Q, quality factor, of a resonant circuit is a measure of
the “goodness” or quality of a resonant circuit. A higher
value for this figure of merit corresponds to a more narrow
bandwith, which is desirable in many applications. More
formally, Q is the ratio of power stored to power dissipated in
the circuit reactance and resistance, respectively:
Qs Pstored/Paissipated = I1°X/I?R
Q = X/R
where: X = Capacitive or Inductive reactance at
resonance
R = Series resistance.
This formula is applicable to series resonant circuits, and
also parallel resonant circuits if the resistance is in series
with the inductor. This is the case in practical applications,
as we are mostly concerned with the resistance of the
inductor limiting the Q. Note: Some text may show X and R
interchanged in the “Q” formula for a parallel resonant
circuit. This is correct for a large value of R in parallel with C
and L. Our formula is correct for a small R in series with L.
A practical application of “Q” is that voltage across L or Cin
a series resonant circuit is Q times total applied voltage. Ina
parallel resonant circuit, current through L or C is Q times
the total applied current.
Series resonant circuits
A series resonant circuit looks like a resistance at the
resonant frequency. (Figure below) Since the definition of
resonance is X,=Xc, the reactive components cancel,
leaving only the resistance to contribute to the impedance.
The impedance is also at a minimum at resonance. (Figure
below) Below the resonant frequency, the series resonant
circuit looks capacitive since the impedance of the capacitor
increases to a value greater than the decreasing inducitve
reactance, leaving a net capacitive value. Above resonance,
the inductive reactance increases, capacitive reactance
decreases, leaving a net inductive component.
mA — mag(v3#branch)
400 frequency Hz 10°3
At resonance the series resonant circuit appears purely
resistive. Below resonance it looks capacitive. Above
resonance it appears inductive.
Current is maximum at resonance, impedance at a
minumum. Current is set by the value of the resistance.
Above or below resonance, impedance increases.
Z Ohms — mag(v(L))/mag(v3#branch)
300.0
200.0
100,0
50,0
0,0 ———
100 10°3
frequency Hz
Impedance is at a minumum at resonance in a series
resonant circuit.
The resonant current peak may be changed by varying the
series resistor, which changes the Q. (Figure below) This also
affects the broadness of the curve. A low resistance, high Q
circuit has a narrow bandwidth, as compared to a high
resistance, low Q circuit. Bandwidth in terms of Q and
resonant frequency:
BW = f./Q
Where f, = resonant frquency
Q = quality factor
mA
frequency Hz
A high Q resonant circuit has a narrow bandwidth as
compared to a low Q
Bandwidth is measured between the 0.707 current
amplitude points. The 0.707 current points correspond to
the half power points since P = IR, (0.707)? = (0.5). (Figure
below)
0,0
100 A f=64 1000
df=355-291=64 frequency Hz
Bandwidth, Af is measured between the 70.7% amplitude
points of series resonant circuit.
BW = Af = f,-fi = f,/Q
Where f, = high band edge, f, = low band edge
f, = f. - Af/2
fh, f. + Af/2
Where f,. = center frequency (resonant frequency)
In Figure above, the 100% current point is 50 mA. The
70.7% level is 0.707(50 mA)=35.4 mA. The upper and lower
band edges read from the curve are 291 Hz for f, and 355 Hz
for f,. The bandwidth is 64 Hz, and the half power points are
+ 32 Hz of the center resonant frequency:
BW = Af = fy-f, = 355-291 = 64
f. = f. - Af/2 = 323-32 = 291
fr = fo + Af/2 = 323432 = 355
Since BW = f,/Q:
Q = f./BW = (323 Hz)/(64 Hz) =5
Parallel resonant circuits
A parallel resonant circuit is resistive at the resonant
frequency. (Figure below) At resonance X,=Xc¢, the reactive
components cancel. The impedance is maximum at
resonance. (Figure below) Below the resonant frequency, the
parallel resonant circuit looks inductive since the impedance
of the inductor is lower, drawing the larger proportion of
current. Above resonance, the capacitive reactance
decreases, drawing the larger current, thus, taking ona
Capacitive characteristic.
mA — mag(v3#branch)
30,0
20,0
10,0
0,0
100 1000
frequency Hz
A parallel resonant circuit is resistive at resonance,
inductive below resonance, Capacitive above resonance.
Impedance is maximum at resonance in a parallel resonant
circuit, but decreases above or below resonance. Voltage is
at a peak at resonance since voltage is proportional to
impedance (E=IZ). (Figure below)
— maglv(31)) /mag(v3#branch)
Z Ohms
600,0
4000
200,09
0,0
frequency Hz 1000
Parallel resonant circuit: Impedance peaks at resonance.
A low Q due to a high resistance in series with the inductor
produces a low peak on a broad response curve for a parallel
resonant circuit. (Figure below) conversely, a high Q is due
to a low resistance in series with the inductor. This produces
a higher peak in the narrower response curve. The high Q is
achieved by winding the inductor with larger diameter
(smaller gague), lower resistance wire.
frequency Hz
Parallel resonant response varies with Q.
The bandwidth of the parallel resonant response curve is
measured between the half power points. This corresponds
to the 70.7% voltage points since power is proportional to
E?. ((0.707 )?=0.50) Since voltage is proportional to
impedance, we may use the impedance curve. (Figure
below)
Units = 500 — ,7O07*500
2 Ohms = mag(v(31))/mag(v3#branch)
100 4f=62 1000
frequency Hz
Bandwidth, Af is measured between the 70.7% impedance
points of a parallel resonant circuit.
In Figure above, the 100% impedance point is 500 Q. The
70.7% level is 0.707 (500)=354 QO. The upper and lower
band edges read from the curve are 281 Hz for f; and 343 Hz
for f,. The bandwidth is 62 Hz, and the half power points are
+ 31 Hz of the center resonant frequency:
BW = Af = fy-f, = 343-281 = 62
f. = f. - Af/2 = 312-31 = 281
f, = fo + Af/2 = 312+31 = 343
Q = f./BW = (312 Hz)/(62 Hz) = 5
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] +4] l—
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 7
MIXED-FREQUENCY AC
SIGNALS
Introduction
Square wave signals
Other waveshapes
More on spectrum analysis
Circuit effects
Contributors
Introduction
In our study of AC circuits thus far, we've explored circuits
powered by a single-frequency sine voltage waveform. In
many applications of electronics, though, single-frequency
signals are the exception rather than the rule. Quite often
we may encounter circuits where multiple frequencies of
voltage coexist simultaneously. Also, circuit waveforms may
be something other than sine-wave shaped, in which case
we call them non-sinusoidal waveforms.
Additionally, we may encounter situations where DC is
mixed with AC: where a waveform is superimposed on a
steady (DC) signal. The result of such a mix is a signal
varying in intensity, but never changing polarity, or
changing polarity asymmetrically (spending more time
positive than negative, for example). Since DC does not
alternate as AC does, its “frequency” is said to be zero, and
any signal containing DC along with a signal of varying
intensity (AC) may be rightly called a mixed-frequency
signal as well. In any of these cases where there is a mix of
frequencies in the same circuit, analysis is more complex
than what we've seen up to this point.
Sometimes mixed-frequency voltage and current signals are
created accidentally. This may be the result of unintended
connections between circuits -- called coupling -- made
possible by stray capacitance and/or inductance between
the conductors of those circuits. A classic example of
coupling phenomenon is seen frequently in industry where
DC signal wiring is placed in close proximity to AC power
wiring. The nearby presence of high AC voltages and
currents may cause “foreign” voltages to be impressed upon
the length of the signal wiring. Stray capacitance formed by
the electrical insulation separating power conductors from
signal conductors may cause voltage (with respect to earth
ground) from the power conductors to be impressed upon
the signal conductors, while stray inductance formed by
parallel runs of wire in conduit may cause current from the
power conductors to electromagnetically induce voltage
along the signal conductors. The result is a mix of DC and AC
at the signal load. The following schematic shows how an AC
“noise” source may “couple” to a DC circuit through mutual
inductance (Mctray) and capacitance (C.tray) along the length
of the conductors. (Figure below)
source bees
Moray Cc
Sos “stray
Lire ome Posie Lyi re
"Clean" DC voltage DC voltage + AC "noise”
Stray inductance and capacitance couple stray AC into
desired DC signal.
When stray AC voltages from a “noise” source mix with DC
signals conducted along signal wiring, the results are
usually undesirable. For this reason, power wiring and low-
level signal wiring should a/ways be routed through
separated, dedicated metal conduit, and signals should be
conducted via 2-conductor “twisted pair” cable rather than
through a single wire and ground connection: (Figure below)
e+
Shielded cable
pibeny
Shielded twisted pair minimized noise.
The grounded cable shield -- a wire braid or metal foil
wrapped around the two insulated conductors -- isolates
both conductors from electrostatic (capacitive) coupling by
blocking any external electric fields, while the parallel
proximity of the two conductors effectively cancels any
electromagnetic (mutually inductive) coupling because any
induced noise voltage will be approximately equal in
magnitude and opposite in phase along both conductors,
canceling each other at the receiving end for a net
(differential) noise voltage of almost zero. Polarity marks
placed near each inductive portion of signal conductor
length shows how the induced voltages are phased in such a
way as to cancel one another.
Coupling may also occur between two sets of conductors
carrying AC signals, in which case both signals may become
“mixed” with each other: (Figure below)
Signal B B+A
Coupling of AC signals between parallel conductors.
Coupling is but one example of how signals of different
frequencies may become mixed. Whether it be AC mixed
with DC, or two AC signals mixing with each other, signal
coupling via stray inductance and capacitance is usually
accidental and undesired. In other cases, mixed-frequency
signals are the result of intentional design or they may be an
intrinsic quality of a signal. It is generally quite easy to
create mixed-frequency signal sources. Perhaps the easiest
way is to simply connect voltage sources in series: (Figure
below)
\ 60 Hz ‘
AC+DC mixed-frequency
voltage AC voltage
= J 90 Hz A
|
Series connection of voltage sources mixes signals.
Some computer communications networks operate on the
principle of superimposing high-frequency voltage signals
along 60 Hz power-line conductors, so as to convey
computer data along existing lengths of power cabling. This
technique has been used for years in electric power
distribution networks to communicate load data along high-
voltage power lines. Certainly these are examples of mixed-
frequency AC voltages, under conditions that are
deliberately established.
In some cases, mixed-frequency signals may be produced by
a single voltage source. Such is the case with microphones,
which convert audio-frequency air pressure waves into
corresponding voltage waveforms. The particular mix of
frequencies in the voltage signal output by the microphone
is dependent on the sound being reproduced. If the sound
waves consist of a single, pure note or tone, the voltage
waveform will likewise be a sine wave at a single frequency.
If the sound wave is a chord or other harmony of several
notes, the resulting voltage waveform produced by the
microphone will consist of those frequencies mixed together.
Very few natural sounds consist of single, pure sine wave
vibrations but rather are a mix of different frequency
vibrations at different amplitudes.
Musical chords are produced by blending one frequency with
other frequencies of particular fractional multiples of the
first. However, investigating a little further, we find that
even a single piano note (produced by a plucked string)
consists of one predominant frequency mixed with several
other frequencies, each frequency a whole-number multiple
of the first (called harmonics, while the first frequency is
called the fundamenta/). An illustration of these terms is
shown in Table below with a fundamental frequency of 1000
Hz (an arbitrary figure chosen for this example).
For a “base” frequency of 1000 Hz:
requency (Hz)
000 1st harmonic, or fundamenta
000 2nd harmonic
000 Brd harmonic
000 ath harmonic
000 Bth harmonic
000 6th harmonic
000 7th harmonic
Sometimes the term “overtone” is used to describe the
harmonic frequency produced by a musical instrument. The
“first” overtone is the first harmonic frequency greater than
the fundamental. If we had an instrument producing the
entire range of harmonic frequencies shown in the table
above, the first overtone would be 2000 Hz (the 2nd
harmonic), while the second overtone would be 3000 Hz
(the 3rd harmonic), etc. However, this application of the
term “overtone” is specific to particular instruments.
It so happens that certain instruments are incapable of
producing certain types of harmonic frequencies. For
example, an instrument made from a tube that is open on
one end and closed on the other (such as a bottle, which
produces sound when air is blown across the opening) is
incapable of producing even-numbered harmonics. Such an
instrument set up to produce a fundamental frequency of
1000 Hz would also produce frequencies of 3000 Hz, 5000
Hz, 7000 Hz, etc, but would not produce 2000 Hz, 4000 Hz,
6000 Hz, or any other even-multiple frequencies of the
fundamental. As such, we would say that the first overtone
(the first frequency greater than the fundamental) in such
an instrument would be 3000 Hz (the 3rd harmonic), while
the second overtone would be 5000 Hz (the 5th harmonic),
and so on.
A pure sine wave (single frequency), being entirely devoid of
any harmonics, sounds very “flat” and “featureless” to the
human ear. Most musical instruments are incapable of
producing sounds this simple. What gives each instrument
its distinctive tone is the same phenomenon that gives each
person a distinctive voice: the unique blending of harmonic
waveforms with each fundamental note, described by the
physics of motion for each unique object producing the
sound.
Brass instruments do not possess the same “harmonic
content” as woodwind instruments, and neither produce the
Same harmonic content as stringed instruments. A
distinctive blend of frequencies is what gives a musical
instrument its characteristic tone. As anyone who has
played guitar can tell you, steel strings have a different
sound than nylon strings. Also, the tone produced by a
guitar string changes depending on where along its length it
is plucked. These differences in tone, as well, are a result of
different harmonic content produced by differences in the
mechanical vibrations of an instrument's parts. All these
instruments produce harmonic frequencies (whole-number
multiples of the fundamental frequency) when a single note
is played, but the relative amplitudes of those harmonic
frequencies are different for different instruments. In musical
terms, the measure of a tone's harmonic content is called
timbre or color.
Musical tones become even more complex when the
resonating element of an instrument is a two-dimensional
surface rather than a one-dimensional string. Instruments
based on the vibration of a string (guitar, piano, banjo, lute,
dulcimer, etc.) or of a column of air in a tube (trumpet, flute,
clarinet, tuba, pipe organ, etc.) tend to produce sounds
composed of a single frequency (the “fundamental”) and a
mix of harmonics. Instruments based on the vibration of a
flat plate (steel drums, and some types of bells), however,
produce a much broader range of frequencies, not limited to
whole-number multiples of the fundamental. The result is a
distinctive tone that some people find acoustically offensive.
As you Can see, music provides a rich field of study for
mixed frequencies and their effects. Later sections of this
chapter will refer to musical instruments as sources of
waveforms for analysis in more detail.
e REVIEW:
e A sinusoidal waveform is one shaped exactly like a sine
wave.
e A non-sinusoidal waveform can be anything from a
distorted sine-wave shape to something completely
different like a square wave.
Mixed-frequency waveforms can be accidently created,
purposely created, or simply exist out of necessity. Most
musical tones, for instance, are not composed of a single
frequency sine-wave, but are rich blends of different
frequencies.
e When multiple sine waveforms are mixed together (as Is
often the case in music), the lowest frequency sine-wave
is called the fundamental, and the other sine-waves
whose frequencies are whole-number multiples of the
fundamental wave are called harmonics.
e An overtone is a harmonic produced by a particular
device. The “first” overtone is the first frequency greater
than the fundamental, while the “second” overtone is
the next greater frequency produced. Successive
overtones may or may not correspond to incremental
harmonics, depending on the device producing the
mixed frequencies. Some devices and systems do not
permit the establishment of certain harmonics, and so
their overtones would only include some (not all)
harmonic frequencies.
Square wave Signals
It has been found that any repeating, non-sinusoidal
waveform can be equated to a combination of DC voltage,
sine waves, and/or cosine waves (sine waves with a 90
degree phase shift) at various amplitudes and frequencies.
This is true no matter how strange or convoluted the
waveform in question may be. So long as it repeats itself
regularly over time, it is reducible to this series of sinusoidal
waves. In particular, it has been found that square waves are
mathematically equivalent to the sum of a sine wave at that
same frequency, plus an infinite series of odd-multiple
frequency sine waves at diminishing amplitude:
| V (peak) repeating square wave at 50 Hz is equivalent to:
(2) (1 V peak sine wave at 50 Hz)
+ (2) (1/3 V peak sine wave at 150 Hz)
' (¢
T
' [2
T
“(¢
T
4 ...adinfinitum...
(1/5 V peak sine wave at 250 Hz)
a al
(1/7 V peak sine wave at 350 Hz)
““——_"
(1/9 V peak sine wave at 450 Hz)
i
This truth about waveforms at first may seem too strange to
believe. However, if a square wave is actually an infinite
series of sine wave harmonics added together, it stands to
reason that we should be able to prove this by adding
together several sine wave harmonics to produce a close
approximation of a square wave. This reasoning is not only
sound, but easily demonstrated with SPICE.
The circuit we'll be simulating is nothing more than several
sine wave AC voltage sources of the proper amplitudes and
frequencies connected together in series. We'll use SPICE to
plot the voltage waveforms across successive additions of
voltage sources, like this: (Figure below)
V,=L.27V
50Hz
V3=424mV
L50Hz
V5=255mV
250Hz
V=182mV
350Hz
V,=l4lmV
450Hz
plot voltage waveform
plot voltage waveform
plot voltage waveform
plot voltage waveform
plot voltage waveform
A square wave Is approximated by the sum of harmonics.
In this particular SPICE simulation, I've summed the Lst, 3rd,
5th, 7th, and 9th harmonic voltage sources in series for a
total of five AC voltage sources. The fundamental frequency
is 50 Hz and each harmonic is, of course, an integer multiple
of that frequency. The amplitude (voltage) figures are not
random numbers; rather, they have been arrived at through
the equations shown in the frequency series (the fraction 4/n
multiplied by 1, 1/3, 1/5, 1/7, etc. for each of the increasing
odd harmonics).
lst harmonic (50 Hz)
3rd harmonic
5th harmonic
7th harmonic
9th harmonic
Plot 1st harmonic
building a squarewave
v1 10 sin (0 1.27324 50 0 Q)
v3 2 1 sin (0 424.413m 150 0 0)
v5 3 2 sin (0 254.648m 250 0 0)
v7 4 3 sin (0 181.891m 350 0 0)
v9 5 4 sin (0 141.471m 450 0 0)
rl 5 0 10k
.tran 1m 20m
.plot tran v(1,0)
.plot tran v(2,0) Plot 1st +
.plot tran v(3,0) Plot 1st +
.plot tran v(4,0) Plot 1st +
.plot tran v(5,0) Plot 1st +
end
3rd harmonics
3rd + 5th harmonics
3rd + 5th + 7th harmonics
. + 9th harmonics
I'll narrate the analysis step by step from here, explaining
what it is we're looking at. In this first plot, we see the
fundamental-frequency sine-wave of 50 Hz by itself. It is
nothing but a pure sine shape, with no additional harmonic
content. This is the kind of waveform produced by an ideal
AC power source: (Figure below)
Pure 50 Hz sinewave.
Next, we see what happens when this clean and simple
waveform is combined with the third harmonic (three times
50 Hz, or 150 Hz). Suddenly, it doesn't look like a clean sine
wave any more: (Figure below)
Sum of Ist (50 Hz) and 3rd (150 Hz) harmonics
approximates a 50 Hz square wave.
The rise and fall times between positive and negative cycles
are much steeper now, and the crests of the wave are closer
to becoming flat like a squarewave. Watch what happens as
we add the next odd harmonic frequency: (Figure below)
Vv — v2,1> — vtL>
Sum of 1st, 3rd and 5th harmonics approximates square
wave.
The most noticeable change here is how the crests of the
wave have flattened even more. There are more several dips
and crests at each end of the wave, but those dips and
crests are smaller in amplitude than they were before. Watch
again as we add the next odd harmonic waveform to the
mix: (Figure below)
Sum of Ist, 3rd, 5th, and 7th harmonics approximates
Square wave.
Here we can see the wave becoming flatter at each peak.
Finally, adding the 9th harmonic, the fifth sine wave voltage
source in our circuit, we obtain this result: (Figure below)
v¢C2,1) — v¢1>
— v3,2)
— vt5,4)
Sum of Ist, 3rd, 5th, 7th and 9th harmonics approximates
Square wave.
The end result of adding the first five odd harmonic
waveforms together (all at the proper amplitudes, of course)
iS a close approximation of a square wave. The point in
doing this is to illustrate how we can build a square wave up
from multiple sine waves at different frequencies, to prove
that a pure square wave Is actually equivalent to a series of
sine waves. When a square wave AC voltage is applied toa
circuit with reactive components (capacitors and inductors),
those components react as if they were being exposed to
several sine wave voltages of different frequencies, which in
fact they are.
The fact that repeating, non-sinusoidal waves are equivalent
to a definite series of additive DC voltage, sine waves,
and/or cosine waves is a consequence of how waves work: a
fundamental property of all wave-related phenomena,
electrical or otherwise. The mathematical process of
reducing a non-sinusoidal wave into these constituent
frequencies is called Fourier analysis, the details of which
are well beyond the scope of this text. However, computer
algorithms have been created to perform this analysis at
high speeds on real waveforms, and its application in AC
power quality and signal analysis is widespread.
SPICE has the ability to sample a waveform and reduce it
into its constituent sine wave harmonics by way of a Fourier
Transform algorithm, outputting the frequency analysis as a
table of numbers. Let's try this on a square wave, which we
already know is composed of odd-harmonic sine waves:
Squarewave analysis netlist
v1 10 pulse (-1 10 .1m .1m 10m 20m)
rl 1 0 10k
.tran 1m 40m
.plot tran v(1,0)
.four 50 v(1,0)
.end
The pulse option in the netlist line describing voltage source
v1 instructs SPICE to simulate a square-shaped “pulse”
waveform, in this case one that is symmetrical (equal time
for each half-cycle) and has a peak amplitude of 1 volt. First
we'll plot the square wave to be analyzed: (Figure below)
Squarewave for SPICE Fourier analysis
Next, we'll print the Fourier analysis generated by SPICE for
this square wave:
fourier components of transient response v(1)
dc component = -2.439E-02
harmonic frequency fourier normalized phase
normalized
no (hz) component component (deg) phase
(deg)
1 5.000E+01 1.274E+00 1.000000 -2.195
0.000
2 1.000E+02 4.892E-02 0.038415 -94.390
-92.195
3 1.500E+02 4.253E-01 0.333987 -6.585
-4.390
4 2.Q000E+02 4.936E-02 0.038757 -98.780
-96.585
5 2.500E+02 2.562E-01 0.201179 -10.976
-8.780
6 3.000E+02 5.010E-02 0.039337 -103.171
-100.976
7 3.500E+02 1.841E-01 0.144549 -15.366
-13.171
8 4.000E+02 5.116E-02 0.040175 -107.561
- 105.366
9 4.500E+02 1.443E-01 0.113316 -19.756
-17.561
total harmonic distortion = 43.805747 percent
1.4
“fourier" using 0:3 ii’
1.2
1
0.8
0.6
0.4
0.2 -
Relative Amplitude
0 1 2 3 4 5 6 7 8 9
Harmonic Number
Plot of Fourier analysis esults.
Here, (Figure above) SPICE has broken the waveform down
into a spectrum of sinusoidal frequencies up to the ninth
harmonic, plus a small DC voltage labelled DC component. |
had to inform SPICE of the fundamental frequency (fora
square wave with a 20 millisecond period, this frequency is
50 Hz), so it knew how to classify the harmonics. Note how
small the figures are for all the even harmonics (2nd, 4th,
6th, 8th), and how the amplitudes of the odd harmonics
diminish (1st is largest, 9th is smallest).
This same technique of “Fourier Transformation” is often
used in computerized power instrumentation, sampling the
AC waveform(s) and determining the harmonic content
thereof. A common computer algorithm (sequence of
program steps to perform a task) for this is the Fast Fourier
Transform or FFT function. You need not be concerned with
exactly how these computer routines work, but be aware of
their existence and application.
This same mathematical technique used in SPICE to analyze
the harmonic content of waves can be applied to the
technical analysis of music: breaking up any particular
sound into its constituent sine-wave frequencies. In fact, you
may have already seen a device designed to do just that
without realizing what it was! A graphic equalizer is a piece
of high-fidelity stereo equipment that controls (and
sometimes displays) the nature of music's harmonic content.
Equipped with several knobs or slide levers, the equalizer is
able to selectively attenuate (reduce) the amplitude of
certain frequencies present in music, to “customize” the
sound for the listener's benefit. Typically, there will be a “bar
graph” display next to each control lever, displaying the
amplitude of each particular frequency. (Figure below)
Graphic Equalizer
Bargraph displays the
amplitude of each
frequency
——
Control levers set
= the attenuation factor
for each frequency
; in ——S-— FERRE
rT
~~
i
im)
7o——s— HERG
7 —i— IE
|
Hi-Fi audio graphic equalizer.
A device built strictly to display -- not control -- the
amplitudes of each frequency range for a mixed-frequency
Signal is typically called a spectrum analyzer. The design of
spectrum analyzers may be as simple as a set of “filter”
circuits (see the next chapter for details) designed to
separate the different frequencies from each other, or as
complex as a special-purpose digital computer running an
FFT algorithm to mathematically split the signal into its
harmonic components. Spectrum analyzers are often
designed to analyze extremely high-frequency signals, such
as those produced by radio transmitters and computer
network hardware. In that form, they often have an
appearance like that of an oscilloscope: (Figure below)
Spectrum Analyzer
amplitude
frequency —»
Spectrum analyzer shows amplitude as a function of
frequency.
Like an oscilloscope, the spectrum analyzer uses a CRT (ora
computer display mimicking a CRT) to display a plot of the
signal. Unlike an oscilloscope, this plot is amplitude over
frequency rather than amplitude over time. In essence, a
frequency analyzer gives the operator a Bode plot of the
signal: something an engineer might call a frequency-
domain rather than a time-domain analysis.
The term “domain” is mathematical: a sophisticated word to
describe the horizontal axis of a graph. Thus, an
oscilloscope's plot of amplitude (vertical) over time
(horizontal) is a “time-domain” analysis, whereas a spectrum
analyzer's plot of amplitude (vertical) over frequency
(horizontal) is a “frequency-domain” analysis. When we use
SPICE to plot signal amplitude (either voltage or current
amplitude) over a range of frequencies, we are performing
frequency-domain analysis.
Please take note of how the Fourier analysis from the last
SPICE simulation isn't “perfect.” Ideally, the amplitudes of
all the even harmonics should be absolutely zero, and so
should the DC component. Again, this is not so much a quirk
of SPICE as it is a property of waveforms in general. A
waveform of infinite duration (infinite number of cycles) can
be analyzed with absolute precision, but the less cycles
available to the computer for analysis, the less precise the
analysis. It is only when we have an equation describing a
waveform in its entirety that Fourier analysis can reduce it to
a definite series of sinusoidal waveforms. The fewer times
that a wave cycles, the less certain its frequency is. Taking
this concept to its logical extreme, a short pulse -- a
waveform that doesn't even complete a cycle -- actually has
no frequency, but rather acts as an infinite range of
frequencies. This principle is common to a// wave-based
phenomena, not just AC voltages and currents.
Suffice it to say that the number of cycles and the certainty
of a waveform's frequency component(s) are directly related.
We could improve the precision of our analysis here by
letting the wave oscillate on and on for many cycles, and the
result would be a spectrum analysis more consistent with
the ideal. In the following analysis, I've omitted the
waveform plot for brevity's sake -- its just a really long
Square wave:
Squarewave
v1 10 pulse (-1 10 .1m .1m 10m 20m)
rl 10 10k
.option Limpts=1001
.tran 1m 1
.plot tran v(1,0)
.four 50 v(1,0)
end
fourier components of transient response v(1)
dc component = 9.999E-03
harmonic
normalized
no
(deg)
di
0.000
2
88.182
3
-3.600
4
84.564
5
-7.200
6
80.946
7
-10.800
8
77.329
9
-14.399
14
1.2
1
0.8
0.6
0.4
Relative Amplitude
0.2;
0
frequency
(hz)
.QO00E+01
. Q00E+02
.500E+02
. QO00E+02
. 500E+02
.Q00E+02
. 500E+02
. QO00E+02
. 500E+02
"tj four" using 0:3 ii
2 3
fourier
component
Ls
1.
4,
273E+00
999E-02
238E-01
.997E-02
.536E-01
.994E-02
.804E-01
.989E-02
.396E-01
5 6 7 8
Harmonic Number
Improved fourier analysis.
normalized
component
1.
0.
0.
0.
000000
015704
332897
015688
/199215
.015663
. 141737
015627
. 109662
phase
(deg)
.800
79.
-12.
7D:
-16.
phase
.382
.400
. 764
. 000
146
600
529
199
Notice how this analysis (Figure above) shows less of a DC
component voltage and lower amplitudes for each of the
even harmonic frequency sine waves, all because we let the
computer sample more cycles of the wave. Again, the
imprecision of the first analysis is not So much a flaw in
SPICE as it is a fundamental property of waves and of signal
analysis.
e REVIEW:
e Square waves are equivalent to a sine wave at the same
(fundamental) frequency added to an infinite series of
odd-multiple sine-wave harmonics at decreasing
amplitudes.
e Computer algorithms exist which are able to sample
waveshapes and determine their constituent sinusoidal
components. The Fourier Transform algorithm
(particularly the Fast Fourier Transform, or FFT) is
commonly used in computer circuit simulation programs
such as SPICE and in electronic metering equipment for
determining power quality.
Other waveshapes
As strange as it may seem, any repeating, non-sinusoidal
waveform is actually equivalent to a series of sinusoidal
waveforms of different amplitudes and frequencies added
together. Square waves are a very common and well-
understood case, but not the only one.
Electronic power control devices such as transistors and
silicon-controlled rectifiers (SCRs) often produce voltage and
current waveforms that are essentially chopped-up versions
of the otherwise “clean” (pure) sine-wave AC from the power
supply. These devices have the ability to suddenly change
their resistance with the application of a control signal
voltage or current, thus “turning on” or “turning off” almost
instantaneously, producing current waveforms bearing little
resemblance to the source voltage waveform powering the
circuit. These current waveforms then produce changes in
the voltage waveform to other circuit components, due to
voltage drops created by the non-sinusoidal current through
circuit impedances.
Circuit components that distort the normal sine-wave shape
of AC voltage or current are called nonlinear. Nonlinear
components such as SCRs find popular use in power
electronics due to their ability to regulate large amounts of
electrical power without dissipating much heat. While this is
an advantage from the perspective of energy efficiency, the
waveshape distortions they introduce can cause problems.
These non-sinusoidal waveforms, regardless of their actual
shape, are equivalent to a series of sinusoidal waveforms of
higher (harmonic) frequencies. If not taken into
consideration by the circuit designer, these harmonic
waveforms created by electronic switching components may
cause erratic circuit behavior. It is becoming increasingly
common in the electric power industry to observe
overheating of transformers and motors due to distortions in
the sine-wave shape of the AC power line voltage stemming
from “switching” loads such as computers and high-
efficiency lights. This is no theoretical exercise: it is very real
and potentially very troublesome.
In this section, | will investigate a few of the more common
waveshapes and show their harmonic components by way of
Fourier analysis using SPICE.
One very common way harmonics are generated in an AC
power system is when AC is converted, or “rectified” into DC.
This is generally done with components called diodes, which
only allow the passage of current in one direction. The
simplest type of AC/DC rectification is ha/f-wave, where a
single diode blocks half of the AC current (over time) from
passing through the load. (Figure below) Oddly enough, the
conventional diode schematic symbol is drawn such that
electrons flow aga/nst the direction of the symbol's
arrowhead:
diode
1 2
os
+
(“v) load
QO — — — — + «10
The diode only allows electron
flow in a counter-clockwise
direction.
Half-wave rectifier.
halfwave rectifier
v1 10 sin(0 15 60 0 OQ)
rload 2 0 10k
d1 12 modi
.model modl d
.tran .5m 17m
.plot tran v(1,0) v(2,0)
. four 60 v(1,0) v(2,0)
.end
halfwave rectifier
y — ¥(1)+0,.4— v(2)
Half-wave rectifier waveforms. V(1)+0.4 shifts the sinewave
input V(1) up for clarity. This is not part of the simulation.
First, we'll see how SPICE analyzes the source waveform, a
pure sine wave voltage: (Figure below)
fourier components of transient response v(1)
dc component = 8.016E-04
harmonic frequency fourier normalized phase
normalized
no (hz) component component (deg) phase
(deg)
1 .Q00E+01 1.482E+01 1.000000 -0.005
0.000
2 .200E+02 2.492E-03 0.000168 104.347
-104.342
3 . 800E+02 6.465E-04 0.000044 -86.663
-86.658
4 .400E+02 1.132E-03 0.000076 -61.324
-61.319
5 . Q00E+02 1.185E-03 0.000080 -70.091
-70.086
6 . 600E+02 1.092E-03 0.000074 -63.607
-63.602
yi .200E+02 1.220E-03 0.000082 -56.288
-56.283
8 . 800E+02 1.354E-03 0.000091 -54.669
-54.664
9 5.400E+02 1.467E-03 0.000099 -52.660
-52.655
16
"22022.dat"using0:3
Relative Amplitude
0 1 2 3 4 5 6 F 8 9
Harmonic Number
Fourier analysis of the sine wave input.
Notice the extremely small harmonic and DC components of
this sinusoidal waveform in the table above, though, too
small to show on the harmonic plot above. Ideally, there
would be nothing but the fundamental frequency showing
(being a perfect sine wave), but our Fourier analysis figures
aren't perfect because SPICE doesn't have the luxury of
sampling a waveform of infinite duration. Next, we'll
compare this with the Fourier analysis of the half-wave
“rectified” voltage across the load resistor: (Figure below)
fourier components of transient response v(2)
dc component = 4.456E+00
harmonic frequency fourier normalized phase
normalized
no (hz) component component (deg) phase
(deg)
1 6.000E+01 7 .Q00E+00 1.000000 -0.195
0.000
2 1.200E+02 3.016E+00 0.430849 -89.765
-89.570
3 1.800E+02 1.206E-01 0.017223 -168.005
- 167.810
4 2.400E+02 5.149E-01 0.073556 -87.295
-87.100
5 3.000E+02 6.382E-02 0.009117 -152.790
-152.595
6 3.600E+02 1.727E-01 0.024676 -79.362
-79.167
7 4.200E+02 4.492E-02 0.006417 -132.420
-132.224
8 4.800E+02 7.493E-02 0.010703 -61.479
-61.284
9 5.400E+02 4.051E-02 0.005787 -115.085
-114.889
7
"22023.dat" using0:3
és 6
a2)
2 5
=
eE 4
<
& 4
= 2
or
0 41 2 3 4 5 6 F 8B 9Y9
Harmonic Number
Fourier analysis half-wave output.
Notice the relatively large even-multiple harmonics in this
analysis. By cutting out half of our AC wave, we've
introduced the equivalent of several higher-frequency
sinusoidal (actually, cosine) waveforms into our circuit from
the original, pure sine-wave. Also take note of the large DC
component: 4.456 volts. Because our AC voltage waveform
has been “rectified” (only allowed to push in one direction
across the load rather than back-and-forth), it behaves a lot
more like DC.
Another method of AC/DC conversion is called full-wave
(Figure below), which as you may have guessed utilizes the
full cycle of AC power from the source, reversing the polarity
of half the AC cycle to get electrons to flow through the load
the same direction all the time. | won't bore you with details
of exactly how this is done, but we can examine the
waveform (Figure below) and its harmonic analysis through
SPICE: (Figure below)
Full-wave rectifier circuit.
fullwave bridge rectifier
v1 10 sin(0 15 60 0 0)
rload 2 3 10k
d1 1 2 modl
d2 0 2 modl
d3 3 1 modl
d4 3 0 modl
.model modl d
.tran .5m 17m
.plot tran v(1,0) v(2,3)
. four 60 v(2,3)
.end
y — v(t)
20,0
0.0F
Waveforms for full-wave rectifier
— ¥(2,3)
fourier components of transient response v(2,3)
dc component =
harmonic
normalized
no
(deg)
1 6.
0.000
2 1.
3.289
3 io
-0.267
4 23
1.027
5 3.
-1.507
6 ce
-6.752
7 4.
-0.504
8 4.
-25.319
9 Be
2.612
8.273E+00
frequency fourier
(hz) component
QOO0E+01 7.000E-02
200E+02 5.997E+00
800E+02 7.241E-02
400E+02 1.013E+00
OQO0E+02 7.364E-02
600E+02 3.337E-01
200E+02 7.496E-02
800E+02 1.404E-01
400E+02 7.457E-02
normalized
component
1
85.
1.
14.
. 000000
669415
034465
465161
. 052023
. 767350
070827
. 006043
. 065240
phase
(deg)
-93.519
-90.230
-93.787
-92.492
-95.026
100.271
-94.023
118.839
-90.907
'22025.dat" using 0:3
Relative Amplitude
0 1 2 3 4 5 6 7 8 9
Harmonic Number
Fourier analysis of full-wave rectifier output.
What a difference! According to SPICE's Fourier transform,
we have a 2nd harmonic component to this waveform that's
over 85 times the amplitude of the original AC source
frequency! The DC component of this wave shows up as
being 8.273 volts (almost twice what is was for the half-wave
rectifier circuit) while the second harmonic is almost 6 volts
in amplitude. Notice all the other harmonics further on down
the table. The odd harmonics are actually stronger at some
of the higher frequencies than they are at the lower
frequencies, which is interesting.
As you can see, what may begin as a neat, simple AC sine-
wave may end up as a complex mess of harmonics after
passing through just a few electronic components. While the
complex mathematics behind all this Fourier transformation
is not necessary for the beginning student of electric circuits
to understand, it is of the utmost importance to realize the
principles at work and to grasp the practical effects that
harmonic signals may have on circuits. The practical effects
of harmonic frequencies in circuits will be explored in the
last section of this chapter, but before we do that we'll take
a closer look at waveforms and their respective harmonics.
e REVIEW:
e Any waveform at all, so long as it is repetitive, can be
reduced to a series of sinusoidal waveforms added
together. Different waveshapes consist of different
blends of sine-wave harmonics.
e Rectification of AC to DC is a very common source of
harmonics within industrial power systems.
More on spectrum analysis
Computerized Fourier analysis, particularly in the form of
the FFT algorithm, is a powerful tool for furthering our
understanding of waveforms and their related spectral
components. This same mathematical routine programmed
into the SPICE simulator as the . fourier option is also
programmed into a variety of electronic test instruments to
perform real-time Fourier analysis on measured signals. This
section is devoted to the use of such tools and the analysis
of several different waveforms.
First we have a simple sine wave at a frequency of 523.25
Hz. This particular frequency value is a “C” pitch on a piano
keyboard, one octave above “middle C”. Actually, the signal
measured for this demonstration was created by an
electronic keyboard set to produce the tone of a panflute,
the closest instrument “voice” | could find resembling a
perfect sine wave. The plot below was taken from an
oscilloscope display, showing signal amplitude (voltage)
over time: (Figure below)
Oscilloscope display: voltage vs time.
Viewed with an oscilloscope, a sine wave looks like a wavy
curve traced horizontally on the screen. The horizontal axis
of this oscilloscope display is marked with the word “Time”
and an arrow pointing in the direction of time's progression.
The curve itself, of course, represents the cyclic increase and
decrease of voltage over time.
Close observation reveals imperfections in the sine-wave
Shape. This, unfortunately, is a result of the specific
equipment used to analyze the waveform. Characteristics
like these due to quirks of the test equipment are
technically known as artifacts: phenomena existing solely
because of a peculiarity in the equipment used to perform
the experiment.
If we view this same AC voltage on a spectrum analyzer, the
result is quite different: (Figure below)
melas hclanciales)
!
Frequency —»
Spectrum analyzer display: voltage vs frequency.
As you can see, the horizontal axis of the display is marked
with the word “Frequency,” denoting the domain of this
measurement. The single peak on the curve represents the
predominance of a single frequency within the range of
frequencies covered by the width of the display. If the scale
of this analyzer instrument were marked with numbers, you
would see that this peak occurs at 523.25 Hz. The height of
the peak represents the signal amplitude (voltage).
If we mix three different sine-wave tones together on the
electronic keyboard (C-E-G, a C-major chord) and measure
the result, both the oscilloscope display and the spectrum
analyzer display reflect this increased complexity: (Figure
below)
Oscilloscape display: three tones.
The oscilloscope display (time-domain) shows a waveform
with many more peaks and valleys than before, a direct
result of the mixing of these three frequencies. As you will
notice, some of these peaks are higher than the peaks of the
original single-pitch waveform, while others are lower. This is
a result of the three different waveforms alternately
reinforcing and canceling each other as their respective
phase shifts change in time.
Frequency —»
Spectrum analyzer display: three tones.
The spectrum display (frequency-domain) is much easier to
interpret: each pitch is represented by its own peak on the
curve. (Figure above) The difference in height between
these three peaks is another artifact of the test equipment: a
consequence of limitations within the equipment used to
generate and analyze these waveforms, and not a necessary
characteristic of the musical chord itself.
As was Stated before, the device used to generate these
waveforms is an electronic keyboard: a musical instrument
designed to mimic the tones of many different instruments.
The panflute “voice” was chosen for the first demonstrations
because it most closely resembled a pure sine wave (a single
frequency on the spectrum analyzer display). Other musical
instrument “voices” are not as simple as this one, though. In
fact, the unique tone produced by any instrument is a
function of its waveshape (or spectrum of frequencies). For
example, let's view the signal for a trumpet tone: (Figure
below)
Time
Oscilloscope display: waveshape of a trumpet tone.
The fundamental frequency of this tone is the same as in the
first panflute example: 523.25 Hz, one octave above “middle
C.” The waveform itself is far from a pure and simple sine-
wave form. Knowing that any repeating, non-sinusoidal
waveform is equivalent to a series of sinusoidal waveforms
at different amplitudes and frequencies, we should expect to
see multiple peaks on the spectrum analyzer display: (Figure
below)
Spectrum of a trumpet tone.
Indeed we do! The fundamental frequency component of
523.25 Hz is represented by the left-most peak, with each
successive harmonic represented as its own peak along the
width of the analyzer screen. The second harmonic is twice
the frequency of the fundamental (1046.5 Hz), the third
harmonic three times the fundamental (1569.75 Hz), and so
on. This display only shows the first six harmonics, but there
are many more comprising this complex tone.
Trying a different instrument voice (the accordion) on the
keyboard, we obtain a similarly complex oscilloscope (time-
domain) plot (Figure below) and spectrum analyzer
(frequency-domain) display: (Figure below)
Oscilloscope display: waveshape of accordion tone.
Frequency —»
Spectrum of accordion tone.
Note the differences in relative harmonic amplitudes (peak
heights) on the spectrum displays for trumpet and
accordion. Both instrument tones contain harmonics all the
way from 1st (fundamental) to 6th (and beyond!), but the
proportions aren't the same. Each instrument has a unique
harmonic “signature” to its tone. Bear in mind that all this
complexity is in reference to a single note played with these
two instrument “voices.” Multiple notes played on an
accordion, for example, would create a much more complex
mixture of frequencies than what is seen here.
The analytical power of the oscilloscope and spectrum
analyzer permit us to derive general rules about waveforms
and their harmonic spectra from real waveform examples.
We already know that any deviation from a pure sine-wave
results in the equivalent of a mixture of multiple sine-wave
waveforms at different amplitudes and frequencies.
However, close observation allows us to be more specific
than this. Note, for example, the time- (Figure below) and
frequency-domain (Figure below) plots for a waveform
approximating a square wave:
Oscilloscope time-domain display of a square wave
| Fundamental
7 requency —+
Spectrum (frequency-domain) of a square wave.
According to the spectrum analysis, this waveform contains
no even harmonics, only odd. Although this display doesn't
show frequencies past the sixth harmonic, the pattern of
odd-only harmonics in descending amplitude continues
indefinitely. This should come as no Surprise, as we've
already seen with SPICE that a square wave is comprised of
an infinitude of odd harmonics. The trumpet and accordion
tones, however, contained both even and odd harmonics.
This difference in harmonic content is noteworthy. Let's
continue our investigation with an analysis of a triangle
wave: (Figure below)
Oscilloscope time-domain display of a triangle wave.
Fundamental
ang 4!"
Frequency —»
Spectrum of a triangle wave.
In this waveform there are practically no even harmonics:
(Figure above) the only significant frequency peaks on the
spectrum analyzer display belong to odd-numbered
multiples of the fundamental frequency. Tiny peaks can be
seen for the second, fourth, and sixth harmonics, but this is
due to imperfections in this particular triangle waveshape
(once again, artifacts of the test equipment used in this
analysis). A perfect triangle waveshape produces no even
harmonics, just like a perfect square wave. It should be
obvious from inspection that the harmonic spectrum of the
triangle wave is not identical to the spectrum of the square
wave: the respective harmonic peaks are of different
heights. However, the two different waveforms are common
in their lack of even harmonics.
Let's examine another waveform, this one very similar to the
triangle wave, except that its rise-time is not the same as its
fall-time. Known as a sawtooth wave, its oscilloscope plot
reveals it to be aptly named: (Figure below)
Time-domain display of a sawtooth wave.
When the spectrum analysis of this waveform is plotted, we
see a result that is quite different from that of the regular
triangle wave, for this analysis shows the strong presence of
even-numbered harmonics (second and fourth): (Figure
below)
| Fundamenta
Frequency
Frequency-domain display of a sawtooth wave.
The distinction between a waveform having even harmonics
versus no even harmonics resides in the difference between
a triangle waveshape and a sawtooth waveshape. That
difference is symmetry above and below the horizontal
centerline of the wave. A waveform that is symmetrical
above and below its centerline (the shape on both sides
mirror each other precisely) will contain no even-numbered
harmonics. (Figure below)
Pure sine wave =
1°" harmonic only
Waveforms symmetric about their x-axis center line contain
only odd harmonics.
Square waves, triangle waves, and pure sine waves all
exhibit this symmetry, and all are devoid of even harmonics.
Waveforms like the trumpet tone, the accordion tone, and
the sawtooth wave are unsymmetrical around their
centerlines and therefore do contain even harmonics.
(Figure below)
/ \/™
Asymmetric waveforms contain even harmonics.
This principle of centerline symmetry should not be
confused with symmetry around the zero line. In the
examples shown, the horizontal centerline of the waveform
happens to be zero volts on the time-domain graph, but this
has nothing to do with harmonic content. This rule of
harmonic content (even harmonics only with unsymmetrical
waveforms) applies whether or not the waveform is shifted
above or below zero volts with a “DC component.” For
further clarification, | will show the same sets of waveforms,
shifted with DC voltage, and note that their harmonic
contents are unchanged. (Figure below)
Pure sine wave =
15" harmonic only
These waveforms are composed exclusively of odd
harmonics.
Again, the amount of DC voltage present in a waveform has
nothing to do with that waveform's harmonic frequency
content. (Figure below)
These waveforms contain even harmonics.
Why is this harmonic rule-of-thumb an important rule to
know? It can help us comprehend the relationship between
harmonics in AC circuits and specific circuit components.
Since most sources of sine-wave distortion in AC power
circuits tend to be symmetrical, even-numbered harmonics
are rarely seen in those applications. This is good to know if
you're a power system designer and are planning ahead for
harmonic reduction: you only have to concern yourself with
mitigating the odd harmonic frequencies, even harmonics
being practically nonexistent. Also, if you happen to
measure even harmonics in an AC circuit with a spectrum
analyzer or frequency meter, you know that something in
that circuit must be unsymmetrically distorting the sine-
wave voltage or current, and that clue may be helpful in
locating the source of a problem (look for components or
conditions more likely to distort one half-cycle of the AC
waveform more than the other).
Now that we have this rule to guide our interpretation of
nonsinusoidal waveforms, it makes more sense that a
waveform like that produced by a rectifier circuit should
contain such strong even harmonics, there being no
symmetry at all above and below center.
e REVIEW:
e Waveforms that are symmetrical above and below their
horizontal centerlines contain no even-numbered
harmonics.
e The amount of DC “bias” voltage present (a waveform's
“DC component”) has no impact on that wave's
harmonic frequency content.
Circuit effects
The principle of non-sinusoidal, repeating waveforms being
equivalent to a series of sine waves at different frequencies
is a fundamental property of waves in general and it has
great practical import in the study of AC circuits. It means
that any time we have a waveform that isn't perfectly sine-
wave-shaped, the circuit in question will react as though its
having an array of different frequency voltages imposed on
it at once.
When an AC circuit is subjected to a source voltage
consisting of a mixture of frequencies, the components in
that circuit respond to each constituent frequency ina
different way. Any reactive component such as a capacitor or
an inductor will simultaneously present a unique amount of
impedance to each and every frequency present in a circuit.
Thankfully, the analysis of such circuits is made relatively
easy by applying the Superposition Theorem, regarding the
multiple-frequency source as a set of single-frequency
voltage sources connected in series, and analyzing the
circuit for one source at a time, Summing the results at the
end to determine the aggregate total:
5V 2.2 kQ
60 Hz
5V
90 Hz
Circuit driven by a combination of frequencies: 60 Hz and 90
Hz.
Analyzing circuit for 60 Hz source alone:
R
2.2 kQ
ca 1nF
5V
60 Hz
Xe = 2.653 kQ
Circuit for solving 60 Hz.
c | 2.0377 + j2.4569 2.9623 - j2.4569
3.1919 Z 50.328° | 3.8486 2 -39.6716°
926.221 + jl.LL68m
926.226 + jl.LL68m | 926.22 4 jl.LL68m
L.4509m 2 50.328° L.4509m 2 50.328° L.4509m 2 50.328°
. 2.2k + j0 0 - j2.653k 2.2k - j2.653k
2.2k Z 0° 2.653k Z -90° 3.446k Z -50.328°
Analyzing the circuit for 90 Hz source alone:
R
2.2kQ
Xo = 1.768 kQ
90 Hz
Circuit of solving 90 Hz.
Cc
0375 + j2.4415 1.9625 - j24415
8971 Z 38.793° 3.1325 Z -51.207°
ny
be |
+
a |
L.3807m + jL.L098m | L.3807m+ jl.L098m | 1.3807m + jl. L098m
L.77 14m Z 38.793° L.7714m Z 38.793° L.7714m Z 38.793°
0 - jL.768k
L.768k Z -90°
Superimposing the voltage drops across R and C, we get:
Ep = [3.1919 V Z 50.328° (60 Hz)] + [3.8971 V Z 38.793° (90 Hz)]
Ex = [3.8486 V Z -39.6716° (60 Hz)] + [3.1325 V Z -51.207° (90 Hz)]
Because the two voltages across each component are at
different frequencies, we cannot consolidate them into a
single voltage figure as we could if we were adding together
two voltages of different amplitude and/or phase angle at
the same frequency. Complex number notation give us the
ability to represent waveform amplitude (polar magnitude)
and phase angle (polar angle), but not frequency.
What we can tell from this application of the superposition
theorem is that there will be a greater 60 Hz voltage
dropped across the capacitor than a 90 Hz voltage. Just the
opposite is true for the resistor's voltage drop. This is worthy
to note, especially in light of the fact that the two source
voltages are equal. It is this kind of unequal circuit response
to signals of differing frequency that will be our specific
focus in the next chapter.
We can also apply the superposition theorem to the analysis
of a circuit powered by a non-sinusoidal voltage, such as a
Square wave. If we know the Fourier series (multiple
sine/cosine wave equivalent) of that wave, we can regard it
as originating from a series-connected string of multiple
sinusoidal voltage sources at the appropriate amplitudes,
frequencies, and phase shifts. Needless to say, this can bea
laborious task for some waveforms (an accurate square-wave
Fourier Series is considered to be expressed out to the ninth
harmonic, or five sine waves in all!), but it is possible. |
mention this not to scare you, but to inform you of the
potential complexity lurking behind seemingly simple
waveforms. A real-life circuit will respond just the same to
being powered by a square wave as being powered by an
infinite series of sine waves of odd-multiple frequencies and
diminishing amplitudes. This has been known to translate
into unexpected circuit resonances, transformer and
inductor core overheating due to eddy currents,
electromagnetic noise over broad ranges of the frequency
spectrum, and the like. Technicians and engineers need to
be made aware of the potential effects of non-sinusoidal
waveforms in reactive circuits.
Harmonics are known to manifest their effects in the form of
electromagnetic radiation as well. Studies have been
performed on the potential hazards of using portable
computers aboard passenger aircraft, citing the fact that
computers’ high frequency square-wave “clock” voltage
signals are capable of generating radio waves that could
interfere with the operation of the aircraft's electronic
navigation equipment. It's bad enough that typical
microprocessor clock signal frequencies are within the range
of aircraft radio frequency bands, but worse yet is the fact
that the harmonic multiples of those fundamental
frequencies span an even larger range, due to the fact that
clock signal voltages are square-wave in shape and not sine-
wave.
Electromagnetic “emissions” of this nature can be a problem
in industrial applications, too, with harmonics abounding in
very large quantities due to (nonlinear) electronic control of
motor and electric furnace power. The fundamental power
line frequency may only be 60 Hz, but those harmonic
frequency multiples theoretically extend into infinitely high
frequency ranges. Low frequency power line voltage and
current doesn't radiate into space very well as
electromagnetic energy, but high frequencies do.
Also, capacitive and inductive “coupling” caused by close-
proximity conductors is usually more severe at high
frequencies. Signal wiring nearby power wiring will tend to
“pick up” harmonic interference from the power wiring toa
far greater extent than pure sine-wave interference. This
problem can manifest itself in industry when old motor
controls are replaced with new, solid-state electronic motor
controls providing greater energy efficiency. Suddenly there
may be weird electrical noise being impressed upon signal
wiring that never used to be there, because the old controls
never generated harmonics, and those high-frequency
harmonic voltages and currents tend to inductively and
Capacitively “couple” better to nearby conductors than any
60 Hz signals from the old controls used to.
e REVIEW:
e Any regular (repeating), non-sinusoidal waveform is
equivalent to a particular series of sine/cosine waves of
different frequencies, phases, and amplitudes, plus a DC
offset voltage if necessary. The mathematical process for
determining the sinusoidal waveform equivalent for any
waveform is called Fourier analysis.
Multiple-frequency voltage sources can be simulated for
analysis by connecting several single-frequency voltage
sources in series. Analysis of voltages and currents is
accomplished by using the superposition theorem.
NOTE: superimposed voltages and currents of different
frequencies cannot be added together in complex
number form, since complex numbers only account for
amplitude and phase shift, not frequency!
e Harmonics can cause problems by impressing unwanted
(“noise”) voltage signals upon nearby circuits. These
unwanted signals may come by way of capacitive
coupling, inductive coupling, electromagnetic radiation,
or a combination thereof.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4/l—
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 8
FILTERS
e What is a filter?
Low-pass filters
High-pass filters
Band-pass filters
Band-stop filters
Resonant filters
Summary
Contributors
What is a filter?
It is sometimes desirable to have circuits capable of
selectively filtering one frequency or range of frequencies
out of a mix of different frequencies in a circuit. A circuit
designed to perform this frequency selection is called a fi/ter
circuit, or simply a filter. A common need for filter circuits is
in high-performance stereo systems, where certain ranges of
audio frequencies need to be amplified or suppressed for
best sound quality and power efficiency. You may be familiar
with equalizers, which allow the amplitudes of several
frequency ranges to be adjusted to suit the listener's taste
and acoustic properties of the listening area. You may also
be familiar with crossover networks, which block certain
ranges of frequencies from reaching speakers. A tweeter
(high-frequency speaker) is inefficient at reproducing low-
frequency signals such as drum beats, so a crossover circuit
is connected between the tweeter and the stereo's output
terminals to block low-frequency signals, only passing high-
frequency signals to the speaker's connection terminals.
This gives better audio system efficiency and thus better
performance. Both equalizers and crossover networks are
examples of filters, designed to accomplish filtering of
certain frequencies.
Another practical application of filter circuits is in the
“conditioning” of non-sinusoidal voltage waveforms in power
circuits. Some electronic devices are sensitive to the
presence of harmonics in the power supply voltage, and so
require power conditioning for proper operation. If a
distorted sine-wave voltage behaves like a series of
harmonic waveforms added to the fundamental frequency,
then it should be possible to construct a filter circuit that
only allows the fundamental waveform frequency to pass
through, blocking all (higher-frequency) harmonics.
We will be studying the design of several elementary filter
circuits in this lesson. To reduce the load of math on the
reader, | will make extensive use of SPICE as an analysis
tool, displaying Bode plots (amplitude versus frequency) for
the various kinds of filters. Bear in mind, though, that these
circuits can be analyzed over several points of frequency by
repeated series-parallel analysis, much like the previous
example with two sources (60 and 90 Hz), if the student is
willing to invest a lot of time working and re-working circuit
calculations for each frequency.
e REVIEW:
e A filteris an AC circuit that separates some frequencies
from others within mixed-frequency signals.
e Audio egualizers and crossover networks are two well-
known applications of filter circuits.
e A Bode plotis a graph plotting waveform amplitude or
phase on one axis and frequency on the other.
Low-pass filters
By definition, a low-pass filter is a circuit offering easy
passage to low-frequency signals and difficult passage to
high-frequency signals. There are two basic kinds of circuits
capable of accomplishing this objective, and many
variations of each one: The inductive low-pass filter in Figure
below and the capacitive low-pass filter in Figure below
Inductive low-pass filter
The inductor's impedance increases with increasing
frequency. This high impedance in series tends to block
high-frequency signals from getting to the load. This can be
demonstrated with a SPICE analysis: (Figure below)
inductive lowpass filter
vl 10 ac 1 sin
tL. 1.23
rload 2 © 1k
.ac Lin 20 1 200
.plot ac v(2)
end
frequency Hz
The response of an inductive low-pass filter falls off with
increasing frequency.
Capacitive low-pass filter.
The capacitor's impedance decreases with increasing
frequency. This low impedance in parallel with the load
resistance tends to short out high-frequency signals,
dropping most of the voltage across series resistor Rj.
(Figure below)
Capacitive lowpass filter
v1 10 ac 1 sin
rl 1 2 500
cl 2 0 7u
rload 2 0 1k
.ac Lin 20 30 150
.plot ac v(2)
end
200.0 Vrvsessersenseen fasnnananannnnmunnvanenennns
0,0 50,0 100.0 150.0
frequency Hz
The response of a capacitive low-pass filter falls off with
increasing frequency.
The inductive low-pass filter is the pinnacle of simplicity,
with only one component comprising the filter. The
Capacitive version of this filter is not that much more
complex, with only a resistor and capacitor needed for
operation. However, despite their increased complexity,
Capacitive filter designs are generally preferred over
inductive because capacitors tend to be “purer” reactive
components than inductors and therefore are more
predictable in their behavior. By “pure” | mean that
capacitors exhibit little resistive effects than inductors,
making them almost 100% reactive. Inductors, on the other
hand, typically exhibit significant dissipative (resistor-like)
effects, both in the long lengths of wire used to make them,
and in the magnetic losses of the core material. Capacitors
also tend to participate less in “coupling” effects with other
components (generate and/or receive interference from
other components via mutual electric or magnetic fields)
than inductors, and are less expensive.
However, the inductive low-pass filter is often preferred in
AC-DC power supplies to filter out the AC “ripple” waveform
created when AC is converted (rectified) into DC, passing
only the pure DC component. The primary reason for this is
the requirement of low filter resistance for the output of such
a power supply. A capacitive low-pass filter requires an extra
resistance in series with the source, whereas the inductive
low-pass filter does not. In the design of a high-current
circuit like a DC power supply where additional series
resistance is undesirable, the inductive low-pass filter is the
better design choice. On the other hand, if low weight and
compact size are higher priorities than low internal supply
resistance in a power supply design, the capacitive low-pass
filter might make more sense.
All low-pass filters are rated at a certain cutoff frequency.
That is, the frequency above which the output voltage falls
below 70.7% of the input voltage. This cutoff percentage of
70.7 is not really arbitrary, all though it may seem so at first
glance. In a simple capacitive/resistive low-pass filter, it is
the frequency at which capacitive reactance in ohms equals
resistance in ohms. In a simple capacitive low-pass filter
(one resistor, one capacitor), the cutoff frequency is given
as:
es |
cutott — ITRC
Inserting the values of R and C from the last SPICE
simulation into this formula, we arrive at a cutoff frequency
of 45.473 Hz. However, when we look at the plot generated
by the SPICE simulation, we see the load voltage well below
70.7% of the source voltage (1 volt) even at a frequency as
low as 30 Hz, below the calculated cutoff point. What's
wrong? The problem here is that the load resistance of 1 kO
affects the frequency response of the filter, skewing it down
from what the formula told us it would be. Without that load
resistance in place, SPICE produces a Bode plot whose
numbers make more sense: (Figure below)
Capacitive lowpass filter
vl 10 ac 1 sin
rl 12 500
cl 2 0 7u
* note: no load resistor!
.ac Lin 20 40 50
.plot ac v(2)
end
frequency Hz
For the capacitive low-pass filter with R = 500 QandC =7
UF, the Output should be 70.7% at 45.473 Hz.
fcutore = 1/(2mMRC) = 1/(2n(500 9)(7 WF)) = 45.473
Hz
When dealing with filter circuits, it is always important to
note that the response of the filter depends on the filter's
component values and the impedance of the load. If a cutoff
frequency equation fails to give consideration to load
impedance, it assumes no load and will fail to give accurate
results for a real-life filter conducting power to a load.
One frequent application of the capacitive low-pass filter
principle is in the design of circuits having components or
sections sensitive to electrical “noise.” As mentioned at the
beginning of the last chapter, sometimes AC signals can
“couple” from one circuit to another via capacitance (Cctray)
and/or mutual inductance (Mgt;ay) between the two sets of
conductors. A prime example of this is unwanted AC signals
(“noise”) becoming impressed on DC power lines supplying
sensitive circuits: (Figure below)
Lwi re : : : Lwire Lwi re
"Clean" DC power oe ae eee ee
Eons Dirty” or oe DC power
load
Noise Is coupled by stray capacitance and mutual
inductance into “clean” DC power.
The oscilloscope-meter on the left shows the “clean” power
from the DC voltage source. After coupling with the AC noise
source via stray mutual inductance and stray capacitance,
though, the voltage as measured at the load terminals is
now a mix of AC and DC, the AC being unwanted. Normally,
one would expect Ejgag to be precisely identical to Exource,
because the uninterrupted conductors connecting them
should make the two sets of points electrically common.
However, power conductor impedance allows the two
voltages to differ, which means the noise magnitude can
vary at different points in the DC system.
If we wish to prevent such “noise” from reaching the DC
load, all we need to do is connect a low-pass filter near the
load to block any coupled signals. In its simplest form, this is
nothing more than a capacitor connected directly across the
power terminals of the load, the capacitor behaving asa
very low impedance to any AC noise, and shorting it out.
Such a capacitor is called a decoupling capacitor. (Figure
below)
"Clean" DC power
E "Cleaner" DC power with
supply decoupling capacitor
Broad
Decoupling capacitor, applied to load, filters noise from DC
power supply.
A cursory glance at a crowded printed-circuit board (PCB)
will typically reveal decoupling capacitors scattered
throughout, usually located as close as possible to the
sensitive DC loads. Capacitor size is usually 0.1 UF or more,
a minimum amount of capacitance needed to produce a low
enough impedance to short out any noise. Greater
Capacitance will do a better job at filtering noise, but size
and economics limit decoupling capacitors to meager
values.
e REVIEW:
e A low-pass filter allows for easy passage of low-
frequency signals from source to load, and difficult
passage of high-frequency signals.
Inductive low-pass filters insert an inductor in series
with the load; capacitive low-pass filters insert a resistor
in series and a capacitor in parallel with the load. The
former filter design tries to “block” the unwanted
frequency signal while the latter tries to short it out.
The cutoff frequency for a low-pass filter is that
frequency at which the output (load) voltage equals
70.7% of the input (Source) voltage. Above the cutoff
frequency, the output voltage is lower than 70.7% of the
input, and vice versa.
High-pass filters
A high-pass filter's task is just the opposite of a low-pass
filter: to offer easy passage of a high-frequency signal and
difficult passage to a low-frequency signal. As one might
expect, the inductive (Figure below) and capacitive (Figure
below) versions of the high-pass filter are just the opposite
of their respective low-pass filter designs:
Capacitive high-pass filter.
The capacitor's impedance (Figure above) increases with
decreasing frequency. (Figure below) This high impedance in
series tends to block low-frequency signals from getting to
load.
Capacitive highpass filter
vl 10 ac 1 sin
cl 1 2 0.5u
rload 2 0 1k
.ac Lin 20 1 200
.plot ac v(2)
end
mV = ym(2)
B00,0 presses vvtaesenonaesnnonsnn ;
400 0 PEE UUE EOE EERE AREER EER EU HE ECE REOOHEEHEESMED GHD OU HASH EEO H RENE
= - =
0,0 100,0
frequency Hz
The response of the capacitive high-pass filter increases
with frequency.
Inductive high-pass filter.
The inductor's impedance (Figure above) decreases with
decreasing frequency. (Figure below) This low impedance in
parallel tends to short out low-frequency signals from
getting to the load resistor. As a consequence, most of the
voltage gets dropped across series resistor Rj.
inductive highpass filter
vl 10 aci1 sin
rl 1 2 200
Ll1 2 0 100m
rload 2 0 1k
.ac Lin 20 1 200
.plot ac v(2)
.end
mV = ym(2)
BO0,0 presses sennnnsonsennenoniny
400.0 [trsrsreneernmmnadonnnsgefinnnnaun
200.0 Jrrinnnnnnnyonnmndinmnnannnnninnn i
200.0
0,0 100,0
frequency Hz
The response of the inductive high-pass filter increases with
frequency.
This time, the capacitive design is the simplest, requiring
only one component above and beyond the load. And,
again, the reactive purity of capacitors over inductors tends
to favor their use in filter design, especially with high-pass
filters where high frequencies commonly cause inductors to
behave strangely due to the skin effect and electromagnetic
core losses.
As with low-pass filters, high-pass filters have a rated cutoff
frequency, above which the output voltage increases above
70.7% of the input voltage. Just as in the case of the
Capacitive low-pass filter circuit, the capacitive high-pass
filter's cutoff frequency can be found with the same formula:
l
cutoff — OTRO
In the example circuit, there is no resistance other than the
load resistor, so that is the value for R in the formula.
Using a stereo system as a practical example, a capacitor
connected in series with the tweeter (treble) speaker will
serve as a high-pass filter, imposing a high impedance to
low-frequency bass signals, thereby preventing that power
from being wasted on a speaker inefficient for reproducing
such sounds. In like fashion, an inductor connected in series
with the woofer (bass) speaker will serve as a low-pass filter
for the low frequencies that particular speaker is designed to
reproduce. In this simple example circuit, the midrange
speaker is subjected to the full spectrum of frequencies from
the stereo's output. More elaborate filter networks are
sometimes used, but this should give you the general idea.
Also bear in mind that I'm only showing you one channel
(either left or right) on this stereo system. A real stereo
would have six speakers: 2 woofers, 2 midranges, and 2
tweeters.
low-pass
Woofer
Midrange
Stereo ;
high-pass
Tweeter
High-pass filter routes high frequencies to tweeter, while
low-pass filter routes lows to woofer.
For better performance yet, we might like to have some kind
of filter circuit capable of passing frequencies that are
between low (bass) and high (treble) to the midrange
Speaker so that none of the low- or high-frequency signal
power is wasted on a speaker incapable of efficiently
reproducing those sounds. What we would be looking for is
called a band-pass filter, which is the topic of the next
section.
e REVIEW:
e A high-pass filter allows for easy passage of high-
frequency signals from source to load, and difficult
passage of low-frequency signals.
e Capacitive high-pass filters insert a capacitor in series
with the load; inductive high-pass filters insert a resistor
in series and an inductor in parallel with the load. The
former filter design tries to “block” the unwanted
frequency signal while the latter tries to short it out.
The cutoff frequency for a high-pass filter is that
frequency at which the output (load) voltage equals
70.7% of the input (source) voltage. Above the cutoff
frequency, the output voltage is greater than 70.7% of
the input, and vice versa.
Band-pass filters
There are applications where a particular band, or spread, or
frequencies need to be filtered from a wider range of mixed
signals. Filter circuits can be designed to accomplish this
task by combining the properties of low-pass and high-pass
into a single filter. The result is called a band-pass filter.
Creating a bandpass filter from a low-pass and high-pass
filter can be illustrated using block diagrams: (Figure below)
Signal —> Low-pass filter |—| High-pass filter meet
blocks frequencies blocks frequencies
that are too high that are too low
System level block diagram of a band-pass filter.
What emerges from the series combination of these two
filter circuits is a circuit that will only allow passage of those
frequencies that are neither too high nor too low. Using real
components, here is what a typical schematic might look
like Figure below. The response of the band-pass filter is
shown in (Figure below)
Source _Low-pass _High-pass
filter section filter section
Capacitive band-pass filter.
Capacitive bandpass filter
vl 10 ac 1 sin
rl 12 200
cl 2 0 2.5u
c2 2 3 lu
rload 3 0 1k
.ac Lin 20 100 500
.plot ac v(3)
.end
0,0 200,0 400,0 600,0
frequency Hz
The response of a capacitive bandpass filter peaks within a
narrow frequency range.
Band-pass filters can also be constructed using inductors,
but as mentioned before, the reactive “purity” of capacitors
gives them a design advantage. If we were to design a
bandpass filter using inductors, it might look something like
Figure below.
Source _High-pass Low-pass
filter section filter section
Inductive band-pass filter.
The fact that the high-pass section comes “first” in this
design instead of the low-pass section makes no difference
In its overall operation. It will still filter out all frequencies
too high or too low.
While the general idea of combining low-pass and high-pass
filters together to make a bandpass filter is sound, it is not
without certain limitations. Because this type of band-pass
filter works by relying on either section to block unwanted
frequencies, it can be difficult to design such a filter to allow
unhindered passage within the desired frequency range.
Both the low-pass and high-pass sections will always be
blocking signals to some extent, and their combined effort
makes for an attenuated (reduced amplitude) signal at best,
even at the peak of the “pass-band” frequency range. Notice
the curve peak on the previous SPICE analysis: the load
voltage of this filter never rises above 0.59 volts, although
the source voltage is a full volt. This signal attenuation
becomes more pronounced if the filter is designed to be
more selective (steeper curve, narrower band of passable
frequencies).
There are other methods to achieve band-pass operation
without sacrificing signal strength within the pass-band. We
will discuss those methods a little later in this chapter.
e REVIEW:
e A band-pass filter works to screen out frequencies that
are too low or too high, giving easy passage only to
frequencies within a certain range.
e Band-pass filters can be made by stacking a low-pass
filter on the end of a high-pass filter, or vice versa.
“Attenuate” means to reduce or diminish in amplitude.
When you turn down the volume control on your stereo,
you are “attenuating” the signal being sent to the
speakers.
Band-stop filters
Also called band-elimination, band-reject, or notch filters,
this kind of filter passes all frequencies above and below a
particular range set by the component values. Not
surprisingly, it can be made out of a low-pass and a high-
pass filter, just like the band-pass design, except that this
time we connect the two filter sections in parallel with each
other instead of in series. (Figure below)
passes low frequencies
i Low-pass filter |
Signal _, = sone
input output
_ High-pass filter 7
passes high frequencies
System level block diagram of a bana-stop filter.
Constructed using two capacitive filter sections, it looks
something like (Figure below).
R, R,
source (~v) = | Ricad
“Twin-T” band-stop filter.
The low-pass filter section is comprised of Rj, Ro, and C, ina
“T” configuration. The high-pass filter section is comprised
of C5, C3, and R3 in a “T” configuration as well. Together, this
arrangement is commonly known as a “Twin-T” filter, giving
sharp response when the component values are chosen in
the following ratios:
Component value ratios for
the "Twin-T" band-stop filter
R, = R, = 2(R;)
C=C =5K,
Given these component ratios, the frequency of maximum
rejection (the “notch frequency”) can be calculated as
follows:
I
4nR5C;
notch —
The impressive band-stopping ability of this filter is
illustrated by the following SPICE analysis: (Figure below)
twin-t bandstop filter
vl 10 ac1 sin
rl 1 2 200
cl 2 0 2u
r2 2 3 200
c2 14 lu
r3 4 0 100
c3 4 3 lu
rload 3 0 1k
.ac Lin 20 200 1.5k
.plot ac v(3)
end
my
“0,0 0,5 1,0 1,5
frequency kHz
Response of “twin-T” band-stop filter.
REVIEW:
A band-stop filter works to screen out frequencies that
are within a certain range, giving easy passage only to
frequencies outside of that range. Also known as band-
elimination, band-reject, or notch filters.
Band-stop filters can be made by placing a low-pass
filter in parallel with a high-pass filter. Commonly, both
the low-pass and high-pass filter sections are of the “T”
configuration, giving the name “Twin-T” to the band-stop
combination.
The frequency of maximum attenuation is called the
notch frequency.
Resonant filters
So far, the filter designs we've concentrated on have
employed e/ther capacitors or inductors, but never both at
the same time. We should know by now that combinations of
L and C will tend to resonate, and this property can be
exploited in designing band-pass and band-stop filter
circuits.
Series LC circuits give minimum impedance at resonance,
while parallel LC (“tank”) circuits give maximum impedance
at their resonant frequency. Knowing this, we have two basic
strategies for designing either band-pass or band-stop
filters.
For band-pass filters, the two basic resonant strategies are
this: series LC to pass a signal (Figure below), or parallel LC
(Figure below) to short a signal. The two schemes will be
contrasted and simulated here:
+ filter —-
Series resonant LC band-pass filter.
Series LC components pass signal at resonance, and block
signals of any other frequencies from getting to the load.
(Figure below)
series resonant bandpass filter
vl 10 aci1 sin
11121
cl 2 3 lu
rload 3 0 1k
ac Lin 20 50 250
.plot ac v(3)
.end
0:99 Nhinscinsiomiin S adenuanaa ee re
0,0 100,0 2000 300,0
frequency Hz
Series resonant band-pass filter: voltage peaks at resonant
frequency of 159.15 Hz.
A couple of points to note: see how there is virtually no
signal attenuation within the “pass band” (the range of
frequencies near the load voltage peak), unlike the band-
pass filters made from capacitors or inductors alone. Also,
since this filter works on the principle of series LC resonance,
the resonant frequency of which is unaffected by circuit
resistance, the value of the load resistor will not skew the
peak frequency. However, different values for the load
resistor wi// change the “steepness” of the Bode plot (the
“selectivity” of the filter).
The other basic style of resonant band-pass filters employs a
tank circuit (parallel LC combination) to short out signals too
high or too low in frequency from getting to the load: (Figure
below)
Parallel resonant band-pass filter.
The tank circuit will have a lot of impedance at resonance,
allowing the signal to get to the load with minimal
attenuation. Under or over resonant frequency, however, the
tank circuit will have a low impedance, shorting out the
signal and dropping most of it across series resistor Rj.
(Figure below)
parallel resonant bandpass filter
vl 10 ac1 sin
rl 1 2 500
l1 2 0 100m
cl 2 0 10u
rload 2 0 1k
.ac Lin 20 50 250
.plot ac v(2)
.end
mY = ym(2)
BOO ,0 possesses Se :
0 = = =
0,0 100,0 200.0 3000
frequency Hz
Parallel resonant filter: voltage peaks a resonant frequency
of 159.15 Hz.
Just like the low-pass and high-pass filter designs relying on
a series resistance and a parallel “shorting” component to
attenuate unwanted frequencies, this resonant circuit can
never provide full input (Source) voltage to the load. That
series resistance will always be dropping some amount of
voltage so long as there is a load resistance connected to
the output of the filter.
It should be noted that this form of band-pass filter circuit is
very popular in analog radio tuning circuitry, for selecting a
particular radio frequency from the multitudes of
frequencies available from the antenna. In most analog radio
tuner circuits, the rotating dial for station selection moves a
variable capacitor in a tank circuit.
SINGLE-TUBE RADIO
Variable capacitor tunes radio receiver tank circuit to select
one out of many broadcast stations.
The variable capacitor and air-core inductor shown in Figure
above photograph of a simple radio comprise the main
elements in the tank circuit filter used to discriminate one
radio station's signal from another.
Just as we can use series and parallel LC resonant circuits to
pass only those frequencies within a certain range, we can
also use them to block frequencies within a certain range,
creating a band-stop filter. Again, we have two major
strategies to follow in doing this, to use either series or
parallel resonance. First, we'll look at the series variety:
(Figure below)
Series resonant band-stop filter.
When the series LC combination reaches resonance, its very
low impedance shorts out the signal, dropping it across
resistor R, and preventing its passage on to the load. (Figure
below)
series resonant bandstop filter
vl 10 ac 1 sin
rl 1 2 500
11 2 3 100m
cl 3 0 10u
rload 2 0 1k
.ac Lin 20 70 230
.plot ac v(2)
end
mY = ym{2)
400.0 prvvesresensnn peveversensnnorigusnonsensaann
3000 frvsseedoon Bvsennennnenaguninnnesensin
200.0 frvrrsessessnen BrvststasssneBasenenennnniins
100.0 rrsrsrsrrenen Bugental gp nanonsle
0 = = =
0,0 100,0 200.0 3000
frequency Hz
Series resonant band-stop filter: Notch frequency = LC
resonant frequency (159.15 Hz).
Next, we will examine the parallel resonant band-stop filter:
(Figure below)
C, ,, lO UF
Vi (IV 100mH Rw 1kQ
0 0
Parallel resonant band-stop filter.
The parallel LC components present a high impedance at
resonant frequency, thereby blocking the signal from the
load at that frequency. Conversely, it passes signals to the
load at any other frequencies. (Figure below)
parallel resonant bandstop filter
vl 10 ac1 sin
11 1 2 100m
cl 1 2 10u
rload 2 0 1k
.ac Lin 20 100 200
.plot ac v(2)
.end
0,00 —$— $<
100,0 150,0 200.0
frequency Hz
Parallel resonant band-stop filter: Notch frequency = LC
resonant frequency (159.15 Hz).
Once again, notice how the absence of a series resistor
makes for minimum attenuation for all the desired (passed)
signals. The amplitude at the notch frequency, on the other
hand, is very low. In other words, this is a very “selective”
filter.
In all these resonant filter designs, the selectivity depends
greatly upon the “purity” of the inductance and capacitance
used. If there is any stray resistance (especially likely in the
inductor), this will diminish the filter's ability to finely
discriminate frequencies, as well as introduce antiresonant
effects that will skew the peak/notch frequency.
A word of caution to those designing low-pass and high-pass
filters is in order at this point. After assessing the standard
RC and LR low-pass and high-pass filter designs, it might
occur to a student that a better, more effective design of
low-pass or high-pass filter might be realized by combining
Capacitive and inductive elements together like Figure
below.
~—— filer ——-~
L, 2 L, 3
| |
100 mH 100 mH
Capacitive Inductive low-pass filter.
The inductors should block any high frequencies, while the
Capacitor should short out any high frequencies as well, both
working together to allow only low frequency signals to
reach the load.
At first, this seems to be a good strategy, and eliminates the
need for a series resistance. However, the more insightful
student will recognize that any combination of capacitors
and inductors together in a circuit is likely to cause resonant
effects to happen at a certain frequency. Resonance, as we
have seen before, can cause strange things to happen. Let's
plot a SPICE analysis and see what happens over a wide
frequency range: (Figure below)
lc lowpass filter
vl 10 ac1 sin
11 1 2 100m
cl 2 0 lu
12 2 3 100m
rload 3 0 1k
.ac Lin 20 100 1k
.plot ac v(3)
end
0,00 0,50 1,00
frequency kHz
Unexpected response of L-C low-pass filter.
What was supposed to be a low-pass filter turns out to be a
band-pass filter with a peak somewhere around 526 Hz! The
Capacitance and inductance in this filter circuit are attaining
resonance at that point, creating a large voltage drop
around C, which is seen at the load, regardless of L,'s
attenuating influence. The output voltage to the load at this
point actually exceeds the input (source) voltage! A little
more reflection reveals that if L] and C, are at resonance,
they will impose a very heavy (very low impedance) load on
the AC source, which might not be good either. We'll run the
same analysis again, only this time plotting C,'s voltage,
vm(2) in Figure below, and the source current, I(v1), along
with load voltage, vm(3):
Units vm(2) — ym(3) Units
— 100*mag(v1#branch)
= I(v1)
0,00 0,50 1,00
(vi) frequency kHz
Current inceases at the unwanted resonance of the L-C low-
pass filter.
Sure enough, we see the voltage across C, and the source
Current spiking to a high point at the same frequency where
the load voltage is maximum. If we were expecting this filter
to provide a simple low-pass function, we might be
disappointed by the results.
The problem is that an L-C filter has an input impedance and
an output impedance which must be matched. The voltage
source impedance must match the input impedance of the
filter, and the filter output impedance must be matched by
“rload” for a flat response. The input and output impedance
is given by the square root of (L/C).
Z.= (LC)
Taking the component values from (Figure below), we can
find the impedance of the filter, and the required , Rg and
Rioag to match it.
For L= 100 mH, C= 1uF
Z = (L/C)2=((100 mH)/(1 uF))”? = 316 0
In Figure below we have added Rg = 316 Q to the generator,
and changed the load Rjgag from 1000 O to 316 Q. Note that
if we needed to drive a 1000 Q load, the L/C ratio could have
been adjusted to match that resistance.
—— 1
3169 , 0OmH , 1l0OmH
3162
Circuit of source and load matched L-C low-pass filter.
LC matched lowpass filter
V1 10 ac 1 SIN
Rg 1 4 316
L1 4 2 100m
C1 2 0 1.0u
L2 2 3 100m
Rload 3 0 316
.ac Lin 20 100 1k
.plot ac v(3)
.end
Figure below shows the “flat” response of the L-C low pass
filter when the source and load impedance match the filter
input and output impedances.
frequency kHz
The response of impedance matched L-C low-pass filter is
nearly flat up to the cut-off frequency.
The point to make in comparing the response of the
unmatched filter (Figure above) to the matched filter (Figure
above) is that variable load on the filter produces a
considerable change in voltage. This property is directly
applicable to L-C filtered power supplies- the regulation is
poor. The power supply voltage changes with a change in
load. This is undesirable.
This poor load regulation can be mitigated by a swinging
choke. This is a choke, inductor, designed to saturate when
a large DC current passes through it. By saturate, we mean
that the DC current creates a “too” high level of flux in the
magnetic core, so that the AC component of current cannot
vary the flux. Since induction is proportional to d®/dt, the
inductance is decreased by the heavy DC current. The
decrease in inductance decreases reactance X,. Decreasing
reactance, reduces the voltage drop across the inductor;
thus, increasing the voltage at the filter output. This
improves the voltage regulation with respect to variable
loads.
Despite the unintended resonance, low-pass filters made up
of capacitors and inductors are frequently used as final
stages in AC/DC power supplies to filter the unwanted AC
“ripple” voltage out of the DC converted from AC. Why is
this, if this particular filter design possesses a potentially
troublesome resonant point?
The answer lies in the selection of filter component sizes and
the frequencies encountered from an AC/DC converter
(rectifier). What we're trying to do in an AC/DC power supply
filter is separate DC voltage from a small amount of
relatively high-frequency AC voltage. The filter inductors
and capacitors are generally quite large (Several Henrys for
the inductors and thousands of uF for the capacitors is
typical), making the filter's resonant frequency very, very
low. DC of course, has a “frequency” of zero, so there's no
way it can make an LC circuit resonate. The ripple voltage,
on the other hand, is a non-sinusoidal AC voltage consisting
of a fundamental frequency at least twice the frequency of
the converted AC voltage, with harmonics many times that
in addition. For plug-in-the-wall power supplies running on
60 Hz AC power (60 Hz United States; 50 Hz in Europe), the
lowest frequency the filter will ever see is 120 Hz (100 Hz in
Europe), which is well above its resonant point. Therefore,
the potentially troublesome resonant point in a such a filter
is completely avoided.
The following SPICE analysis calculates the voltage output
(AC and DC) for such a filter, with series DC and AC (120 Hz)
voltage sources providing a rough approximation of the
mixed-frequency output of an AC/DC converter.
LkQ
AC/DC power suppply filter provides “ripple free” DC power.
ac/dc power supply filter
vl 10 ac 1 sin
v2 2 1 dc
Li- 2.3 3
cl 3 0 9500u
23 42
rload 4 0 1k
.dc v2 12 12 1
.ac Lin 1 120 120
print dc v(4)
.print ac v(4)
.end
v2 v(4)
1.200E+01 1.200E+01 DC voltage at load = 12 volts
freq v(4)
1.200E+02 3.412E-05 AC voltage at load = 34.12
microvolts
With a full 12 volts DC at the load and only 34.12 uV of AC
left from the 1 volt AC source imposed across the load, this
circuit design proves itself to be a very effective power
supply filter.
The lesson learned here about resonant effects also applies
to the design of high-pass filters using both capacitors and
inductors. So long as the desired and undesired frequencies
are well to either side of the resonant point, the filter will
work OK. But if any signal of significant magnitude close to
the resonant frequency is applied to the input of the filter,
strange things will happen!
e REVIEW:
e Resonant combinations of capacitance and inductance
can be employed to create very effective band-pass and
band-stop filters without the need for added resistance
in a circuit that would diminish the passage of desired
frequencies.
f l
resonant 7
: 2n \/ LC
Summary
As lengthy as this chapter has been up to this point, it only
begins to scratch the surface of filter design. A quick perusal
of any advanced filter design textbook is sufficient to prove
my point. The mathematics involved with component
selection and frequency response prediction is daunting to
say the least -- well beyond the scope of the beginning
electronics student. It has been my intent here to present
the basic principles of filter design with as little math as
possible, leaning on the power of the SPICE circuit analysis
program to explore filter performance. The benefit of such
computer simulation software cannot be understated, for the
beginning student or for the working engineer.
Circuit simulation software empowers the student to explore
circuit designs far beyond the reach of their math skills. With
the ability to generate Bode plots and precise figures, an
intuitive understanding of circuit concepts can be attained,
which is something often lost when a student is burdened
with the task of solving lengthy equations by hand. If you
are not familiar with the use of SPICE or other circuit
simulation programs, take the time to become so! It will be
of great benefit to your study. To see SPICE analyses
presented in this book is an aid to understanding circuits,
but to actually set up and analyze your own circuit
simulations is a much more engaging and worthwhile
endeavor as a student.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+4]l—
—|}|4/l—
Lessons In Electric Circuits
-- Volume Il
Chapter 9
TRANSFORMERS
Mutual inductance and basic operation
Step-up_ and step-down transformers
Electrical isolation
Phasing
Winding_configurations
Voltage regulation
Special transformers and applications
Impedance matching
Potential transformers
Current transformers
Air core transformers
Tesla Coil
Saturable reactors
o Scott-T transformer
o Linear Variable Differential Transformer
Practical considerations
o Power capacity
Energy losses
Stray capacitance and inductance
Core saturation
Inrush current
Heat and Noise
Contributors
Bibliography
o O 0 0 0 °O
o Oo O08 0 O
Mutual inductance and basic
operation
Suppose we were to wrap a coil of insulated wire around a
loop of ferromagnetic material and energize this coil with an
AC voltage source: (Figure below (a))
(b)
Insulated winding on ferromagnetic loop has inductive
reactance, limiting AC current.
As an inductor, we would expect this iron-core coil to oppose
the applied voltage with its inductive reactance, limiting
current through the coil as predicted by the equations X, =
2nfL and |I=E/X (or l=E/Z). For the purposes of this example,
though, we need to take a more detailed look at the
interactions of voltage, current, and magnetic flux in the
device.
Kirchhoff's voltage law describes how the algebraic sum of all
voltages in a loop must equal zero. In this example, we could
apply this fundamental law of electricity to describe the
respective voltages of the source and of the inductor coil.
Here, aS in any one-source, one-load circuit, the voltage
dropped across the load must equal the voltage supplied by
the source, assuming zero voltage dropped along the
resistance of any connecting wires. In other words, the load
(inductor coil) must produce an opposing voltage equal in
magnitude to the source, in order that it may balance against
the source voltage and produce an algebraic loop voltage
sum of zero. From where does this opposing voltage arise? If
the load were a resistor (Figure above (b)), the voltage drop
originates from electrical energy loss, the “friction” of
electrons flowing through the resistance. With a perfect
inductor (no resistance in the coil wire), the opposing voltage
comes from another mechanism: the reaction to a changing
magnetic flux in the iron core. When AC current changes, flux
® changes. Changing flux induces a counter EMF.
Michael Faraday discovered the mathematical relationship
between magnetic flux (®) and induced voltage with this
equation:
e= N—
dt
Where,
e = (Instantaneous) induced voltage in volts
N= Number of turns in wire coil (straight wire = 1)
® = Magnetic flux in Webers
t= Time in seconds
The instantaneous voltage (voltage dropped at any instant in
time) across a wire coil is equal to the number of turns of that
coil around the core (N) multiplied by the instantaneous rate-
of-change in magnetic flux (d®/dt) linking with the coil.
Graphed, (Figure below) this shows itself as a set of sine
waves (assuming a sinusoidal voltage source), the flux wave
90° lagging behind the voltage wave:
e = voltage ® = magnetic flux
e ®
Magnetic flux lags applied voltage by 90° because flux is
proportional to a rate of change, d®/dt.
Magnetic flux through a ferromagnetic material is analogous
to current through a conductor: it must be motivated by
some force in order to occur. In electric circuits, this
motivating force is voltage (a.k.a. electromotive force, or
EMF). In magnetic “circuits,” this motivating force is
magnetomotive force, or mmf. Magnetomotive force (mmf)
and magnetic flux (®) are related to each other by a property
of magnetic materials known as re/uctance (the latter
quantity symbolized by a strange-looking letter “R”):
A comparison of "Ohm's Law" for
electric and magnetic circuits:
E=I1R mmf = DR
Electrical Magnetic
In our example, the mmf required to produce this changing
magnetic flux (®) must be supplied by a changing current
through the coil. Magnetomotive force generated by an
electromagnet coil is equal to the amount of current through
that coil (in amps) multiplied by the number of turns of that
coil around the core (the SI unit for mmf is the amp-turn).
Because the mathematical relationship between magnetic
flux and mmf is directly proportional, and because the
mathematical relationship between mmf and current is also
directly proportional (no rates-of-change present in either
equation), the current through the coil will be in-phase with
the flux wave as in (Figure below)
e=voltage &=magnetic flux i= coil current
: P
Magnetic flux, like current, lags applied voltage by 90°.
This is why alternating current through an inductor lags the
applied voltage waveform by 90°: because that is what is
required to produce a changing magnetic flux whose rate-of-
change produces an opposing voltage in-phase with the
applied voltage. Due to its function in providing magnetizing
force (mmf) for the core, this current is sometimes referred to
as the magnetizing current.
It should be mentioned that the current through an iron-core
inductor is not perfectly sinusoidal (Sine-wave shaped), due
to the nonlinear B/H magnetization curve of iron. In fact, if
the inductor is cheaply built, using as little iron as possible,
the magnetic flux density might reach high levels
(approaching saturation), resulting in a magnetizing current
waveform that looks something like Figure below
e = voltage
® = magnetic flux
i= coil current
: ®
As flux density approaches saturation, the magnetizing
current waveform becomes distorted.
When a ferromagnetic material approaches magnetic flux
saturation, disproportionately greater levels of magnetic field
force (mmf) are required to deliver equal increases in
magnetic field flux (®). Because mmf is proportional to
current through the magnetizing coil (mmf = NI, where “N” is
the number of turns of wire in the coil and “I” is the current
through it), the large increases of mmf required to supply the
needed increases in flux results in large increases in coil
current. Thus, coil current increases dramatically at the
peaks in order to maintain a flux waveform that isn't
distorted, accounting for the bell-shaped half-cycles of the
current waveform in the above plot.
The situation is further complicated by energy losses within
the iron core. The effects of hysteresis and eddy currents
conspire to further distort and complicate the current
waveform, making it even less sinusoidal and altering its
phase to be lagging slightly less than 90° behind the applied
voltage waveform. This coil current resulting from the sum
total of all magnetic effects in the core (d®/dt magnetization
plus hysteresis losses, eddy current losses, etc.) is called the
exciting current. The distortion of an iron-core inductor's
exciting current may be minimized if it is designed for and
operated at very low flux densities. Generally speaking, this
requires a core with large cross-sectional area, which tends to
make the inductor bulky and expensive. For the sake of
simplicity, though, we'll assume that our example core is far
from saturation and free from all losses, resulting in a
perfectly sinusoidal exciting current.
As we've seen already in the inductors chapter, having a
current waveform 90° out of phase with the voltage
waveform creates a condition where power is alternately
absorbed and returned to the circuit by the inductor. If the
inductor is perfect (no wire resistance, no magnetic core
losses, etc.), it will dissipate zero power.
Let us now consider the same inductor device, except this
time with a second coil (Figure below) wrapped around the
Same iron core. The first coil will be labeled the primary coil,
while the second will be labeled the secondary:
Ferromagnetic core with primary coil (AC driven) and
secondary coil.
If this secondary coil experiences the same magnetic flux
change as the primary (which it should, assuming perfect
containment of the magnetic flux through the common core),
and has the same number of turns around the core, a voltage
of equal magnitude and phase to the applied voltage will be
induced along its length. In the following graph, (Figure
below) the induced voltage waveform is drawn slightly
smaller than the source voltage waveform simply to
distinguish one from the other:
e, = primary coil voltage i, = primary coil current
& = magnetic flux e, = secondary coil voltage
Open circuited secondary sees the same flux ® as the
primary. Therefore induced secondary voltage e, is the same
magnitude and phase as the primary voltage ep,
This effect is called mutual inductance: the induction of a
voltage in one coil in response to a change in current in the
other coil. Like normal (self-) inductance, it is measured in
the unit of Henrys, but unlike normal inductance it is
symbolized by the capital letter “M” rather than the letter “L’:
Inductance Mutual inductance
e= i cal e&= iy
dt 7 dt
Where,
e, = voltage induced in
secondary coil
i; = Current in primary
coil
No current will exist in the secondary coil, since it is open-
circuited. However, if we connect a load resistor to it, an
alternating current will go through the coil, in-phase with the
induced voltage (because the voltage across a resistor and
the current through it are a/ways in-phase with each other).
(Figure below)
Resistive load on secondary has voltage and current in-
phase.
At first, one might expect this secondary coil current to cause
additional magnetic flux in the core. In fact, it does not. If
more flux were induced in the core, it would cause more
voltage to be induced voltage in the primary coil (remember
that e = d®/dt). This cannot happen, because the primary
coil's induced voltage must remain at the same magnitude
and phase in order to balance with the applied voltage, in
accordance with Kirchhoff's voltage law. Consequently, the
magnetic flux in the core cannot be affected by secondary
coil current. However, what does change is the amount of
mmf in the magnetic circuit.
Magnetomotive force is produced any time electrons move
through a wire. Usually, this mmf is accompanied by
magnetic flux, in accordance with the mmf=OR “magnetic
Ohm's Law” equation. In this case, though, additional flux is
not permitted, so the only way the secondary coil's mmf may
exist is if a counteracting mmf is generated by the primary
coil, of equal magnitude and opposite phase. Indeed, this is
what happens, an alternating current forming in the primary
coil -- 180° out of phase with the secondary coil's current -- to
generate this counteracting mmf and prevent additional core
flux. Polarity marks and current direction arrows have been
added to the illustration to clarify phase relations: (Figure
below)
m mf, mary
m MT econdary
Flux remains constant with application of a load. However, a
counteracting mmf is produced by the loaded secondary.
If you find this process a bit confusing, do not worry.
Transformer dynamics is a complex subject. What is
important to understand is this: when an AC voltage is
applied to the primary coil, it creates a magnetic flux in the
core, which induces AC voltage in the secondary coil in-
phase with the source voltage. Any current drawn through
the secondary coil to power a load induces a corresponding
current in the primary coil, drawing current from the source.
Notice how the primary coil is behaving as a load with
respect to the AC voltage source, and how the secondary coil
is behaving as a source with respect to the resistor. Rather
than energy merely being alternately absorbed and returned
the primary coil circuit, energy is now being coupled to the
secondary coil where it is delivered to a dissipative (energy-
consuming) load. As far as the source “knows,” its directly
powering the resistor. Of course, there is also an additional
primary coil current lagging the applied voltage by 90°, just
enough to magnetize the core to create the necessary
voltage for balancing against the source (the exciting
current).
We call this type of device a transformer, because it
transforms electrical energy into magnetic energy, then back
into electrical energy again. Because its operation depends
on electromagnetic induction between two stationary coils
and a magnetic flux of changing magnitude and “polarity,”
transformers are necessarily AC devices. Its schematic
symbol looks like two inductors (coils) sharing the same
magnetic core: (Figure below)
Transformer
s|lé
Schematic symbol for transformer consists of two inductor
symbols, separated by lines indicating a ferromagnetic core.
The two inductor coils are easily distinguished in the above
symbol. The pair of vertical lines represent an iron core
common to both inductors. While many transformers have
ferromagnetic core materials, there are some that do not,
their constituent inductors being magnetically linked
together through the air.
The following photograph shows a power transformer of the
type used in gas-discharge lighting. Here, the two inductor
coils can be clearly seen, wound around an iron core. While
most transformer designs enclose the coils and coreina
metal frame for protection, this particular transformer is open
for viewing and so serves its illustrative purpose well: (Figure
below)
Example of a gas-discharge lighting transformer.
Both coils of wire can be seen here with copper-colored
varnish insulation. The top coil is larger than the bottom coil,
having a greater number of “turns” around the core. In
transformers, the inductor coils are often referred to as
windings, in reference to the manufacturing process where
wire iS wound around the core material. As modeled in our
initial example, the powered inductor of a transformer is
called the primary winding, while the unpowered coil is
called the secondary winding.
In the next photograph, Figure below, a transformer is shown
cut in half, exposing the cross-section of the iron core as well
as both windings. Like the transformer shown previously, this
unit also utilizes primary and secondary windings of differing
turn counts. The wire gauge can also be seen to differ
between primary and secondary windings. The reason for this
disparity in wire gauge will be made clear in the next section
of this chapter. Additionally, the iron core can be seen in this
photograph to be made of many thin sheets (laminations)
rather than a solid piece. The reason for this will also be
explained in a later section of this chapter.
Transformer cross-section cut shows core and windings.
It is easy to demonstrate simple transformer action using
SPICE, setting up the primary and secondary windings of the
simulated transformer as a pair of “mutual” inductors. (Figure
2low) The coefficient of magnetic field coupling is given at
the end of the “k” line in the SPICE circuit description, this
example being set very nearly at perfection (1.000). This
coefficient describes how closely “linked” the two inductors
are, magnetically. The better these two inductors are
magnetically coupled, the more efficient the energy transfer
between them should be.
(for SPICE to measure current)
1 Rrosus1 o 3 4
(very smal)
Rpogus2
Spice circuit for coupled inductors.
transformer
v1 10 ac 10 sin
rbogus1 1 2 le-12
rbogus2 5 0 9e12
l1 2 0 100
12 3 5 100
** This Line tells SPICE that the two inductors
** Ll and 12 are magnetically “Linked” together
k l1 12 0.999
vil 3 4 ac 0
rload 4 5 1k
.ac lin 1 60 60
.print ac v(2,0) i(vl)
.print ac v(3,5) i(vil)
.end
Note: the Rpogus resistors are required to satisfy certain
quirks of SPICE. The first breaks the otherwise continuous
loop between the voltage source and L, which would not be
permitted by SPICE. The second provides a path to ground
(node O) from the secondary circuit, necessary because
SPICE cannot function with any ungrounded circuits.
freq v(2) i(vl)
6.000E+01 1.000E+01 9.975E-03 Primary winding
freq v(3,5) i(vil)
6.000E+01 9.962E+00 9.962E-03 Secondary winding
Note that with equal inductances for both windings (100
Henrys each), the AC voltages and currents are nearly equal
for the two. The difference between primary and secondary
currents is the magnetizing current spoken of earlier: the 90°
lagging current necessary to magnetize the core. As is seen
here, it is usually very small compared to primary current
induced by the load, and so the primary and secondary
currents are almost equal. What you are seeing here is quite
typical of transformer efficiency. Anything less than 95%
efficiency is considered poor for modern power transformer
designs, and this transfer of power occurs with no moving
parts or other components subject to wear.
If we decrease the load resistance so as to draw more current
with the same amount of voltage, we see that the current
through the primary winding increases in response. Even
though the AC power source is not directly connected to the
load resistance (rather, it is electromagnetically “coupled”),
the amount of current drawn from the source will be almost
the same as the amount of current that would be drawn if the
load were directly connected to the source. Take a close look
at the next two SPICE simulations, showing what happens
with different values of load resistors:
transformer
v1 10 ac 10 sin
rbogusl 1 2 le-12
rbogus2 5 0 9e12
l1 2 0 100
12 3 5 100
k L1 12 0.999
vil 3 4 ac 0
** Note load resistance value of 200 ohms
rload 4 5 200
.ac Lin 1 60 60
.print ac v(2,0) i(vl)
.print ac v(3,5) i(vil)
.end
freq v(2) i(vl1)
6.000E+01 1.000E+01 4.679E-02
freq v(3,5) i(vil)
6.000E+01 9.348E+00 4.674E-02
Notice how the primary current closely follows the secondary
current. In our first simulation, both currents were
approximately 10 mA, but now they are both around 47 mA.
In this second simulation, the two currents are closer to
equality, because the magnetizing current remains the same
as before while the load current has increased. Note also how
the secondary voltage has decreased some with the heavier
(greater current) load. Let's try another simulation with an
even lower value of load resistance (15 Q):
transformer
v1 10 ac 10 sin
rbogus1l 1 2 le-12
rbogus2 5 0 9e12
l1 2 0 100
12 3 5 100
k L1 12 0.999
vil 3 4 ac 0
rload 4 5 15
.ac lin 1 60 60
.print ac v(2,0) i(vl1)
.print ac v(3,5) i(vil)
.end
freq v(2) i(vl)
6.000E+01 1.000E+01 1.301E-01
freq v(3,5) i(vil)
6.000E+01 1.950E+00 1.300E-01
Our load current is now 0.13 amps, or 130 mA, which is
substantially higher than the last time. The primary current is
very close to being the same, but notice how the secondary
voltage has fallen well below the primary voltage (1.95 volts
versus 10 volts at the primary). The reason for this is an
imperfection in our transformer design: because the primary
and secondary inductances aren't perfectly linked (a k factor
of 0.999 instead of 1.000) there is “stray” or “/eakage”
inductance. In other words, some of the magnetic field isn't
linking with the secondary coil, and thus cannot couple
energy to it: (Figure below)
leakage
flux
leakage
flux
Leakage inductance is due to magnetic flux not cutting both
windings.
Consequently, this “leakage” flux merely stores and returns
energy to the source circuit via self-inductance, effectively
acting as a series impedance in both primary and secondary
circuits. Voltage gets dropped across this series impedance,
resulting in a reduced load voltage: voltage across the load
“sags” as load current increases. (Figure below)
ideal
transformer _------~
“sss:
ss ~
>
Equivalent circuit models leakage inductance as series
inductors independent of the “ideal transformer”.
If we change the transformer design to have better magnetic
coupling between the primary and secondary coils, the
figures for voltage between primary and secondary windings
will be much closer to equality again:
transformer
v1 10 ac 10 sin
rbogus1 1 2 le-12
rbogus2 5 0 9e12
ll 2 0 100
12 3 5 100
** Coupling factor = 0.99999 instead of 0.999
k 11 12 0.99999
vil 3 4 ac 0
rload 4 5 15
.ac Lin 1 60 60
.print ac v(2,0) i(vl1)
.print ac v(3,5) i(vil)
.end
freq v(2) i(vl)
6.000E+01 1.000E+01 6.658E-01
freq v(3,5) i(vil)
6.000E+01 9.987E+00 6.658E-01
Here we see that our secondary voltage is back to being
equal with the primary, and the secondary current is equal to
the primary current as well. Unfortunately, building a real
transformer with coupling this complete is very difficult. A
compromise solution is to design both primary and secondary
coils with less inductance, the strategy being that less
inductance overall leads to less “leakage” inductance to
cause trouble, for any given degree of magnetic coupling
inefficiency. This results in a load voltage that is closer to
ideal with the same (high current heavy) load and the same
coupling factor:
transformer
vl 10 ac 10 sin
rbogusl 1 2 le-12
rbogus2 5 0 9e12
** inductance = 1 henry instead of 100 henrys
11201
12°33. 5.1
k Ll 12 0.999
vil 3 4 ac 0
rload 4 5 15
.ac lin 1 60 60
.print ac v(2,0) i(vl)
.print ac v(3,5) i(vil)
.end
freq v(2) i(vl)
6.000E+01 1.000E+01 6.664E-01
freq v(3,5) i(vil)
6.000E+01 9.977E+00 6.652E-01
Simply by using primary and secondary coils of less
inductance, the load voltage for this heavy load (high
current) has been brought back up to nearly ideal levels
(9.977 volts). At this point, one might ask, “If less inductance
is all that's needed to achieve near-ideal performance under
heavy load, then why worry about coupling efficiency at all?
If its impossible to build a transformer with perfect coupling,
but easy to design coils with low inductance, then why not
just build all transformers with low-inductance coils and have
excellent efficiency even with poor magnetic coupling?”
The answer to this question is found in another simulation:
the same low-inductance transformer, but this time with a
lighter load (less current) of 1 kQ instead of 15 Q:
transformer
v1 10 ac 10 sin
rbogus1 1 2 le-12
rbogus2 5 0 9e12
11201
TZ. 3: Be
k L1 12 0.999
vil 3 4 ac 0
rload 4 5 1k
.ac lin 1 60 60
.print ac v(2,0) i(vl1)
.print ac v(3,5) i(vil)
.end
freq v(2) i(vl)
6.000E+01 1.000E+01 2.835E-02
freq v(3,5) i(vil)
6.000E+01 9.990E+00 9.990E-03
With lower winding inductances, the primary and secondary
voltages are closer to being equal, but the primary and
secondary currents are not. In this particular case, the
primary current is 28.35 mA while the secondary current is
only 9.990 mA: almost three times as much current in the
primary as the secondary. Why is this? With less inductance
in the primary winding, there is less inductive reactance, and
consequently a much larger magnetizing current. A
substantial amount of the current through the primary
winding merely works to magnetize the core rather than
transfer useful energy to the secondary winding and load.
An ideal transformer with identical primary and secondary
windings would manifest equal voltage and current in both
sets of windings for any load condition. In a perfect world,
transformers would transfer electrical power from primary to
secondary as smoothly as though the load were directly
connected to the primary power source, with no transformer
there at all. However, you can see this ideal goal can only be
met if there is perfect coupling of magnetic flux between
primary and secondary windings. Being that this is
impossible to achieve, transformers must be designed to
operate within certain expected ranges of voltages and loads
in order to perform as close to ideal as possible. For now, the
most important thing to keep in mind is a transformer's basic
operating principle: the transfer of power from the primary to
the secondary circuit via electromagnetic coupling.
e REVIEW:
e Mutual inductance is where the magnetic flux of two or
more inductors are “linked” so that voltage is induced in
one coil proportional to the rate-of-change of current in
another.
e A transformer is a device made of two or more inductors,
one of which is powered by AC, inducing an AC voltage
across the second inductor. If the second inductor is
connected to a load, power will be electromagnetically
coupled from the first inductor's power source to that
load.
e The powered inductor in a transformer is called the
primary winding. The unpowered inductor in a
transformer is called the secondary winding.
e Magnetic flux in the core (®) lags 90° behind the source
voltage waveform. The current drawn by the primary coil
from the source to produce this flux is called the
magnetizing current, and it also lags the supply voltage
by 90°.
e Total primary current in an unloaded transformer is called
the exciting current, and is comprised of magnetizing
current plus any additional current necessary to
overcome core losses. It is never perfectly sinusoidal ina
real transformer, but may be made more so if the
transformer is designed and operated so that magnetic
flux density is kept to a minimum.
e Core flux induces a voltage in any coil wrapped around
the core. The induces voltage(s) are ideally in- phase
with the primary winding source voltage and share the
Same waveshape.
e Any current drawn through the secondary winding by a
load will be “reflected” to the primary winding and drawn
from the voltage source, as if the source were directly
powering a similar load.
Step-up and step-down transformers
So far, we've observed simulations of transformers where the
primary and secondary windings were of identical
inductance, giving approximately equal voltage and current
levels in both circuits. Equality of voltage and current
between the primary and secondary sides of a transformer,
however, is not the norm for all transformers. If the
inductances of the two windings are not equal, something
interesting happens:
transformer
v1 10 ac 10 sin
rbogusl 1 2 le-12
rbogus2 5 0 9e12
l1 2 0 10000
12 3 5 100
k Ll1 12 0.999
vil 3 4 ac 0
rload 4 5 1k
.ac lin 1 60 60
.print ac v(2,0) i(vl)
.print ac v(3,5) i(vil)
.end
freq v(2) i(vl1)
6.000E+01 1.000E+01 9.975E-05 Primary winding
freq v(3,5) i(vil)
6.000E+01 9.962E-01 9.962E-04 Secondary winding
Notice how the secondary voltage is approximately ten times
less than the primary voltage (0.9962 volts compared to 10
volts), while the secondary current is approximately ten
times greater (0.9962 mA compared to 0.09975 mA). What
we have here is a device that steps voltage down by a factor
of ten and current up by a factor of ten: (Figure below)
Primary | Secondary
winding : winding
Turns ratio of 10:1 yields 10:1 primary:secondary voltage
ratio and 1:10 primary:secondary current ratio.
This is a very useful device, indeed. With it, we can easily
multiply or divide voltage and current in AC circuits. Indeed,
the transformer has made long-distance transmission of
electric power a practical reality, as AC voltage can be
“stepped up” and current “stepped down” for reduced wire
resistance power losses along power lines connecting
generating stations with loads. At either end (both the
generator and at the loads), voltage levels are reduced by
transformers for safer operation and less expensive
equipment. A transformer that increases voltage from
primary to secondary (more secondary winding turns than
primary winding turns) is called a step-up transformer.
Conversely, a transformer designed to do just the opposite is
called a step-down transformer.
Let's re-examine a photograph shown in the previous section:
(Figure below)
Transformer cross-section showing primary and secondary
windings Is a few inches tall (approximately 10 cm).
This is a step-down transformer, as evidenced by the high
turn count of the primary winding and the low turn count of
the secondary. As a step-down unit, this transformer converts
high-voltage, low-current power into low-voltage, high-
current power. The larger-gauge wire used in the secondary
winding is necessary due to the increase in current. The
primary winding, which doesn't have to conduct as much
current, may be made of smaller-gauge wire.
In case you were wondering, it /s possible to operate either of
these transformer types backwards (powering the secondary
winding with an AC source and letting the primary winding
power a load) to perform the opposite function: a step-up can
function as a step-down and viSa-versa. However, as we saw
in the first section of this chapter, efficient operation of a
transformer requires that the individual winding inductances
be engineered for specific operating ranges of voltage and
current, so if a transformer is to be used “backwards” like this
it must be employed within the original design parameters of
voltage and current for each winding, lest it prove to be
inefficient (or lest it be damaged by excessive voltage or
current!).
Transformers are often constructed in such a way that it is
not obvious which wires lead to the primary winding and
which lead to the secondary. One convention used in the
electric power industry to help alleviate confusion is the use
of “H” designations for the higher-voltage winding (the
primary winding in a step-down unit; the secondary winding
in a step-up) and “X” designations for the lower-voltage
winding. Therefore, a simple power transformer will have
wires labeled “H,”, “H5”, “X,”, and “X5”. There is usually
significance to the numbering of the wires (H; versus H>,
etc.), which we'll explore a little later in this chapter.
The fact that voltage and current get “stepped” in opposite
directions (one up, the other down) makes perfect sense
when you recall that power is equal to voltage times current,
and realize that transformers cannot produce power, only
convert it. Any device that could output more power than it
took in would violate the Law of Energy Conservation in
physics, namely that energy cannot be created or destroyed,
only converted. As with the first transformer example we
looked at, power transfer efficiency is very good from the
primary to the secondary sides of the device.
The practical significance of this is made more apparent
when an alternative is considered: before the advent of
efficient transformers, voltage/current level conversion could
only be achieved through the use of motor/generator sets. A
drawing of a motor/generator set reveals the basic principle
involved: (Figure below)
A motor/generator set
Power Power
In out
Shaft
coupling
Generator
Motor generator i/lustrates the basic principle of the
transformer.
In such a machine, a motor is mechanically coupled to a
generator, the generator designed to produce the desired
levels of voltage and current at the rotating speed of the
motor. While both motors and generators are fairly efficient
devices, the use of both in this fashion compounds their
inefficiencies so that the overall efficiency is in the range of
90% or less. Furthermore, because motor/generator sets
obviously require moving parts, mechanical wear and
balance are factors influencing both service life and
performance. Transformers, on the other hand, are able to
convert levels of AC voltage and current at very high
efficiencies with no moving parts, making possible the
widespread distribution and use of electric power we take for
granted.
In all fairness it should be noted that motor/generator sets
have not necessarily been obsoleted by transformers for a//
applications. While transformers are clearly Superior over
motor/generator sets for AC voltage and current level
conversion, they cannot convert one frequency of AC power
to another, or (by themselves) convert DC to AC or visa-
versa. Motor/generator sets can do all these things with
relative simplicity, albeit with the limitations of efficiency
and mechanical factors already described. Motor/generator
sets also have the unique property of kinetic energy storage:
that is, if the motor's power supply is momentarily
interrupted for any reason, its angular momentum (the
inertia of that rotating mass) will maintain rotation of the
generator for a short duration, thus isolating any loads
powered by the generator from “glitches” in the main power
system.
Looking closely at the numbers in the SPICE analysis, we
should see a correspondence between the transformer's ratio
and the two inductances. Notice how the primary inductor
(IL) has 100 times more inductance than the secondary
inductor (10000 H versus 100 H), and that the measured
voltage step-down ratio was 10 to 1. The winding with more
inductance will have higher voltage and less current than the
other. Since the two inductors are wound around the same
core material in the transformer (for the most efficient
magnetic coupling between the two), the parameters
affecting inductance for the two coils are equal except for the
number of turns in each coil. lf we take another look at our
inductance formula, we see that inductance is proportional to
the square of the number of coil turns:
N7A
|
Where,
L= Inductance of coil in Henrys
N= Number of turns in wire coil (straight wire = 1)
u= Permeability of core material (absolute, not relative)
A = Area of coil in square meters
| = Average length of coil in meters
L=
So, it should be apparent that our two inductors in the last
SPICE transformer example circuit -- with inductance ratios of
100:1 -- should have coil turn ratios of 10:1, because 10
squared equals 100. This works out to be the same ratio we
found between primary and secondary voltages and currents
(10:1), so we can say as a rule that the voltage and current
transformation ratio is equal to the ratio of winding turns
between primary and secondary.
Step-down transformer
many turnss few turns
load
(V) high voltage S || = low voltage
low current 3
~ high current
Step-down transformer: (many turns :few turns).
The step-up/step-down effect of coil turn ratios ina
transformer (Figure above) is analogous to gear tooth ratios
in mechanical gear systems, transforming values of speed
and torque in much the same way: (Figure below)
LARGE GEAR
(many teeth)
SMALL GEAR
(few teeth)
low torque
high torque high speed
low speed
Torque reducing gear train steps torque down, while stepping
speed up.
Step-up and step-down transformers for power distribution
purposes can be gigantic in proportion to the power
transformers previously shown, some units standing as tall as
a home. The following photograph shows a substation
transformer standing about twelve feet tall: (Figure below)
b
=»
=
=
=
=
=
=
-—
cs
c—
ie
™
.
Substation transformer.
e REVIEW:
e Transformers “step up” or “step down” voltage according
to the ratios of primary to secondary wire turns.
N .
Voltage transformation ratio = —e
primacy
Current transformation ratio = primary
‘seco nary
Where,
e N=number of turns in winding
e A transformer designed to increase voltage from primary
to secondary is called a step-up transformer. A
transformer designed to reduce voltage from primary to
secondary is called a step-down transformer.
e The transformation ratio of a transformer will be equal to
the square root of its primary to secondary inductance (L)
ratio.
, : : : ; L omiay
V oltage transtormation ratio = L
e proagy
Electrical isolation
Aside from the ability to easily convert between different
levels of voltage and current in AC and DC circuits,
transformers also provide an extremely useful feature called
isolation, which is the ability to couple one circuit to another
without the use of direct wire connections. We can
demonstrate an application of this effect with another SPICE
simulation: this time showing “ground” connections for the
two circuits, imposing a high DC voltage between one circuit
and ground through the use of an additional voltage source:
(Figure below)
(for SPICE to measure current) Vi
1 Riogus 2 3
Lo
Transformer isolates 10 V;- at V; from 250 Vpc at V>.
v1 10 ac 10 sin
rbogusl 1 2 le-12
v2 5 0 dc 250
l1 2 0 10000
12 3 5 100
k 11 12 0.999
vil 3 4 ac 0
rload 4 5 1k
.ac lin 1 60 60
.print ac v(2,0) i(vl1)
.print ac v(3,5) i(vil)
.end
DC voltages referenced to ground (node 0):
(1) 0.0000 (2) 0.0000 (3) 250.0000
(4) 250.0000 (5) 250.0000
AC voltages:
freq v(2) i(vl)
6.000E+01 1.000E+01 9.975E-05 Primary winding
freq v(3,5) i(vil)
6.000E+01 9.962E-01 9.962E-04 Secondary winding
SPICE shows the 250 volts DC being impressed upon the
secondary circuit elements with respect to ground, (Figure
above) but as you can see there is no effect on the primary
circuit (zero DC voltage) at nodes 1 and 2, and the
transformation of AC power from primary to secondary
circuits remains the same as before. The impressed voltage in
this example is often called a common-mode voltage
because it is seen at more than one point in the circuit with
reference to the common point of ground. The transformer
isolates the common-mode voltage so that it is not impressed
upon the primary circuit at all, but rather isolated to the
secondary side. For the record, it does not matter that the
common-mode voltage is DC, either. It could be AC, even ata
different frequency, and the transformer would isolate it from
the primary circuit all the same.
There are applications where electrical isolation is needed
between two AC circuit without any transformation of voltage
or current levels. In these instances, transformers called
isolation transformers having 1:1 transformation ratios are
used. A benchtop isolation transformer is shown in Figure
below.
Isolation transformer isolates power out from the power line.
« REVIEW:
e By being able to transfer power from one circuit to
another without the use of interconnecting conductors
between the two circuits, transformers provide the useful
feature of e/ectrical isolation.
e Transformers designed to provide electrical isolation
without stepping voltage and current either up or down
are called isolation transformers.
Phasing
Since transformers are essentially AC devices, we need to be
aware of the phase relationships between the primary and
secondary circuits. Using our SPICE example from before, we
can plot the waveshapes (Figure below) for the primary and
secondary circuits and see the phase relations for ourselves:
Spice transient analysis file for use with nutmeg:
transformer
v1 10 sin(@ 15 60 0 0)
rbogus1 1 2 le-12
v2 5 0 dc 250
l1 2 0 10000
12 3 5 100
k 11 12 0.999
vil 3 4 ac 0
rload 4 5 1k
.tran 0.5m 17m
end
nutmeg commands:
setplot tranl
plot v(2) v(3,5)
y = y(2) — (3,5)
v(2) :
Secondary voltage V(3,5) is in-phase with primary voltage
V(2), and stepped down by factor of ten.
In going from primary, V(2), to secondary, V(3,5), the voltage
was stepped down by a factor of ten, (Figure above) , and the
current was stepped up by a factor of 10. (Figure below) Both
current (Figure below) and voltage (Figure above) waveforms
are in-phase in going from primary to secondary.
nutmeg commands:
setplot tranl
plot I(L1#branch) I(L2#branch)
mUnits- I(L1#branch I(L2#branch)
1.0 povessereagangestsesstnssernnsetnssanaen sresnnsennnse :
: I(L2#branch)} : :
ee
1(L1#branctr
es
0,0
HOG freeeeeeeessndessseeessneen
-1,0 CeeeeeeeeneeecceceveccceeeuscecesescevceseessWmmMs cceceveeeeeeeeeeeeeusener
0,0
time mS
Primary and secondary currents are in-phase. Secondary
current is stepped up by a factor of ten.
It would appear that both voltage and current for the two
transformer windings are in-phase with each other, at least
for our resistive load. This is simple enough, but it would be
nice to know which way we should connect a transformer in
order to ensure the proper phase relationships be kept. After
all, a transformer is nothing more than a set of magnetically-
linked inductors, and inductors don't usually come with
polarity markings of any kind. If we were to look at an
unmarked transformer, we would have no way of knowing
which way to hook it up to a circuit to get in-phase (or 180°
out-of-phase) voltage and current: (Figure below)
+ + =
‘E or 2???
. - +
As a practical matter, the polarity of a transformer can be
ambiguous.
Since this is a practical concern, transformer manufacturers
have come up with a sort of polarity marking standard to
denote phase relationships. It is called the dot convention,
and is nothing more than a dot placed next to each
corresponding leg of a transformer winding: (Figure below)
a _
a UNE
A pair of dots indicates like polarity.
Typically, the transformer will come with some kind of
schematic diagram labeling the wire leads for primary and
secondary windings. On the diagram will be a pair of dots
similar to what is seen above. Sometimes dots will be
omitted, but when “H” and “X” labels are used to label
transformer winding wires, the subscript numbers are
Supposed to represent winding polarity. The “1” wires (H,
and Xj) represent where the polarity-marking dots would
normally be placed.
The similar placement of these dots next to the top ends of
the primary and secondary windings tells us that whatever
instantaneous voltage polarity seen across the primary
winding will be the same as that across the secondary
winding. In other words, the phase shift from primary to
secondary will be zero degrees.
On the other hand, if the dots on each winding of the
transformer do not match up, the phase shift will be 180°
between primary and secondary, like this: (Figure below)
ma sa
NS
Out of phase: primary red to dot, secondary black to dot.
Of course, the dot convention only tells you which end of
each winding is which, relative to the other winding(s). If you
want to reverse the phase relationship yourself, all you have
to do is swap the winding connections like this: (Figure
below)
- ae
=o HE UNS
In phase: primary red to dot, secondary red to dot.
e REVIEW:
e The phase relationships for voltage and current between
primary and secondary circuits of a transformer are
direct: ideally, zero phase shift.
e The dot convention is a type of polarity marking for
transformer windings showing which end of the winding
is which, relative to the other windings.
Winding configurations
Transformers are very versatile devices. The basic concept of
energy transfer between mutual inductors is useful enough
between a single primary and single secondary coil, but
transformers don't have to be made with just two sets of
windings. Consider this transformer circuit: (Figure below)
D5 Eg load #2
Transformer with multiple secondaries, provides multiple
output voltages.
Here, three inductor coils share a common magnetic core,
magnetically “coupling” or “linking” them together. The
relationship of winding turn ratios and voltage ratios seen
with a single pair of mutual inductors still holds true here for
multiple pairs of coils. It is entirely possible to assemble a
transformer such as the one above (one primary winding, two
secondary windings) in which one secondary winding is a
step-down and the other is a step-up. In fact, this design of
transformer was quite common in vacuum tube power supply
circuits, which were required to supply low voltage for the
tubes' filaments (typically 6 or 12 volts) and high voltage for
the tubes' plates (several hundred volts) from a nominal
primary voltage of 110 volts AC. Not only are voltages and
currents of completely different magnitudes possible with
such a transformer, but all circuits are electrically isolated
from one another.
Photograph of multiple-winding transformer with six
windings, a primary and five secondaries.
The transformer in Figure above is intended to provide both
high and low voltages necessary in an electronic system
using vacuum tubes. Low voltage is required to power the
filaments of vacuum tubes, while high voltage is required to
create the potential difference between the plate and
cathode elements of each tube. One transformer with
multiple windings suffices elegantly to provide all the
necessary voltage levels from a single 115 V source. The
wires for this transformer (15 of them!) are not shown in the
photograph, being hidden from view.
If electrical isolation between secondary circuits is not of
great importance, a similar effect can be obtained by
“tapping” a single secondary winding at multiple points
along its length, like Figure below.
load #1
» 3 | load #2
A single tapped secondary provides multiple voltages.
A tap is nothing more than a wire connection made at some
point on a winding between the very ends. Not surprisingly,
the winding turn/voltage magnitude relationship of a normal
transformer holds true for all tapped segments of windings.
This fact can be exploited to produce a transformer capable
of multiple ratios: (Figure below)
multi-pole
switch
load
A tapped secondary using a switch to select one of many
possible voltages.
Carrying the concept of winding taps further, we end up with
a “variable transformer,” where a sliding contact is moved
along the length of an exposed secondary winding, able to
connect with it at any point along its length. The effect is
equivalent to having a winding tap at every turn of the
winding, and a switch with poles at every tap position:
(Figure below)
Variable transformer
load
A sliding contact on the secondary continuously varies the
secondary voltage.
One consumer application of the variable transformer is in
speed controls for model train sets, especially the train sets
of the 1950's and 1960's. These transformers were
essentially step-down units, the highest voltage obtainable
from the secondary winding being substantially less than the
primary voltage of 110 to 120 volts AC. The variable-sweep
contact provided a simple means of voltage control with little
wasted power, much more efficient than control using a
variable resistor!
Moving-slide contacts are too impractical to be used in large
industrial power transformer designs, but multi-pole switches
and winding taps are common for voltage adjustment.
Adjustments need to be made periodically in power systems
to accommodate changes in loads over months or years in
time, and these switching circuits provide a convenient
means. Typically, such “tap switches” are not engineered to
handle full-load current, but must be actuated only when the
transformer has been de-energized (no power).
Seeing as how we can tap any transformer winding to obtain
the equivalent of several windings (albeit with loss of
electrical isolation between them), it makes sense that it
should be possible to forego electrical isolation altogether
and build a transformer from a single winding. Indeed this is
possible, and the resulting device is called an
autotransformer. (Figure below)
Autotransformer
load
This autotransformer steps voltage up with a single tapped
winding, saving copper, sacrificing isolation.
The autotransformer depicted above performs a voltage step-
up function. A step-down autotransformer would look
something like Figure below.
Autotransformer
load
This auto transformer steps voltage down with a single
copper-saving tapped winding.
Autotransformers find popular use in applications requiring a
Slight boost or reduction in voltage to a load. The alternative
with a normal (isolated) transformer would be to either have
just the right primary/secondary winding ratio made for the
job or use a step-down configuration with the secondary
winding connected in series-aiding (“boosting”) or series-
opposing (“bucking”) fashion. Primary, secondary, and load
voltages are given to illustrate how this would work.
First, the “boosting” configuration. In Figure below the
secondary coil's polarity is oriented so that its voltage
directly adds to the primary voltage.
"boosting"
Ordinary transformer wired as an autotransformer to boost
the line voltage.
Next, the “bucking” configuration. In Figure below the
secondary coil's polarity is oriented so that its voltage
directly subtracts from the primary voltage:
"bucking"
Ordinary transformer wired as an autotransformer to buck
the line voltage down.
The prime advantage of an autotransformer is that the same
boosting or bucking function is obtained with only a single
winding, making it cheaper and lighter to manufacture than
a regular (isolating) transformer having both primary and
secondary windings.
Like regular transformers, autotransformer windings can be
tapped to provide variations in ratio. Additionally, they can
be made continuously variable with a sliding contact to tap
the winding at any point along its length. The latter
configuration is popular enough to have earned itself its own
name: the Variac. (Figure below)
The "Variac"
variable autotransformer
load
A variac is an autotransformer with a sliding tap.
Small variacs for benchtop use are popular pieces of
equipment for the electronics experimenter, being able to
step household AC voltage down (or sometimes up as well)
with a wide, fine range of control by a simple twist of a knob.
REVIEW:
Transformers can be equipped with more than just a
single primary and single secondary winding pair. This
allows for multiple step-up and/or step-down ratios in the
same device.
Transformer windings can also be “tapped:” that is,
intersected at many points to segment a single winding
into sections.
Variable transformers can be made by providing a
movable arm that sweeps across the length of a winding,
making contact with the winding at any point along its
length. The winding, of course, has to be bare (no
insulation) in the area where the arm sweeps.
An autotransformer is a single, tapped inductor coil used
to step up or step down voltage like a transformer,
except without providing electrical isolation.
A Variac is a variable autotransformer.
Voltage regulation
As we Saw in a few SPICE analyses earlier in this chapter, the
output voltage of a transformer varies some with varying
load resistances, even with a constant voltage input. The
degree of variance is affected by the primary and secondary
winding inductances, among other factors, not the least of
which includes winding resistance and the degree of mutual
inductance (magnetic coupling) between the primary and
secondary windings. For power transformer applications,
where the transformer is seen by the load (ideally) asa
constant source of voltage, it is good to have the secondary
voltage vary as little as possible for wide variances in load
current.
The measure of how well a power transformer maintains
constant secondary voltage over a range of load currents is
called the transformer's vo/tage regulation. |It can be
calculated from the following formula:
E, load ~ -lo
Regulation percentage = _Pro-toad ~ Frul-oxt _(19qq%)
Feut-toad
“Full-load” means the point at which the transformer is
operating at maximum permissible secondary current. This
operating point will be determined primarily by the winding
wire size (ampacity) and the method of transformer cooling.
Taking our first SPICE transformer simulation as an example,
let's compare the output voltage with a 1 kQ load versus a
200 Q load (assuming that the 200 Q load will be our “full
load” condition). Recall if you will that our constant primary
voltage was 10.00 volts AC:
freq v(3,5) i(vil)
6.000E+01 9.962E+00 9.962E-03 Output with 1k ohm
load
freq v(3,5) i(vil)
6.000E+01 9.348E+00 4.674E-02 Output with 200 ohm
load
Notice how the output voltage decreases as the load gets
heavier (more current). Now let's take that same transformer
circuit and place a load resistance of extremely high
magnitude across the secondary winding to simulate a “no-
load” condition: (See "transformer" spice list")
transformer
v1 10 ac 10 sin
rbogusl 1 2 le-12
rbogus2 5 0 9e12
ll 2 0 100
12 3 5 100
k L1 12 0.999
vil 3 4 ac 0
rload 4 5 9e12
.ac lin 1 60 60
.print ac v(2,0) i(vl1)
.print ac v(3,5) i(vil)
.end
freq v(2) i(vl)
6.000E+01 1.Q00E+01 2.653E-04
freq v(3,5) i(vil)
6.000E+01 9.990E+00 1.110E-12 Output with (almost) no
load
So, we see that our output (Secondary) voltage spans a range
of 9.990 volts at (virtually) no load and 9.348 volts at the
point we decided to call “full load.” Calculating voltage
regulation with these figures, we get:
. 9.990 V - 9.348 V
Regulation percentage = ————————————_ (100%)
9.348 V
Regulation percentage = 6.8678 %
Incidentally, this would be considered rather poor (or “loose”)
regulation for a power transformer. Powering a simple
resistive load like this, a good power transformer should
exhibit a regulation percentage of less than 3%. Inductive
loads tend to create a condition of worse voltage regulation,
so this analysis with purely resistive loads was a “best-case”
condition.
There are some applications, however, where poor regulation
is actually desired. One such case is in discharge lighting,
where a step-up transformer is required to initially generate a
high voltage (necessary to “ignite” the lamps), then the
voltage is expected to drop off once the lamp begins to draw
current. This is because discharge lamps' voltage
requirements tend to be much lower after a current has been
established through the arc path. In this case, a step-up
transformer with poor voltage regulation suffices nicely for
the task of conditioning power to the lamp.
Another application is in current control for AC arc welders,
which are nothing more than step-down transformers
supplying low-voltage, high-current power for the welding
process. A high voltage Is desired to assist in “striking” the
arc (getting it started), but like the discharge lamp, an arc
doesn't require aS much voltage to sustain itself once the air
has been heated to the point of ionization. Thus, a decrease
of secondary voltage under high load current would be a
good thing. Some arc welder designs provide arc current
adjustment by means of a movable iron core in the
transformer, cranked in or out of the winding assembly by
the operator. Moving the iron slug away from the windings
reduces the strength of magnetic coupling between the
windings, which diminishes no-load secondary voltage and
makes for poorer voltage regulation.
No exposition on transformer regulation could be called
complete without mention of an unusual device called a
ferroresonant transformer. “Ferroresonance” is a
phenomenon associated with the behavior of iron cores while
operating near a point of magnetic saturation (where the
core is so strongly magnetized that further increases in
winding current results in little or no increase in magnetic
flux).
While being somewhat difficult to describe without going
deep into electromagnetic theory, the ferroresonant
transformer is a power transformer engineered to operate in
a condition of persistent core saturation. That is, its iron core
is “stuffed full” of magnetic lines of flux for a large portion of
the AC cycle so that variations in supply voltage (primary
winding current) have little effect on the core's magnetic flux
density, which means the secondary winding outputs a
nearly constant voltage despite significant variations in
supply (primary winding) voltage. Normally, core saturation
in a transformer results in distortion of the sinewave shape,
and the ferroresonant transformer is no exception. To combat
this side effect, ferroresonant transformers have an auxiliary
secondary winding paralleled with one or more capacitors,
forming a resonant circuit tuned to the power supply
frequency. This “tank circuit” serves as a filter to reject
harmonics created by the core saturation, and provides the
added benefit of storing energy in the form of AC oscillations,
which is available for sustaining output winding voltage for
brief periods of input voltage loss (milliseconds' worth of
time, but certainly better than nothing). (Figure below)
AC power
output
AC power
Resonant LC circuit
Ferroresonant transformer provides voltage regulation of the
output.
In addition to blocking harmonics created by the saturated
core, this resonant circuit also “filters out” harmonic
frequencies generated by nonlinear (switching) loads in the
secondary winding circuit and any harmonics present in the
source voltage, providing “clean” power to the load.
Ferroresonant transformers offer several features useful in AC
power conditioning: constant output voltage given
substantial variations in input voltage, harmonic filtering
between the power source and the load, and the ability to
“ride through” brief losses in power by keeping a reserve of
energy in its resonant tank circuit. These transformers are
also highly tolerant of excessive loading and transient
(momentary) voltage surges. They are so tolerant, in fact,
that some may be briefly paralleled with unsynchronized AC
power sources, allowing a load to be switched from one
source of power to another in a “make-before-break” fashion
with no interruption of power on the secondary side!
Unfortunately, these devices have equally noteworthy
disadvantages: they waste a lot of energy (due to hysteresis
losses in the saturated core), generating significant heat in
the process, and are intolerant of frequency variations, which
means they don't work very well when powered by small
engine-driven generators having poor speed regulation.
Voltages produced in the resonant winding/capacitor circuit
tend to be very high, necessitating expensive capacitors and
presenting the service technician with very dangerous
working voltages. Some applications, though, may prioritize
the ferroresonant transformer's advantages over its
disadvantages. Semiconductor circuits exist to “condition”
AC power as an alternative to ferroresonant devices, but
none can compete with this transformer in terms of sheer
simplicity.
¢ REVIEW:
e Voltage regulation is the measure of how well a power
transformer can maintain constant secondary voltage
given a constant primary voltage and wide variance in
load current. The lower the percentage (closer to zero),
the more stable the secondary voltage and the better the
regulation it will provide.
e A ferroresonant transformer is a special transformer
designed to regulate voltage at a stable level despite
wide variation in input voltage.
Special transformers and applications
Impedance matching
Because transformers can step voltage and current to
different levels, and because power is transferred
equivalently between primary and secondary windings, they
can be used to “convert” the impedance of a load toa
different level. That last phrase deserves some explanation,
so let's investigate what it means.
The purpose of a load (usually) is to do something productive
with the power it dissipates. In the case of a resistive heating
element, the practical purpose for the power dissipated is to
heat something up. Loads are engineered to safely dissipate
a certain maximum amount of power, but two loads of equal
power rating are not necessarily identical. Consider these two
1000 watt resistive heating elements: (Figure below)
15.625 Q
Pi. = L000 W
Heating elements dissipate 1000 watts, at different voltage
and current ratings.
Both heaters dissipate exactly 1000 watts of power, but they
do so at different voltage and current levels (either 250 volts
and 4 amps, or 125 volts and 8 amps). Using Ohm's Law to
determine the necessary resistance of these heating
elements (R=E/I), we arrive at figures of 62.5 O and 15.625
Q, respectively. If these are AC loads, we might refer to their
opposition to current in terms of impedance rather than plain
resistance, although in this case that's all they're composed
of (no reactance). The 250 volt heater would be said to bea
higher impedance load than the 125 volt heater.
If we desired to operate the 250 volt heater element directly
on a125 volt power system, we would end up being
disappointed. With 62.5 O of impedance (resistance), the
current would only be 2 amps (I=E/R; 125/62.5), and the
power dissipation would only be 250 watts (P=IE; 125 x 2),
or one-quarter of its rated power. The impedance of the
heater and the voltage of our source would be mismatched,
and we couldn't obtain the full rated power dissipation from
the heater.
All hope is not lost, though. With a step-up transformer, we
could operate the 250 volt heater element on the 125 volt
power system like Figure below.
125 V :
1000 watts dissipation at the load resistor !
Step-up transformer operates 1000 watt 250 V heater from
125 V power source
The ratio of the transformer's windings provides the voltage
step-up and current step-down we need for the otherwise
mismatched load to operate properly on this system. Take a
close look at the primary circuit figures: 125 volts at 8 amps.
As far as the power supply “knows,” its powering a 15.625 QO
(R=E/I) load at 125 volts, not a 62.5 O load! The voltage and
current figures for the primary winding are indicative of
15.625 Q load impedance, not the actual 62.5 Q of the load
itself. In other words, not only has our step-up transformer
transformed voltage and current, but it has transformed
impedance as well.
The transformation ratio of impedance is the square of the
voltage/current transformation ratio, the same as the winding
inductance ratio:
N
secondary
Voltage transformation ratio =
primary
N
primary
Current transformation ratio =
secondary
. . Nyecondary ’
Impedance transformation ratio = {| —————
primary
N
: secondary
Inductance ratio = (soe
primary
Where,
N = number of turns in winding
This concurs with our example of the 2:1 step-up transformer
and the impedance ratio of 62.5 Q to 15.625 O (a 4:1 ratio,
which is 2:1 squared). Impedance transformation is a highly
useful ability of transformers, for it allows a load to dissipate
its full rated power even if the power system is not at the
proper voltage to directly do so.
Recall from our study of network analysis the Maximum
Power Transfer Theorem, which states that the maximum
amount of power will be dissipated by a load resistance when
that load resistance is equal to the Thevenin/Norton
resistance of the network supplying the power. Substitute the
word “impedance” for “resistance” in that definition and you
have the AC version of that Theorem. If we're trying to obtain
theoretical maximum power dissipation from a load, we must
be able to properly match the load impedance and source
(Thevenin/Norton) impedance together. This is generally
more of a concern in specialized electric circuits such as
radio transmitter/antenna and audio amplifier/speaker
systems. Let's take an audio amplifier system and see how it
works: (Figure below)
Audio amplifier
. Speaker
Thevenin/Norton 7-30
Z=500 2
... equivalent to. ..
Prtevetin
500 2 Speaker
Ethevenin 7-30
Amplifier with impedance of 500 Q drives 8 Q at much less
than maximum power.
With an internal impedance of 500 Q, the amplifier can only
deliver full power to a load (speaker) also having 500 Q of
impedance. Such a load would drop higher voltage and draw
less current than an 8 QO speaker dissipating the same
amount of power. If an 8 O speaker were connected directly
to the 500 Q amplifier as shown, the impedance mismatch
would result in very poor (low peak power) performance.
Additionally, the amplifier would tend to dissipate more than
its fair share of power in the form of heat trying to drive the
low impedance speaker.
To make this system work better, we can use a transformer to
match these mismatched impedances. Since we're going
from a high impedance (high voltage, low current) supply to
a low impedance (low voltage, high current) load, we'll need
to use a step-down transformer: (Figure below)
impedance "matching"
transformer
Audio amplifier
. Speaker
Thevenin/Norton 7-89
Z = 500 2
impedance ratio = 500 : 8 winding ratio = 7.906: 1
Impedance matching transformer matches 500 Q amplifier to
8 Q speaker for maximum efficiency.
To obtain an impedance transformation ratio of 500:8, we
would need a winding ratio equal to the square root of 500:8
(the square root of 62.5:1, or 7.906:1). With such a
transformer in place, the speaker will load the amplifier to
just the right degree, drawing power at the correct voltage
and current levels to satisfy the Maximum Power Transfer
Theorem and make for the most efficient power delivery to
the load. The use of a transformer in this capacity is called
impedance matching.
Anyone who has ridden a multi-speed bicycle can intuitively
understand the principle of impedance matching. A human's
legs will produce maximum power when spinning the bicycle
crank at a particular speed (about 60 to 90 revolution per
minute). Above or below that rotational soeed, human leg
muscles are less efficient at generating power. The purpose
of the bicycle's “gears” is to impedance-match the rider's
legs to the riding conditions so that they always spin the
crank at the optimum speed.
If the rider attempts to start moving while the bicycle is
shifted into its “top” gear, he or she will find it very difficult
to get moving. Is it because the rider is weak? No, its
because the high step-up ratio of the bicycle's chain and
sprockets in that top gear presents a mismatch between the
conditions (lots of inertia to overcome) and their legs
(needing to spin at 60-90 RPM for maximum power output).
On the other hand, selecting a gear that is too low will enable
the rider to get moving immediately, but limit the top speed
they will be able to attain. Again, is the lack of speed an
indication of weakness in the bicyclist's legs? No, its because
the lower speed ratio of the selected gear creates another
type of mismatch between the conditions (low load) and the
rider's legs (losing power if spinning faster than 90 RPM). It is
much the same with electric power sources and loads: there
must be an impedance match for maximum system
efficiency. In AC circuits, transformers perform the same
matching function as the sprockets and chain (“gears”) ona
bicycle to match otherwise mismatched sources and loads.
Impedance matching transformers are not fundamentally
different from any other type of transformer in construction
or appearance. A small impedance-matching transformer
(about two centimeters in width) for audio-frequency
applications is shown in the following photograph: (Figure
below)
Audio frequency impedance matching transformer.
Another impedance-matching transformer can be seen on
this printed circuit board, in the upper right corner, to the
immediate left of resistors Ro and Rj. It is labeled “TL”:
(Figure below)
Printed circuit board mounted audio impedance matching
transformer, top right.
Potential transformers
Transformers can also be used in electrical instrumentation
systems. Due to transformers’ ability to step up or step down
voltage and current, and the electrical isolation they provide,
they can serve as a way of connecting electrical
instrumentation to high-voltage, high current power systems.
Suppose we wanted to accurately measure the voltage of a
13.8 kV power system (a very common power distribution
voltage in American industry): (Figure below)
high-voltage load
power source
Direct measurement of high voltage by a voltmeter Is a
potential safety hazard.
Designing, installing, and maintaining a voltmeter capable of
directly measuring 13,800 volts AC would be no easy task.
The safety hazard alone of bringing 13.8 kV conductors into
an instrument panel would be severe, not to mention the
design of the voltmeter itself. However, by using a precision
step-down transformer, we can reduce the 13.8 kV down toa
safe level of voltage at a constant ratio, and isolate it from
the instrument connections, adding an additional level of
safety to the metering system: (Figure below)
high-voltage load
power source
fuse
D00000000 recision
step-down
ratio
PT
grounded for
safety
0-120 VAC voltmeter range
Instrumentation application: “Potential transformer” precisely
scales dangerous high voltage to a safe value applicable to a
conventional voltmeter.
Now the voltmeter reads a precise fraction, or ratio, of the
actual system voltage, its scale set to read as though it were
measuring the voltage directly. The transformer keeps the
instrument voltage at a safe level and electrically isolates it
from the power system, so there is no direct connection
between the power lines and the instrument or instrument
wiring. When used in this capacity, the transformer is called
a Potential Transformer, or simply PT.
Potential transformers are designed to provide as accurate a
voltage step-down ratio as possible. To aid in precise voltage
regulation, loading is kept to a minimum: the voltmeter is
made to have high input impedance so as to draw as little
current from the PT as possible. As you can see, a fuse has
been connected in series with the PTs primary winding, for
safety and ease of disconnecting the PT from the circuit.
A standard secondary voltage for a PT is 120 volts AC, for
full-rated power line voltage. The standard voltmeter range
to accompany a PT is 150 volts, full-scale. PTs with custom
winding ratios can be manufactured to suit any application.
This lends itself well to industry standardization of the actual
voltmeter instruments themselves, since the PT will be sized
to step the system voltage down to this standard instrument
level.
Current transformers
Following the same line of thinking, we can use a transformer
to step down current through a power line so that we are able
to safely and easily measure high system currents with
inexpensive ammeters. Of course, such a transformer would
be connected in series with the power line, like (Figure
below).
grounded for 0-5 A ammeter range
f
safety
Instrument application: the "Current Transformer’ a
CAAT
CT
fuse
precision
step-down
ratio
grounded for
safety —_L
0-120 VAC voltmeter range
Instrumentation application: “Currrent transformer” steps
high current down to a value applicable to a conventional
ammeter.
Note that while the PT is a step-down device, the Current
Transformer (or CT) is a step-up device (with respect to
voltage), which is what is needed to step down the power
line current. Quite often, CTs are built as donut-shaped
devices through which the power line conductor is run, the
power line itself acting as a single-turn primary winding:
(Figure below)
Current conductor to be measured Is threaded through the
opening. Scaled down current is available on wire leads.
Some CTs are made to hinge open, allowing insertion around
a power conductor without disturbing the conductor at all.
The industry standard secondary current for a CT is a range
of 0 to 5 amps AC. Like PTs, CTs can be made with custom
winding ratios to fit almost any application. Because their
“full load” secondary current is 5 amps, CT ratios are usually
described in terms of full-load primary amps to 5 amps, like
this:
600: 5 ratio (for measuring up to 600 A line current)
100: 5ratio (for measuring up to 100 A line current)
lk: 5ratio (for measuring up to 1000 A line current)
The “donut” CT shown in the photograph has a ratio of 50:5.
That is, when the conductor through the center of the torus is
carrying 50 amps of current (AC), there will be 5 amps of
Current induced in the CT's winding.
Because CTs are designed to be powering ammeters, which
are low-impedance loads, and they are wound as voltage
step-up transformers, they should never, ever be operated
with an open-circuited secondary winding. Failure to heed
this warning will result in the CT producing extremely high
secondary voltages, dangerous to equipment and personnel
alike. To facilitate maintenance of ammeter instrumentation,
Short-circuiting switches are often installed in parallel with
the CT's secondary winding, to be closed whenever the
ammeter is removed for service: (Figure below)
power conductor — current ---->
ground conn eciion
(for safety) close switch BEFORE
disconnecting ammeter!
0-5 A meter movement range
Short-circuit switch allows ammeter to be removed from an
active current transformer circuit.
Though it may seem strange to intentionally short-circuit a
power system component, it is perfectly proper and quite
necessary when working with current transformers.
Air core transformers
Another kind of special transformer, seen often in radio-
frequency circuits, is the a/r core transformer. (Figure below)
True to its name, an air core transformer has its windings
wrapped around a nonmagnetic form, usually a hollow tube
of some material. The degree of coupling (mutual
inductance) between windings in such a transformer is many
times less than that of an equivalent iron-core transformer,
but the undesirable characteristics of a ferromagnetic core
(eddy current losses, hysteresis, saturation, etc.) are
completely eliminated. It is in high-frequency applications
that these effects of iron cores are most problematic.
(a) 7 (b)
Air core transformers may be wound on cylindrical (a) or
toroidal (b) forms. Center tapped primary with secondary (a).
Bifilar winding on toroidal form (b).
The inside tapped solenoid winding, (Figure (a) above),
without the over winding, could match unequal impedances
when DC isolation is not required. When isolation is required
the over winding is added over one end of the main winding.
Air core transformers are used at radio frequencies when iron
core losses are too high. Frequently air core transformers are
paralleled with a capacitor to tune it to resonance. The over
winding is connected between a radio antenna and ground
for one such application. The secondary is tuned to
resonance with a variable capacitor. The output may be
taken from the tap point for amplification or detection. Small
millimeter size air core transformers are used in radio
receivers. The largest radio transmitters may use meter sized
coils. Unshielded air core solenoid transformers are mounted
at right angles to each other to prevent stray coupling.
Stray coupling is minimized when the transformer is wound
on a toroid form. (Figure (b) above) Toroidal air core
transformers also show a higher degree of coupling,
particularly for bifilar windings. Bifilar windings are wound
from a slightly twisted pair of wires. This implies a 1:1 turns
ratio. Three or four wires may be grouped for 1:2 and other
integral ratios. Windings do not have to be bifilar. This allows
arbitrary turns ratios. However, the degree of coupling
suffers. Toroidal air core transformers are rare except for VHF
(Very High Frequency) work. Core materials other than air
such as powdered iron or ferrite are preferred for lower radio
frequencies.
Tesla Coil
One notable example of an air-core transformer is the 7es/a
Coil, named after the Serbian electrical genius Nikola Tesla,
who was also the inventor of the rotating magnetic field AC
motor, polyphase AC power systems, and many elements of
radio technology. The Tesla Coil is a resonant, high-frequency
step-up transformer used to produce extremely high
voltages. One of Tesla's dreams was to employ his coil
technology to distribute electric power without the need for
wires, simply broadcasting it in the form of radio waves which
could be received and conducted to loads by means of
antennas. The basic schematic for a Tesla Coil is shown in
Figure below.
discharge terminal
"Tesla Coil”
Tesla Coil: A few heavy primary turns, many secondary turns.
The capacitor, in conjunction with the transformer's primary
winding, forms a tank circuit. The secondary winding is
wound in close proximity to the primary, usually around the
Same nonmagnetic form. Several options exist for “exciting”
the primary circuit, the simplest being a high-voltage, low-
frequency AC source and spark gap: (Figure below)
HIGH voltage!
HIGH frequency!
RFC
high voltage spark gap
low frequency
RFC
System level diagram of Tesla coil with spark gap drive.
The purpose of the high-voltage, low-frequency AC power
source is to “charge” the primary tank circuit. When the
Spark gap fires, its low impedance acts to complete the
Capacitor/primary coil tank circuit, allowing it to oscillate at
its resonant frequency. The “RFC” inductors are “Radio
Frequency Chokes,” which act as high impedances to prevent
the AC source from interfering with the oscillating tank
circuit.
The secondary side of the Tesla coil transformer is also a tank
circuit, relying on the parasitic (stray) capacitance existing
between the discharge terminal and earth ground to
complement the secondary winding's inductance. For
optimum operation, this secondary tank circuit is tuned to
the same resonant frequency as the primary circuit, with
energy exchanged not only between capacitors and
inductors during resonant oscillation, but also back-and-forth
between primary and secondary windings. The visual results
are spectacular: (Figure below)
High voltage high frequency discharge from Tesla coil.
Tesla Coils find application primarily as novelty devices,
showing up in high school science fairs, basement
workshops, and the occasional low budget science-fiction
movie.
It should be noted that Tesla coils can be extremely
dangerous devices. Burns caused by radio-frequency (“RF”)
current, like all electrical burns, can be very deep, unlike skin
burns caused by contact with hot objects or flames. Although
the high-frequency discharge of a Tesla coil has the curious
property of being beyond the “shock perception” frequency
of the human nervous system, this does not mean Tesla coils
cannot hurt or even kill you! | strongly advise seeking the
assistance of an experienced Tesla coil experimenter if you
would embark on building one yourself.
Saturable reactors
So far, we've explored the transformer as a device for
converting different levels of voltage, current, and even
impedance from one circuit to another. Now we'll take a look
at it as a completely different kind of device: one that allows
a small electrical signal to exert contro/ over a much larger
quantity of electrical power. In this mode, a transformer acts
as an amplifier.
The device I'm referring to is called a saturable-core reactor,
or simply saturable reactor. Actually, it is not really a
transformer at all, but rather a special kind of inductor whose
inductance can be varied by the application of a DC current
through a second winding wound around the same iron core.
Like the ferroresonant transformer, the saturable reactor
relies on the principle of magnetic saturation. When a
material such as iron is completely saturated (that is, all its
magnetic domains are lined up with the applied magnetizing
force), additional increases in current through the
magnetizing winding will not result in further increases of
magnetic flux.
Now, inductance is the measure of how well an inductor
opposes changes in current by developing a voltage in an
opposing direction. The ability of an inductor to generate this
opposing voltage is directly connected with the change in
magnetic flux inside the inductor resulting from the change
in current, and the number of winding turns in the inductor. If
an inductor has a saturated core, no further magnetic flux
will result from further increases in current, and so there will
be no voltage induced in opposition to the change in current.
In other words, an inductor loses its inductance (ability to
oppose changes in current) when its core becomes
magnetically saturated.
If an inductor's inductance changes, then its reactance (and
impedance) to AC current changes as well. In a circuit with a
constant voltage source, this will result in a change in
current: (Figure below)
load
If L changes in inductance, Z, will correspondingly change,
thus changing the circuit current.
A saturable reactor capitalizes on this effect by forcing the
core into a state of saturation with a strong magnetic field
generated by current through another winding. The reactor's
“power” winding is the one carrying the AC load current, and
the “control” winding is one carrying a DC current strong
enough to drive the core into saturation: (Figure below)
saturable reactor
load
DC, via the control winding, saturates the core. Thus,
modulating the power winding inductance, impedance, and
current.
The strange-looking transformer symbol shown in the above
schematic represents a saturable-core reactor, the upper
winding being the DC control winding and the lower being
the “power” winding through which the controlled AC current
goes. Increased DC control current produces more magnetic
flux in the reactor core, driving it closer to a condition of
saturation, thus decreasing the power winding's inductance,
decreasing its impedance, and increasing current to the load.
Thus, the DC control current is able to exert contro/ over the
AC current delivered to the load.
The circuit shown would work, but it would not work very
well. The first problem is the natural transformer action of the
saturable reactor: AC current through the power winding will
induce a voltage in the control winding, which may cause
trouble for the DC power source. Also, saturable reactors tend
to regulate AC power only in one direction: in one half of the
AC cycle, the mmf's from both windings add; in the other
half, they subtract. Thus, the core will have more flux in it
during one half of the AC cycle than the other, and will
saturate first in that cycle half, passing load current more
easily in one direction than the other. Fortunately, both
problems can be overcome with a little ingenuity: (Figure
below)
load
Out of phase DC control windings allow symmetrical of
control AC.
Notice the placement of the phasing dots on the two
reactors: the power windings are “in phase” while the control
windings are “out of phase.” If both reactors are identical,
any voltage induced in the control windings by load current
through the power windings will cancel out to zero at the
battery terminals, thus eliminating the first problem
mentioned. Furthermore, since the DC control current
through both reactors produces magnetic fluxes in different
directions through the reactor cores, one reactor will saturate
more in one cycle of the AC power while the other reactor will
saturate more in the other, thus equalizing the control action
through each half-cycle so that the AC power is “throttled”
symmetrically. This phasing of control windings can be
accomplished with two separate reactors as shown, or ina
single reactor design with intelligent layout of the windings
and core.
Saturable reactor technology has even been miniaturized to
the circuit-board level in compact packages more generally
known as magnetic amplifiers. | personally find this to be
fascinating: the effect of amplification (one electrical signal
controlling another), normally requiring the use of physically
fragile vacuum tubes or electrically “fragile” semiconductor
devices, can be realized in a device both physically and
electrically rugged. Magnetic amplifiers do have
disadvantages over their more fragile counterparts, namely
size, weight, nonlinearity, and bandwidth (frequency
response), but their utter simplicity still commands a certain
degree of appreciation, if not practical application.
Saturable-core reactors are less commonly known as
“saturable-core inductors” or transductors.
Scott-T transformer
Nikola Tesla's original polyphase power system was based on
simple to build 2-phase components. However, as
transmission distances increased, the more transmission line
efficient 3-phase system became more prominent. Both 2-9
and 3-@ components coexisted for a number of years. The
Scott-T transformer connection allowed 2-@ and 3-@
components like motors and alternators to be
interconnected. Yamamoto and Yamaguchi:
In 1896, General Electric built a 35.5 km (22 mi) three-
phase transmission line operated at 11 kV to transmit
power to Buffalo, New York, from the Niagara Falls Project.
The two-phase generated power was changed to three-
phase by the use of Scott-T transformations. [MYA]
R, 2-phase, = VZ0° R,
Scott-T transformer converts 2-9 to 3-9, or vice versa.
The Scott-T transformer set, Figure above, consists of a
center tapped transformer T1 and an 86.6% tapped
transformer T2 on the 3-q9 side of the circuit. The primaries of
both transformers are connected to the 2-9 voltages. One
end of the T2 86.6% secondary winding is a 3-@ output, the
other end is connected to the Tl secondary center tap. Both
ends of the T1 secondary are the other two 3-@ connections.
Application of 2-9 Niagara generator power produced a 3-9
output for the more efficient 3-9 transmission line. More
common these days is the application of 3-@ power to
produce a 2-@ output for driving an old 2-@ motor.
In Figure below, we use vectors in both polar and complex
notation to prove that the Scott-T converts a pair of 2-@
voltages to 3-q. First, one of the 3-@ voltages is identical to a
2-m voltage due to the 1:1 transformer T1 ratio, Vp}5= Vp}.
The T1 center tapped secondary produces opposite polarities
of 0.5V>5p, on the secondary ends. This Z0° is vectorially
subtracted from T2 secondary voltage due to the KVL
equations V31, Vo3. The T2 secondary voltage is 0.866V>p>
due to the 86.6% tap. Keep in mind that this 2nd phase of
the 2-9 is 290°. This 0.866V>p,5 is added at V3,, subtracted at
V3 In the KVL equations.
Given two 90° phased voltages:
Vop, =Vsin(6+0°)=V.20°=V(1 +0)
Vop2 =Vsin(6+90°)=Vcos(8)=V 290°=V(0+)1)
Derive the three phase voltages V,>, Vas, Vz :
V ,2=Vop, =Vsin(6+0°)=V 40°=V(1+j0)
1) KVL: -Vj2 +Vac =0
2) KVL: V5, -Vop +Vep= 0
3) KVL: V5; = -Vop = Vea =0
1) KVL: Vi5 = Vac
2) KVL: V3 = -VeptVep
3) KVL: V3 = -Vos = Vea
Vopg = 0.866V.p. = 0.866V 790° = 0. 866V(0+)1)
Vop = Vea = 0.5Vap, = 0.5V.20° = 0.5V(1+j0)
(
(
(
(
(
(
Vi2 = Vap, = VZ0°
V3, = (-0.5)V.20°+0.866V.290°=V(-0.5(1+j0)+0.866(0+j1))=V(-0.5+j0.866)=V.2120°
V3 =(-0.5)V20°-0.866V.290°=V(-0.5(1 +j0)-0.866(0+)1})=V(-0.5+40.866)=V.2-120°=V.4240°
Scott-T transformer 2-9 to 3-g conversion equations.
We show “DC” polarities all over this AC only circuit, to keep
track of the Kirchhoff voltage loop polarities. Subtracting Z0°
Is equivalent to adding Z180°. The bottom line is when we
add 86.6% of 290° to 50% of Z2180° we get 21202.
Subtracting 86.6% of 290° from 50% of 2180° yields Z-120°
or Z240°.
0.866V.290°
—_ 12120°
120° = 0.520"
-0.520°
-0.866V.290° 12240)
120°, 1.290° yields 14-120° ,12240°
Graphical explanation of equations in Figure previous.
In Figure above we graphically show the 2-@ vectors at (a). At
(b) the vectors are scaled by transformers T1 and T2 to 0.5
and 0.866 respectively. At (c) 12120° = -0.5Z20° +
0.866290°, and 12240° = -0.5Z0° - 0.866290°. The three
output phases are 1Z2120° and 12240° from (c), along with
input 120° (a).
Linear Variable Differential Transformer
A linear variable differential transformer (LVDT) has an AC
driven primary wound between two secondaries on a
cylindrical air core form. (Figure below) A movable
ferromagnetic slug converts displacement to a variable
voltage by changing the coupling between the driven
primary and secondary windings. The LVDT is a displacement
or distance measuring transducer. Units are available for
measuring displacement over a distance of a fraction of a
millimeter to a half a meter. LVDT's are rugged and dirt
resistant compared to linear optical encoders.
center down
“OA A a at
YY ACS RSS
Ya NE et Ht FO} ORG AG
LVDT: linear variable differential transformer.
The excitation voltage is in the range of 0.5 to10 VAC ata
frequency of 1 to 200 Khz. A ferrite core is suitable at these
frequencies. It is extended outside the body by an non-
magnetic rod. As the core is moved toward the top winding,
the voltage across this coil increases due to increased
coupling, while the voltage on the bottom coil decreases. If
the core is moved toward the bottom winding, the voltage on
this coil increases as the voltage decreases across the top
coil. Theoretically, a centered slug yields equal voltages
across both coils. In practice leakage inductance prevents
the null from dropping all the way to 0 V.
With a centered slug, the series-opposing wired secondaries
cancel yielding V3 = 0. Moving the slug up increases Vj3.
Note that it is in-phase with with V,, the top winding, and
180° out of phase with V3, bottom winding.
Moving the slug down from the center position increases V;3.
However, it is 180° out of phase with with V,, the top
winding, and in-phase with V3, bottom winding. Moving the
slug from top to bottom shows a minimum at the center
point, with a 180° phase reversal in passing the center.
e REVIEW:
e Transformers can be used to transform impedance as well
as voltage and current. When this is done to improve
power transfer to a load, it is called impedance matching.
e A Potential Transformer (PT) is a special instrument
transformer designed to provide a precise voltage step-
down ratio for voltmeters measuring high power system
voltages.
e A Current Transformer (CT) is another special instrument
transformer designed to step down the current through a
power line to a safe level for an ammeter to measure.
e An air-core transformer is one lacking a ferromagnetic
core.
e A Tesla Coilis a resonant, air-core, step-up transformer
designed to produce very high AC voltages at high
frequency.
e A saturable reactor is a special type of inductor, the
inductance of which can be controlled by the DC current
through a second winding around the same core. With
enough DC current, the magnetic core can be saturated,
decreasing the inductance of the power winding in a
controlled fashion.
e A Scott-T transformer converts 3-@ power to 2-@ power
and vice versa.
e A linear variable differential transformer, also Known as
an LVDT, is a distance measuring device. It has a
movable ferromagnetic core to vary the coupling
between the excited primary and a pair of secondaries.
Practical considerations
Power capacity
As has already been observed, transformers must be well
designed in order to achieve acceptable power coupling,
tight voltage regulation, and low exciting current distortion.
Also, transformers must be designed to carry the expected
values of primary and secondary winding current without any
trouble. This means the winding conductors must be made of
the proper gauge wire to avoid any heating problems. An
ideal transformer would have perfect coupling (no leakage
inductance), perfect voltage regulation, perfectly sinusoidal
exciting current, no hysteresis or eddy current losses, and
wire thick enough to handle any amount of current.
Unfortunately, the ideal transformer would have to be
infinitely large and heavy to meet these design goals. Thus,
in the business of practica/ transformer design, compromises
must be made.
Additionally, winding conductor insulation is a concern where
high voltages are encountered, as they often are in step-up
and step-down power distribution transformers. Not only do
the windings have to be well insulated from the iron core, but
each winding has to be sufficiently insulated from the other
in order to maintain electrical isolation between windings.
Respecting these limitations, transformers are rated for
certain levels of primary and secondary winding voltage and
current, though the current rating is usually derived from a
volt-amp (VA) rating assigned to the transformer. For
example, take a step-down transformer with a primary
voltage rating of 120 volts, a secondary voltage rating of 48
volts, and a VA rating of 1 kVA (1000 VA). The maximum
winding currents can be determined as such:
rere = 8.333 A (maximum primary winding current)
7;
a = 20.833 A (maximum secondary winding current)
Sometimes windings will bear current ratings in amps, but
this is typically seen on small transformers. Large
transformers are almost always rated in terms of winding
voltage and VA or kVA.
Energy losses
When transformers transfer power, they do so witha
minimum of loss. As it was stated earlier, modern power
transformer designs typically exceed 95% efficiency. It is
good to know where some of this lost power goes, however,
and what causes it to be lost.
There is, of course, power lost due to resistance of the wire
windings. Unless superconducting wires are used, there will
always be power dissipated in the form of heat through the
resistance of current-carrying conductors. Because
transformers require such long lengths of wire, this loss can
be a significant factor. Increasing the gauge of the winding
wire is one way to minimize this loss, but only with
substantial increases in cost, size, and weight.
Resistive losses aside, the bulk of transformer power loss is
due to magnetic effects in the core. Perhaps the most
significant of these “core losses” is eddy-current loss, which
is resistive power dissipation due to the passage of induced
currents through the iron of the core. Because iron is a
conductor of electricity as well as being an excellent
“conductor” of magnetic flux, there will be currents induced
in the iron just as there are currents induced in the
secondary windings from the alternating magnetic field.
These induced currents -- as described by the
perpendicularity clause of Faraday's Law -- tend to circulate
through the cross-section of the core perpendicularly to the
primary winding turns. Their circular motion gives them their
unusual name: like eddies in a stream of water that circulate
rather than move in straight lines.
lron is a fair conductor of electricity, but not as good as the
copper or aluminum from which wire windings are typically
made. Consequently, these “eddy currents” must overcome
significant electrical resistance as they circulate through the
core. In overcoming the resistance offered by the iron, they
dissipate power in the form of heat. Hence, we have a source
of inefficiency in the transformer that is difficult to eliminate.
This phenomenon is so pronounced that it is often exploited
as a means of heating ferrous (iron-containing) materials.
The photograph of (Figure below) shows an “induction
heating” unit raising the temperature of a large pipe section.
Loops of wire covered by high-temperature insulation
encircle the pipe's circumference, inducing eddy currents
within the pipe wall by electromagnetic induction. In order to
maximize the eddy current effect, high-frequency alternating
current is used rather than power line frequency (60 Hz). The
box units at the right of the picture produce the high-
frequency AC and control the amount of current in the wires
to stabilize the pipe temperature at a pre-determined “set-
point.”
Induction heating: Primary insulated winding induces current
into lossy iron pipe (secondary).
The main strategy in mitigating these wasteful eddy currents
in transformer cores is to form the iron core in sheets, each
sheet covered with an insulating varnish so that the core is
divided up into thin slices. The result is very little width in
the core for eddy currents to circulate in: (Figure below)
solid iron core
laminated iron core
Dividing the iron core into thin insulated laminations
minimizes eddy current loss.
Laminated cores like the one shown here are standard in
almost all low-frequency transformers. Recall from the
photograph of the transformer cut in half that the iron core
was composed of many thin sheets rather than one solid
piece. Eddy current losses increase with frequency, so
transformers designed to run on higher-frequency power
(such as 400 Hz, used in many military and aircraft
applications) must use thinner laminations to keep the losses
down to a respectable minimum. This has the undesirable
effect of increasing the manufacturing cost of the
transformer.
Another, similar technique for minimizing eddy current losses
which works better for high-frequency applications is to
make the core out of iron powder instead of thin iron sheets.
Like the lamination sheets, these granules of iron are
individually coated in an electrically insulating material,
which makes the core nonconductive except for within the
width of each granule. Powdered iron cores are often found in
transformers handling radio-frequency currents.
Another “core loss” is that of magnetic hysteresis. All
ferromagnetic materials tend to retain some degree of
magnetization after exposure to an external magnetic field.
This tendency to stay magnetized is called “hysteresis,” and
it takes a certain investment in energy to overcome this
opposition to change every time the magnetic field produced
by the primary winding changes polarity (twice per AC
cycle). This type of loss can be mitigated through good core
material selection (choosing a core alloy with low hysteresis,
as evidenced by a “thin” B/H hysteresis curve), and
designing the core for minimum flux density (large cross-
sectional area).
Transformer energy losses tend to worsen with increasing
frequency. The skin effect within winding conductors reduces
the available cross-sectional area for electron flow, thereby
increasing effective resistance as the frequency goes up and
creating more power lost through resistive dissipation.
Magnetic core losses are also exaggerated with higher
frequencies, eddy currents and hysteresis effects becoming
more severe. For this reason, transformers of significant size
are designed to operate efficiently in a limited range of
frequencies. In most power distribution systems where the
line frequency is very stable, one would think excessive
frequency would never pose a problem. Unfortunately it
does, in the form of harmonics created by nonlinear loads.
As we've seen in earlier chapters, nonsinusoidal waveforms
are equivalent to additive series of multiple sinusoidal
waveforms at different amplitudes and frequencies. In power
systems, these other frequencies are whole-number multiples
of the fundamental (line) frequency, meaning that they will
always be higher, not lower, than the design frequency of the
transformer. In significant measure, they can cause severe
transformer overheating. Power transformers can be
engineered to handle certain levels of power system
harmonics, and this capability is sometimes denoted with a
“K factor” rating.
Stray capacitance and inductance
Aside from power ratings and power losses, transformers
often harbor other undesirable limitations which circuit
designers must be made aware of. Like their simpler
counterparts -- inductors -- transformers exhibit capacitance
due to the insulation dielectric between conductors: from
winding to winding, turn to turn (in a single winding), and
winding to core. Usually this capacitance is of no concern ina
power application, but small signal applications (especially
those of high frequency) may not tolerate this quirk well.
Also, the effect of having capacitance along with the
windings’ designed inductance gives transformers the ability
to resonate at a particular frequency, definitely a design
concern in signal applications where the applied frequency
may reach this point (usually the resonant frequency of a
power transformer is well beyond the frequency of the AC
power it was designed to operate on).
Flux containment (making sure a transformer's magnetic flux
doesn't escape so as to interfere with another device, and
making sure other devices' magnetic flux is shielded from
the transformer core) is another concern shared both by
inductors and transformers.
Closely related to the issue of flux containment is leakage
inductance. We've already seen the detrimental effects of
leakage inductance on voltage regulation with SPICE
simulations early in this chapter. Because leakage
inductance is equivalent to an inductance connected in
series with the transformer's winding, it manifests itself as a
series impedance with the load. Thus, the more current
drawn by the load, the less voltage available at the
secondary winding terminals. Usually, good voltage
regulation is desired in transformer design, but there are
exceptional applications. As was stated before, discharge
lighting circuits require a step-up transformer with “loose”
(poor) voltage regulation to ensure reduced voltage after the
establishment of an arc through the lamp. One way to meet
this design criterion is to engineer the transformer with flux
leakage paths for magnetic flux to bypass the secondary
winding(s). The resulting leakage flux will produce leakage
inductance, which will in turn produce the poor regulation
needed for discharge lighting.
Core saturation
Transformers are also constrained in their performance by the
magnetic flux limitations of the core. For ferromagnetic core
transformers, we must be mindful of the saturation limits of
the core. Remember that ferromagnetic materials cannot
support infinite magnetic flux densities: they tend to
“saturate” at a certain level (dictated by the material and
core dimensions), meaning that further increases in magnetic
field force (mmf) do not result in proportional increases in
magnetic field flux (®).
When atransformer's primary winding is overloaded from
excessive applied voltage, the core flux may reach saturation
levels during peak moments of the AC sinewave cycle. If this
happens, the voltage induced in the secondary winding will
no longer match the wave-shape as the voltage powering the
primary coil. In other words, the overloaded transformer will
distort the waveshape from primary to secondary windings,
creating harmonics in the secondary winding's output. As we
discussed before, harmonic content in AC power systems
typically causes problems.
Special transformers known as peaking transformers exploit
this principle to produce brief voltage pulses near the peaks
of the source voltage waveform. The core is designed to
saturate quickly and sharply, at voltage levels well below
peak. This results in a severely cropped sine-wave flux
waveform, and secondary voltage pulses only when the flux
Is changing (below saturation levels): (Figure below)
e,=primary voltage e,=secondary voltage &=magnetic flux
Voltage and flux waveforms for a peaking transformer.
Another cause of abnormal transformer core saturation Is
operation at frequencies lower than normal. For example, if a
power transformer designed to operate at 60 Hz is forced to
operate at 50 Hz instead, the flux must reach greater peak
levels than before in order to produce the same opposing
voltage needed to balance against the source voltage. This is
true even if the source voltage is the same as before. (Figure
below)
60 Hz
e = voltage
® = magnetic flux
oy
50 Hz
Magnetic flux is higher in a transformer core driven by 50 Hz
as compared to 60 Hz for the same voltage.
Since instantaneous winding voltage is proportional to the
instantaneous magnetic flux's rate of change ina
transformer, a voltage waveform reaching the same peak
value, but taking a longer amount of time to complete each
half-cycle, demands that the flux maintain the same rate of
change as before, but for longer periods of time. Thus, if the
flux has to climb at the same rate as before, but for longer
periods of time, it will climb to a greater peak value. (Figure
below)
Mathematically, this is another example of calculus in action.
Because the voltage is proportional to the flux's rate-of-
change, we say that the voltage waveform is the derivative of
the flux waveform, “derivative” being that calculus operation
defining one mathematical function (waveform) in terms of
the rate-of-change of another. If we take the opposite
perspective, though, and relate the original waveform to its
derivative, we may call the original waveform the integral of
the derivative waveform. In this case, the voltage waveform
is the derivative of the flux waveform, and the flux waveform
is the integral of the voltage waveform.
The integral of any mathematical function is proportional to
the area accumulated underneath the curve of that function.
Since each half-cycle of the 50 Hz waveform accumulates
more area between it and the zero line of the graph than the
60 Hz waveform will -- and we know that the magnetic flux is
the integral of the voltage -- the flux will attain higher values
in Figure below.
e
60 Hz less height
less area |
|
50 Hz more height
Af more area |
Flux changing at the same rate rises to a higher level at 50
Hz than at 60 Hz.
Yet another cause of transformer saturation is the presence of
DC current in the primary winding. Any amount of DC voltage
dropped across the primary winding of a transformer will
cause additional magnetic flux in the core. This additional
flux “bias” or “offset” will push the alternating flux waveform
closer to saturation in one half-cycle than the other. (Figure
below)
DC in primary, shifts the waveform peaks toward the upper
saturation limit.
For most transformers, core saturation is a very undesirable
effect, and it is avoided through good design: engineering
the windings and core so that magnetic flux densities remain
well below the saturation levels. This ensures that the
relationship between mmf and © is more linear throughout
the flux cycle, which is good because it makes for less
distortion in the magnetization current waveform. Also,
engineering the core for low flux densities provides a safe
margin between the normal flux peaks and the core
saturation limits to accommodate occasional, abnormal
conditions such as frequency variation and DC offset.
Inrush current
When a transformer is initially connected to a source of AC
voltage, there may be a substantial surge of current through
the primary winding called inrush current. (Figure below)
This is analogous to the inrush current exhibited by an
electric motor that is started up by sudden connection toa
power source, although transformer inrush is caused by a
different phenomenon.
We know that the rate of change of instantaneous flux ina
transformer core is proportional to the instantaneous voltage
drop across the primary winding. Or, as stated before, the
voltage waveform is the derivative of the flux waveform, and
the flux waveform is the integral of the voltage waveform. In
a continuously-operating transformer, these two waveforms
are phase-shifted by 90°. (Figure below) Since flux (®) is
proportional to the magnetomotive force (mmf) in the core,
and the mmf is proportional to winding current, the current
waveform will be in-phase with the flux waveform, and both
will be lagging the voltage waveform by 90°:
e=voltage &=magnetic flux i=coil current
“ P
Continuous steady-state operation: Magnetic flux, like
current, lags applied voltage by 90°.
Let us suppose that the primary winding of a transformer is
suddenly connected to an AC voltage source at the exact
moment in time when the instantaneous voltage is at its
positive peak value. In order for the transformer to create an
opposing voltage drop to balance against this applied source
voltage, a magnetic flux of rapidly increasing value must be
generated. The result is that winding current increases
rapidly, but actually no more rapidly than under normal
conditions: (Figure below)
e = voltage
® = magnetic flux
i = coil current
P
_Instant in time when transformer
is connected to AC voltage source.
Connecting transformer to line at AC volt peak: Flux
increases rapidly from zero, same as steady-state operation.
Both core flux and coil current start from zero and build up to
the same peak values experienced during continuous
operation. Thus, there is no “surge” or “inrush” or current in
this scenario. (Figure above)
Alternatively, let us consider what happens if the
transformer's connection to the AC voltage source occurs at
the exact moment in time when the instantaneous voltage is
at zero. During continuous operation (when the transformer
has been powered for quite some time), this is the point in
time where both flux and winding current are at their
negative peaks, experiencing zero rate-of-change (d®/dt = 0
and di/dt = 0). As the voltage builds to its positive peak, the
flux and current waveforms build to their maximum positive
rates-of-change, and on upward to their positive peaks as the
voltage descends to a level of zero:
e = voltage
® = magnetic flux
i = coil current
: P
Instant in time when voltage is zero,
during continuous operation.
Starting at e=0 V is not the same as running continuously in
Figure above. These expected waveforms are incorrect- ®
and i should start at zero.
A significant difference exists, however, between continuous-
mode operation and the sudden starting condition assumed
in this scenario: during continuous operation, the flux and
current levels were at their negative peaks when voltage was
at its zero point; in a transformer that has been sitting idle,
however, both magnetic flux and winding current should
start at zero. When the magnetic flux increases in response
to a rising voltage, it will increase from zero upward, not from
a previously negative (magnetized) condition as we would
normally have in a transformer that's been powered for
awhile. Thus, in a transformer that's just “starting,” the flux
will reach approximately twice its normal peak magnitude as
it “integrates” the area under the voltage waveform's first
half-cycle: (Figure below)
flux peak approximately
twice normal height!
Instant in time when voltage is zero,
from a "cold start” condition.
Starting at e=0 V, @ starts at initial condition ®=0,
increasing to twice the normal value, assuming it doesn't
saturate the core.
In an ideal transformer, the magnetizing current would rise to
approximately twice its normal peak value as well,
generating the necessary mmf to create this higher-than-
normal flux. However, most transformers aren't designed
with enough of a margin between normal flux peaks and the
saturation limits to avoid saturating in a condition like this,
and so the core will almost certainly saturate during this first
half-cycle of voltage. During saturation, disproportionate
amounts of mmf are needed to generate magnetic flux. This
means that winding current, which creates the mmf to cause
flux in the core, will disproportionately rise to a value easily
exceeding twice its normal peak: (Figure below)
current peak much
- ~~ greater than normal!
flux peak approximately
twice normal height!
Instant in time when voltage is zero,
from a "cold start" condition.
Starting at e=0 V, Current also increases to twice the normal
value for an unsaturated core, or considerably higher in the
(designed for) case of saturation.
This is the mechanism causing inrush current in a
transformer's primary winding when connected to an AC
voltage source. As you can see, the magnitude of the inrush
current strongly depends on the exact time that electrical
connection to the source is made. If the transformer happens
to have some residual magnetism in its core at the moment
of connection to the source, the inrush could be even more
severe. Because of this, transformer overcurrent protection
devices are usually of the “slow-acting” variety, so as to
tolerate current surges such as this without opening the
circuit.
Heat and Noise
In addition to unwanted electrical effects, transformers may
also exhibit undesirable physical effects, the most notable
being the production of heat and noise. Noise is primarily a
nuisance effect, but heat is a potentially serious problem
because winding insulation will be damaged if allowed to
overheat. Heating may be minimized by good design,
ensuring that the core does not approach saturation levels,
that eddy currents are minimized, and that the windings are
not overloaded or operated too close to maximum ampacity.
Large power transformers have their core and windings
submerged in an oil bath to transfer heat and muffle noise,
and also to displace moisture which would otherwise
compromise the integrity of the winding insulation. Heat-
dissipating “radiator” tubes on the outside of the transformer
case provide a convective oil flow path to transfer heat from
the transformer's core to ambient air: (Figure below)
Primary Secondary
terminals terminals
S 4
Heat ~~ eA, Heat
AN ~
ST aed ~
ST aed ~
nn si
Radiator ~ ™ Radiator
tube ~ tube
= ey
ST aed ~~,
ST aed ~,
neat ie
ST ae ~~
Sy 1
~
~~
rt
$$$
Large power transformers are submerged in heat dissipating
insulating oil.
Oil-less, or “dry,” transformers are often rated in terms of
maximum operating temperature “rise” (temperature
increase beyond ambient) according to a letter-class system:
A, B, F, or H. These letter codes are arranged in order of
lowest heat tolerance to highest:
e Class A: No more than 55° Celsius winding temperature
rise, at 40° Celsius (maximum) ambient air temperature.
e Class B: No more than 80° Celsius winding temperature
rise, at 40° Celsius (maximum)ambient air temperature.
e Class F: No more than 115° Celsius winding temperature
rise, at 40° Celsius (maximum)ambient air temperature.
e Class H: No more than 150° Celsius winding
temperature rise, at 40° Celsius (maximum)ambient air
temperature.
Audible noise is an effect primarily originating from the
phenomenon of magnetostriction: the slight change of
length exhibited by a ferromagnetic object when
magnetized. The familiar “hum” heard around large power
transformers is the sound of the iron core expanding and
contracting at 120 Hz (twice the system frequency, which is
60 Hz in the United States) -- one cycle of core contraction
and expansion for every peak of the magnetic flux waveform
-- plus noise created by mechanical forces between primary
and secondary windings. Again, maintaining low magnetic
flux levels in the core is the key to minimizing this effect,
which explains why ferroresonant transformers -- which must
operate in saturation for a large portion of the current
waveform -- operate both hot and noisy.
Another noise-producing phenomenon in power transformers
is the physical reaction force between primary and secondary
windings when heavily loaded. If the secondary winding is
open-circuited, there will be no current through it, and
consequently no magneto-motive force (mmf) produced by it.
However, when the secondary is “loaded” (current supplied
to a load), the winding generates an mmf, which becomes
counteracted by a “reflected” mmf in the primary winding to
prevent core flux levels from changing. These opposing
mmf's generated between primary and secondary windings
as a result of secondary (load) current produce a repulsive,
physical force between the windings which will tend to make
them vibrate. Transformer designers have to consider these
physical forces in the construction of the winding coils, to
ensure there is adequate mechanical support to handle the
stresses. Under heavy load (high current) conditions, though,
these stresses may be great enough to cause audible noise to
emanate from the transformer.
e REVIEW:
e Power transformers are limited in the amount of power
they can transfer from primary to secondary winding(s).
Large units are typically rated in VA (volt-amps) or kVA
(kilo volt-amps).
e Resistance in transformer windings contributes to
inefficiency, as current will dissipate heat, wasting
energy.
e Magnetic effects in a transformer's iron core also
contribute to inefficiency. Among the effects are eddy
currents (circulating induction currents in the iron core)
and hysteresis (power lost due to overcoming the
tendency of iron to magnetize in a particular direction).
e Increased frequency results in increased power losses
within a power transformer. The presence of harmonics in
a power system is a source of frequencies significantly
higher than normal, which may cause overheating in
large transformers.
e Both transformers and inductors harbor certain
unavoidable amounts of capacitance due to wire
insulation (dielectric) separating winding turns from the
iron core and from each other. This capacitance can be
significant enough to give the transformer a natural
resonant frequency, which can be problematic in signal
applications.
e Leakage inductance is caused by magnetic flux not being
100% coupled between windings in a transformer. Any
flux not involved with transferring energy from one
winding to another will store and release energy, which is
how (self-) inductance works. Leakage inductance tends
to worsen a transformer's voltage regulation (secondary
voltage “sags” more for a given amount of load current).
e Magnetic saturation of a transformer core may be caused
by excessive primary voltage, operation at too low of a
frequency, and/or by the presence of a DC current in any
of the windings. Saturation may be minimized or avoided
by conservative design, which provides an adequate
margin of safety between peak magnetic flux density
values and the saturation limits of the core.
e Transformers often experience significant inrush currents
when initially connected to an AC voltage source. Inrush
Current is most severe when connection to the AC source
is made at the moment instantaneous source voltage Is
zero.
e Noise is a common phenomenon exhibited by
transformers -- especially power transformers -- and is
primarily caused by magnetostriction of the core.
Physical forces causing winding vibration may also
generate noise under conditions of heavy (high current)
secondary winding load.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
Bart Anderson (January 2004): Corrected conceptual errors
regarding Tesla coil operation and safety.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Bibliography
1. [MYA]Mitsuyoshi Yamamoto, Mitsugi Yamaguchi, “Electric
Power In Japan, Rapid Electrification a Century Ago”,
EDN, (4/11/2002).
http://www.ieee.org/organizations/pes/public/2005/mar/p
eshistory.html
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
= 4 —>
Lessons In Electric Circuits -- Volume Il
Chapter 10
POLYPHASE AC CIRCUITS
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Three-phase power systems
Phase rotation
Polyphase motor design
Three-phase Y and Delta configurations
Three-phase transformer circuits
Harmonics in polyphase power systems
Harmonic phase sequences
Contributors
(A) load load
#1 #2
Single phase power system schematic diagram shows little about the wiring of a
practical power circuit.
Depicted above (Figure above) is a very simple AC circuit. If the load resistor's power
dissipation were substantial, we might call this a “power circuit” or “power system”
instead of regarding it as just a regular circuit. The distinction between a “power circuit”
and a “regular circuit” may seem arbitrary, but the practical concerns are definitely not.
One such concern is the size and cost of wiring necessary to deliver power from the AC
source to the load. Normally, we do not give much thought to this type of concern if
we're merely analyzing a circuit for the sake of learning about the laws of electricity.
However, in the real world it can be a major concern. If we give the source in the above
circuit a voltage value and also give power dissipation values to the two load resistors,
we can determine the wiring needs for this particular circuit: (Figure below)
load load
120V (V) #1 #2
P=10kW P=1l0kW
As a practical matter, the wiring for the 20 kW loads at 120 Vac is rather substantial
(167 A).
tos
E
10 kW
120 V
1= 83.33 A (for each load resistor)
Liotat = Load#t + loade2 Protai = (LO KW) + (LO kW)
Lota = (83.33 A) + (83.33 A) Pista! = 20 kW
rota = 166.67 A
83.33 amps for each load resistor in Figure above adds up to 166.66 amps total circuit
current. This is no small amount of current, and would necessitate copper wire
conductors of at least 1/0 gage. Such wire is well over 1/4 inch (6 mm) in diameter,
weighing over 300 pounds per thousand feet. Bear in mind that copper is not cheap
either! It would be in our best interest to find ways to minimize such costs if we were
designing a power system with long conductor lengths.
One way to do this would be to increase the voltage of the power source and use loads
built to dissipate 10 kW each at this higher voltage. The loads, of course, would have to
have greater resistance values to dissipate the same power as before (10 kW each) ata
greater voltage than before. The advantage would be less current required, permitting
the use of smaller, lighter, and cheaper wire: (Figure below)
load load
240 'V (V) #1 #2
P=10kW P=10kW
Same 10 kW loads at 240 Vac requires less substantial wiring than at 120 Vac (83 A).
l= —
E
lO kW
240 V
1=41.67A (for each load resistor)
Liotat = Load#t + loaie2 Protai = (LO KW) + (LO KW)
Lota = (41.67 A) + (41.67 A) Pista = 20 kW
poral = 83-33 A
Now our tota/ circuit current is 83.33 amps, half of what it was before. We can now use
number 4 gage wire, which weighs less than half of what 1/0 gage wire does per unit
length. This is a considerable reduction in system cost with no degradation in
performance. This is why power distribution system designers elect to transmit electric
power using very high voltages (many thousands of volts): to capitalize on the savings
realized by the use of smaller, lighter, cheaper wire.
However, this solution is not without disadvantages. Another practical concern with
power circuits is the danger of electric shock from high voltages. Again, this is not
usually the sort of thing we concentrate on while learning about the laws of electricity,
but it is a very valid concern in the real world, especially when large amounts of power
are being dealt with. The gain in efficiency realized by stepping up the circuit voltage
presents us with increased danger of electric shock. Power distribution companies tackle
this problem by stringing their power lines along high poles or towers, and insulating the
lines from the supporting structures with large, porcelain insulators.
At the point of use (the electric power customer), there is still the issue of what voltage
to use for powering loads. High voltage gives greater system efficiency by means of
reduced conductor current, but it might not always be practical to keep power wiring out
of reach at the point of use the way it can be elevated out of reach in distribution
systems. This tradeoff between efficiency and danger is one that European power system
designers have decided to risk, all their households and appliances operating at a
nominal voltage of 240 volts instead of 120 volts as it is in North America. That is why
tourists from America visiting Europe must carry small step-down transformers for their
portable appliances, to step the 240 VAC (volts AC) power down to a more suitable 120
VAC.
Is there any way to realize the advantages of both increased efficiency and reduced
safety hazard at the same time? One solution would be to install step-down transformers
at the end-point of power use, just as the American tourist must do while in Europe.
However, this would be expensive and inconvenient for anything but very small loads
(where the transformers can be built cheaply) or very large loads (where the expense of
thick copper wires would exceed the expense of a transformer).
An alternative solution would be to use a higher voltage supply to provide power to two
lower voltage loads in series. This approach combines the efficiency of a high-voltage
system with the safety of a low-voltage system: (Figure below)
—~— 83.33 A
83.33 A —
Series connected 120 Vac loads, driven by 240 Vac source at 83.3 A total current.
Notice the polarity markings (+ and -) for each voltage shown, as well as the
unidirectional arrows for current. For the most part, I've avoided labeling “polarities” in
the AC circuits we've been analyzing, even though the notation is valid to provide a
frame of reference for phase. In later sections of this chapter, phase relationships will
become very important, so I'm introducing this notation early on in the chapter for your
familiarity.
The current through each load is the same as it was in the simple 120 volt circuit, but
the currents are not additive because the loads are in series rather than parallel. The
voltage across each load is only 120 volts, not 240, so the safety factor is better. Mind
you, we still have a full 240 volts across the power system wires, but each load is
operating at a reduced voltage. If anyone is going to get shocked, the odds are that it
will be from coming into contact with the conductors of a particular load rather than from
contact across the main wires of a power system.
There's only one disadvantage to this design: the consequences of one load failing open,
or being turned off (assuming each load has a series on/off switch to interrupt current)
are not good. Being a series circuit, if either load were to open, current would stop in the
other load as well. For this reason, we need to modify the design a bit: (Figure below)
—~— 83.33 A
120 V
Z0°
+
Addition of neutral conductor allows loads to be individually driven.
Exotal = (120 V Z 0°) + (120 V 20°)
Br otal =240V 2 0°
l= — Pro = (10 kW) + (10 kW)
Prop = 20 kW
10kW
~ 120V
1= 83.33 A (for each load resistor)
Instead of a single 240 volt power supply, we use two 120 volt supplies (in phase with
each other!) in series to produce 240 volts, then run a third wire to the connection point
between the loads to handle the eventuality of one load opening. This is called a split-
phase power system. Three smaller wires are still cheaper than the two wires needed
with the simple parallel design, so we're still ahead on efficiency. The astute observer will
note that the neutral wire only has to carry the difference of current between the two
loads back to the source. In the above case, with perfectly “balanced” loads consuming
equal amounts of power, the neutral wire carries zero current.
Notice how the neutral wire is connected to earth ground at the power supply end. This
is a common feature in power systems containing “neutral” wires, since grounding the
neutral wire ensures the least possible voltage at any given time between any “hot” wire
and earth ground.
An essential component to a split-phase power system is the dual AC voltage source.
Fortunately, designing and building one is not difficult. Since most AC systems receive
their power from a step-down transformer anyway (stepping voltage down from high
distribution levels to a user-level voltage like 120 or 240), that transformer can be built
with a center-tapped secondary winding: (Figure below)
Step-down transformer with |
center-tapped secondary winding
1
\
American 120/240 Vac power is derived from a center tapped utility transformer.
If the AC power comes directly from a generator (alternator), the coils can be similarly
center-tapped for the same effect. The extra expense to include a center-tap connection
in atransformer or alternator winding is minimal.
Here is where the (+) and (-) polarity markings really become important. This notation is
often used to reference the phasings of multiple AC voltage sources, so it is clear
whether they are aiding (“boosting”) each other or opposing (“bucking”) each other. If
not for these polarity markings, phase relations between multiple AC sources might be
very confusing. Note that the split-phase sources in the schematic (each one 120 volts Z
0°), with polarity marks (+) to (-) just like series-aiding batteries can alternatively be
represented as such: (Figure below)
"hot"
"hot"
Split phase 120/240 Vac source is equivalent to two series aiding 120 Vac sources.
To mathematically calculate voltage between “hot” wires, we must subtract voltages,
because their polarity marks show them to be opposed to each other:
Polar Rectangular
120 20° 120+ joV
- 120 Z 180° - (-120+j0) V
240 Z 0° 240 + jO V
If we mark the two sources' common connection point (the neutral wire) with the same
polarity mark (-), we must express their relative phase shifts as being 180° apart.
Otherwise, we'd be denoting two voltage sources in direct opposition with each other,
which would give 0 volts between the two “hot” conductors. Why am | taking the time to
elaborate on polarity marks and phase angles? It will make more sense in the next
section!
Power systems in American households and light industry are most often of the split-
phase variety, providing so-called 120/240 VAC power. The term “split-phase” merely
refers to the split-voltage supply in such a system. In a more general sense, this kind of
AC power supply is called single phase because both voltage waveforms are in phase, or
in step, with each other.
The term “single phase” is a counterpoint to another kind of power system called
“polyphase” which we are about to investigate in detail. Apologies for the long
introduction leading up to the title-topic of this chapter. The advantages of polyphase
power systems are more obvious if one first has a good understanding of single phase
systems.
REVIEW:
Single phase power systems are defined by having an AC source with only one
voltage waveform.
A split-phase power system is one with multiple (in-phase) AC voltage sources
connected in series, delivering power to loads at more than one voltage, with more
than two wires. They are used primarily to achieve balance between system
efficiency (low conductor currents) and safety (low load voltages).
Split-phase AC sources can be easily created by center-tapping the coil windings of
transformers or alternators.
Three-phase power systems
Split-phase power systems achieve their high conductor efficiency and low safety risk by
splitting up the total voltage into lesser parts and powering multiple loads at those
lesser voltages, while drawing currents at levels typical of a full-voltage system. This
technique, by the way, works just as well for DC power systems as it does for single-
phase AC systems. Such systems are usually referred to as three-wire systems rather
than split-phase because “phase” is a concept restricted to AC.
But we know from our experience with vectors and complex numbers that AC voltages
don't always add up as we think they would if they are out of phase with each other. This
principle, applied to power systems, can be put to use to make power systems with even
greater conductor efficiencies and lower shock hazard than with split-phase.
Suppose that we had two sources of AC voltage connected in series just like the split-
phase system we saw before, except that each voltage source was 120° out of phase
with the other: (Figure below)
—— 83.33A 20°
+
"neutral"
~«— 83.33 A Z 120°
Pair of 120 Vac sources phased 120°, similar to split-phase.
Since each voltage source is 120 volts, and each load resistor is connected directly in
parallel with its respective source, the voltage across each load must be 120 volts as
well. Given load currents of 83.33 amps, each load must still be dissipating 10 kilowatts
of power. However, voltage between the two “hot” wires is not 240 volts (120 Z 0°- 120
Z 180°) because the phase difference between the two sources is not 180°. Instead, the
voltage is:
E, at = (120 V 2 0°) - (120 V Z 120°)
Excrat = 207.85 V Z -30°
Nominally, we say that the voltage between “hot” conductors is 208 volts (rounding up),
and thus the power system voltage is designated as 120/208.
If we calculate the current through the “neutral” conductor, we find that it is not zero,
even with balanced load resistances. Kirchhoff's Current Law tells us that the currents
entering and exiting the node between the two loads must be zero: (Figure below)
<— 33.33A 20°
"neutral"
——_—
Lieuteal
~<— 83.33 A 2 120°
Neutral wire carries a current in the case of a pair of 120° phased sources.
Toads iz Loads? a 1, eutval =0
“Theutal = Loaaet + Loade?
Tjeutral = “lioat#t ~ Loads
Tneural = - (83.33 A Z 0°) - (83.33 A Z 120°)
Ineunal = 83-33 A Z 240° or 83.33 A Z-120°
So, we find that the “neutral” wire is carrying a full 83.33 amps, just like each “hot” wire.
Note that we are still conveying 20 kW of total power to the two loads, with each load's
“hot” wire carrying 83.33 amps as before. With the same amount of current through each
“hot” wire, we must use the same gage copper conductors, so we haven't reduced
system cost over the split-phase 120/240 system. However, we have realized a gain in
safety, because the overall voltage between the two “hot” conductors is 32 volts lower
than it was in the split-phase system (208 volts instead of 240 volts).
The fact that the neutral wire is carrying 83.33 amps of current raises an interesting
possibility: since its carrying current anyway, why not use that third wire as another
“hot” conductor, powering another load resistor with a third 120 volt source having a
phase angle of 240°? That way, we could transmit more power (another 10 kW) without
having to add any more conductors. Let's see how this might look: (Figure below)
With a third load phased 120° to the other two, the currents are the same as for two
loads.
A full mathematical analysis of all the voltages and currents in this circuit would
necessitate the use of a network theorem, the easiest being the Superposition Theorem.
I'll spare you the long, drawn-out calculations because you should be able to intuitively
understand that the three voltage sources at three different phase angles will deliver
120 volts each to a balanced triad of load resistors. For proof of this, we can use SPICE to
do the math for us: (Figure below, SPICE listing: 120/208 polyphase power system)
SPICE circuit: Three 3-® loads phased at 120°.
120/208 polyphase power system
vl 10 ac 120 © sin
v2 2 0 ac 120 120 sin
v3 3 0 ac 120 240 sin
rl 141.44
r2 24 1.44
r3 3 4 1.44
-ac Lin 1 60 60
»print ac v(1,4) v(2,4) v(3,4)
-print ac v(1,2) v(2,3) v(3,1)
-print ac i(vl) i(v2) i(v3)
.end
VOLTAGE ACROSS EACH LOAD
freq v(1,4) v(2,4) v(3,4)
6.000E+01 1.200E+02 1.200E+02 1.200E+02
VOLTAGE BETWEEN “HOT” CONDUCTORS
freq v(1,2) v(2,3) v(3,1)
6.000E+01 2.078E+02 2.078E+02 2.078E+02
CURRENT THROUGH EACH VOLTAGE SOURCE
freq i(vl) i(v2) i(v3)
6.000E+01 8.333E+01 8.333E+01 8.333E+01
Sure enough, we get 120 volts across each load resistor, with (approximately) 208 volts
between any two “hot” conductors and conductor currents equal to 83.33 amps. (Figure
below) At that current and voltage, each load will be dissipating 10 kW of power. Notice
that this circuit has no “neutral” conductor to ensure stable voltage to all loads if one
should open. What we have here is a situation similar to our split-phase power circuit
with no “neutral” conductor: if one load should happen to fail open, the voltage drops
across the remaining load(s) will change. To ensure load voltage stability in the event of
another load opening, we need a neutral wire to connect the source node and load node
together:
—<— 83.33 A Z0°
load
#1
<~—0OA_ "neutral"
SPICE circuit annotated with simulation results: Three 3-® loads phased at 120°.
So long as the loads remain balanced (equal resistance, equal currents), the neutral wire
will not have to carry any current at all. It is there just in case one or more load resistors
should fail open (or be shut off through a disconnecting switch).
This circuit we've been analyzing with three voltage sources is called a polyphase circuit.
The prefix “poly” simply means “more than one,” as in “polytheism” (belief in more than
one deity), “polygon” (a geometrical shape made of multiple line segments: for example,
pentagon and hexagon), and “polyatomic” (a substance composed of multiple types of
atoms). Since the voltage sources are all at different phase angles (in this case, three
different phase angles), this is a “po/yphase” circuit. More specifically, it is a three-phase
circuit, the kind used predominantly in large power distribution systems.
Let's survey the advantages of a three-phase power system over a single-phase system
of equivalent load voltage and power capacity. A single-phase system with three loads
connected directly in parallel would have a very high total current (83.33 times 3, or 250
amps. (Figure below)
load
#3
230A —* 10kW 1l0okW lOokW
For comparison, three 10 Kw loads on a 120 Vac system draw 250 A.
This would necessitate 3/0 gage copper wire (very large!), at about 510 pounds per
thousand feet, and with a considerable price tag attached. If the distance from source to
load was 1000 feet, we would need over a half-ton of copper wire to do the job. On the
other hand, we could build a split-phase system with two 15 kW, 120 volt loads. (Figure
below)
~=«— 125A 2 180°
Split phase system draws half the current of 125 A at 240 Vac compared to 120 Vac
system.
Our current is half of what it was with the simple parallel circuit, which is a great
improvement. We could get away with using number 2 gage copper wire at a total mass
of about 600 pounds, figuring about 200 pounds per thousand feet with three runs of
1000 feet each between source and loads. However, we also have to consider the
increased safety hazard of having 240 volts present in the system, even though each
load only receives 120 volts. Overall, there is greater potential for dangerous electric
shock to occur.
When we contrast these two examples against our three-phase system (Figure above),
the advantages are quite clear. First, the conductor currents are quite a bit less (83.33
amps versus 125 or 250 amps), permitting the use of much thinner and lighter wire. We
can use number 4 gage wire at about 125 pounds per thousand feet, which will total 500
pounds (four runs of 1000 feet each) for our example circuit. This represents a significant
cost savings over the split-phase system, with the additional benefit that the maximum
voltage in the system is lower (208 versus 240).
One question remains to be answered: how in the world do we get three AC voltage
sources whose phase angles are exactly 120° apart? Obviously we can't center-tap a
transformer or alternator winding like we did in the split-phase system, since that can
only give us voltage waveforms that are either in phase or 180° out of phase. Perhaps we
could figure out some way to use capacitors and inductors to create phase shifts of 120°,
but then those phase shifts would depend on the phase angles of our load impedances
as well (substituting a capacitive or inductive load for a resistive load would change
everything!).
The best way to get the phase shifts we're looking for is to generate it at the source:
construct the AC generator (alternator) providing the power in such a way that the
rotating magnetic field passes by three sets of wire windings, each set spaced 120° apart
around the circumference of the machine as in Figure below.
Three-phase alternator (b)
Single-phase alternator (a) mincing meng
winding winding winding os winding
allie ( . ) aati ie
windin N windin
: aby ™ BP 2
(a) Single-phase alternator, (b) Three-phase alternator.
Together, the six “pole” windings of a three-phase alternator are connected to comprise
three winding pairs, each pair producing AC voltage with a phase angle 120° shifted
from either of the other two winding pairs. The interconnections between pairs of
windings (as shown for the single-phase alternator: the jumper wire between windings
la and 1b) have been omitted from the three-phase alternator drawing for simplicity.
In our example circuit, we showed the three voltage sources connected together in a “Y”
configuration (sometimes called the “star” configuration), with one lead of each source
tied to a common point (the node where we attached the “neutral” conductor). The
common way to depict this connection scheme is to draw the windings in the shape of a
“Y” like Figure below.
Alternator "Y" configuration.
The “Y” configuration is not the only option open to us, but it is probably the easiest to
understand at first. More to come on this subject later in the chapter.
REVIEW:
A single-phase power system is one where there is only one AC voltage source (one
source voltage waveform).
A split-phase power system is one where there are two voltage sources, 180° phase-
shifted from each other, powering two series-connected loads. The advantage of this
is the ability to have lower conductor currents while maintaining low load voltages
for safety reasons.
A polyphase power system uses multiple voltage sources at different phase angles
from each other (many “phases” of voltage waveforms at work). A polyphase power
system can deliver more power at less voltage with smaller-gage conductors than
single- or split-phase systems.
e« The phase-shifted voltage sources necessary for a polyphase power system are
created in alternators with multiple sets of wire windings. These winding sets are
spaced around the circumference of the rotor's rotation at the desired angle(s).
Phase rotation
Let's take the three-phase alternator design laid out earlier (Figure below) and watch
what happens as the magnet rotates.
winding windin
2a 9 3a g
in
Three-phase alternator
The phase angle shift of 120° is a function of the actual rotational angle shift of the three
pairs of windings (Figure below). If the magnet is rotating clockwise, winding 3 will
generate its peak instantaneous voltage exactly 120° (of alternator shaft rotation) after
winding 2, which will hits its peak 120° after winding 1. The magnet passes by each pole
pair at different positions in the rotational movement of the shaft. Where we decide to
place the windings will dictate the amount of phase shift between the windings’ AC
voltage waveforms. If we make winding 1 our “reference” voltage source for phase angle
(0°), then winding 2 will have a phase angle of -120° (120° lagging, or 240° leading) and
winding 3 an angle of -240° (or 120° leading).
This sequence of phase shifts has a definite order. For clockwise rotation of the shaft, the
order is 1-2-3 (winding 1 peaks first, them winding 2, then winding 3). This order keeps
repeating itself as long as we continue to rotate the alternator's shaft. (Figure below)
phase sequence:
e238 1s 2= a> 1-2-9
1 2 3
TIME —>
Clockwise rotation phase sequence: 1-2-3.
However, if we reverse the rotation of the alternator's shaft (turn it counter-clockwise),
the magnet will pass by the pole pairs in the opposite sequence. Instead of 1-2-3, we'll
have 3-2-1. Now, winding 2's waveform will be /eading 120° ahead of 1 instead of
lagging, and 3 will be another 120° ahead of 2. (Figure below)
phase sequence:
g-2-1=4-2- 1- 3-2-1
3 2 1
TIME —>
Counterclockwise rotation phase sequence: 3-2-1.
The order of voltage waveform sequences in a polyphase system is called phase rotation
or phase sequence. If we're using a polyphase voltage source to power resistive loads,
phase rotation will make no difference at all. Whether 1-2-3 or 3-2-1, the voltage and
current magnitudes will all be the same. There are some applications of three-phase
power, as we will see shortly, that depend on having phase rotation being one way or the
other. Since voltmeters and ammeters would be useless in telling us what the phase
rotation of an operating power system is, we need to have some other kind of instrument
capable of doing the job.
One ingenious circuit design uses a capacitor to introduce a phase shift between voltage
and current, which is then used to detect the sequence by way of comparison between
the brightness of two indicator lamps in Figure below.
to phase to phase
#1 #2
|
to phase
#3
Phase sequence detector compares brightness of two lamps.
The two lamps are of equal filament resistance and wattage. The capacitor is sized to
have approximately the same amount of reactance at system frequency as each lamp's
resistance. If the capacitor were to be replaced by a resistor of equal value to the lamps'
resistance, the two lamps would glow at equal brightness, the circuit being balanced.
However, the capacitor introduces a phase shift between voltage and current in the third
leg of the circuit equal to 90°. This phase shift, greater than 0° but less than 120°, skews
the voltage and current values across the two lamps according to their phase shifts
relative to phase 3. The following SPICE analysis demonstrates what will happen: (Figure
below), "phase rotation detector -- sequence = v1-v2-v3"
2650 Q
SPICE circuit for phase sequence detector.
phase rotation detector -- sequence = vl-v2-v3
vl 10 ac 120 0 sin
v2 2 0 ac 120 120 sin
v3 3 0 ac 120 240 sin
rl 14 2650
r2 2 4 2650
cl 3 4 lu
-ac lin 1 60 60
-print ac v(1,4) v(2,4) v(3,4)
.end
freq v(1,4) v(2,4) v(3,4)
6.000E+01 4.810E+01 1.795E+02 1.610E+02
The resulting phase shift from the capacitor causes the voltage across phase 1 lamp
(between nodes 1 and 4) to fall to 48.1 volts and the voltage across phase 2 lamp
(between nodes 2 and 4) to rise to 179.5 volts, making the first lamp dim and the second
lamp bright. Just the opposite will happen if the phase sequence is reversed: "phase
rotation detector -- sequence = v3-v2-v1 "
phase rotation detector -- sequence = v3-v2-vl
v1 10 ac 120 240 sin
v2 2 0 ac 120 120 sin
v3 3 0 ac 120 © sin
rl 14 2650
r2 2 4 2650
cl 3 4 lu
-ac Lin 1 60 60
-print ac v(1,4) v(2,4) v(3,4)
.end
freq v(1,4) v(2,4) v(3,4)
6.000E+01 1.795E+02 4.810E+01 1.610E+02
Here,("phase rotation detector -- sequence = v3-v2-v1") the first lamp receives 179.5
volts while the second receives only 48.1 volts.
We've investigated how phase rotation is produced (the order in which pole pairs get
passed by the alternator's rotating magnet) and how it can be changed by reversing the
alternator's shaft rotation. However, reversal of the alternator's shaft rotation is not
usually an option open to an end-user of electrical power supplied by a nationwide grid
(“the” alternator actually being the combined total of all alternators in all power plants
feeding the grid). There is a much easier way to reverse phase sequence than reversing
alternator rotation: just exchange any two of the three “hot” wires going to a three-phase
load.
This trick makes more sense if we take another look at a running phase sequence of a
three-phase voltage source:
1-2-3 rotation: 1-
3-2-1 rotation: 3-
What is commonly designated as a “1-2-3” phase rotation could just as well be called “2-
3-1” or “3-1-2,” going from left to right in the number string above. Likewise, the
opposite rotation (3-2-1) could just as easily be called “2-1-3” or “1-3-2.”
Starting out with a phase rotation of 3-2-1, we can try all the possibilities for swapping
any two of the wires at a time and see what happens to the resulting sequence in Figure
below.
Original 1-2-3
phase rotation
1 2
. < ‘ (wires 1 and 2 swapped)
3
End result
phase rotation = 2-1-3
(wires 2 and 3 swapped)
2 3 ;
XK phase rotation = 1-3-2
3 2
1 3 (wires 1 and 3 swapped)
e e phase rotation = 3-2-1
3 1
All possibilities of swapping any two wires.
No matter which pair of “hot” wires out of the three we choose to swap, the phase
rotation ends up being reversed (1-2-3 gets changed to 2-1-3, 1-3-2 or 3-2-1, all
equivalent).
e REVIEW:
e Phase rotation, or phase sequence, is the order in which the voltage waveforms of a
polyphase AC source reach their respective peaks. For a three-phase system, there
are only two possible phase sequences: 1-2-3 and 3-2-1, corresponding to the two
possible directions of alternator rotation.
e Phase rotation has no impact on resistive loads, but it will have impact on
unbalanced reactive loads, as shown in the operation of a phase rotation detector
circuit.
e Phase rotation can be reversed by swapping any two of the three “hot” leads
supplying three-phase power to a three-phase load.
Polyphase motor design
Perhaps the most important benefit of polyphase AC power over single-phase is the
design and operation of AC motors. As we studied in the first chapter of this book, some
types of AC motors are virtually identical in construction to their alternator (generator)
counterparts, consisting of stationary wire windings and a rotating magnet assembly.
(Other AC motor designs are not quite this simple, but we will leave those details to
another lesson).
Step #1 Step #2 ™
Noms ( [a] )Nern aay
a
; <! {\)} =<! . {)}
Step #8 Step #4 7%
Sm’ ( Je} )) Sam EN]
YY
. — © == : ®
Clockwise AC motor operation.
If the rotating magnet is able to keep up with the frequency of the alternating current
energizing the electromagnet windings (coils), it will continue to be pulled around
clockwise. (Figure above) However, clockwise is not the only valid direction for this
motor's shaft to spin. It could just as easily be powered in a counter-clockwise direction
by the same AC voltage waveform a in Figure below.
Step #1 Step #2
Nom ( |e] )Nerm ws]
\ 4
© ©
Step #4 yw
é ean
Sa
Je @_!== ©
Counterclockwise AC motor operation.
Notice that with the exact same sequence of polarity cycles (voltage, current, and
magnetic poles produced by the coils), the magnetic rotor can spin in either direction.
This is a common trait of all single-phase AC “induction” and “synchronous” motors: they
have no normal or “correct” direction of rotation. The natural question should arise at
this point: how can the motor get started in the intended direction if it can run either
way just as well? The answer is that these motors need a little help getting started. Once
helped to spin in a particular direction. they will continue to spin that way as long as AC
power is maintained to the windings.
Where that “help” comes from for a single-phase AC motor to get going in one direction
can vary. Usually, it comes from an additional set of windings positioned differently from
the main set, and energized with an AC voltage that is out of phase with the main power.
(Figure below)
~«— winding 2's voltage waveform is 90 d
out of phase with Winding 1's voltage w veform
windin
sao
winding
wa (
winding
. )a ae
windin
3 259
winding 2's voltage waveform is 90 di
~— outofp tase with Winding 1's voltage wa ‘storm
Unidirectional-starting AC two-phase motor.
These supplementary coils are typically connected in series with a capacitor to introduce
a phase shift in current between the two sets of windings. (Figure below)
1b 2b
t, ° t
these two branch currents are
out of phase with each other
Capacitor phase shift adds second phase.
That phase shift creates magnetic fields from coils 2a and 2b that are equally out of step
with the fields from coils la and 1b. The result is a set of magnetic fields with a definite
phase rotation. It is this phase rotation that pulls the rotating magnet around in a
definite direction.
Polyphase AC motors require no such trickery to spin in a definite direction. Because
their supply voltage waveforms already have a definite rotation sequence, so do the
respective magnetic fields generated by the motor's stationary windings. In fact, the
combination of all three phase winding sets working together creates what is often
called a rotating magnetic field. It was this concept of a rotating magnetic field that
inspired Nikola Tesla to design the world's first polyphase electrical systems (simply to
make simpler, more efficient motors). The line current and safety advantages of
polyphase power over single phase power were discovered later.
What can be a confusing concept is made much clearer through analogy. Have you ever
seen a row of blinking light bulbs such as the kind used in Christmas decorations? Some
strings appear to “move” in a definite direction as the bulbs alternately glow and darken
in sequence. Other strings just blink on and off with no apparent motion. What makes
the difference between the two types of bulb strings? Answer: phase shift!
Examine a string of lights where every other bulb is lit at any given time as in (Figure
below)
12141212 12
;
al"2"bulbs it @@O@SSCOCOQ
@r
e-
@r
~ 1212412121212
all "1" bulbs lit ee0aeeeeeeeded OO
phase sequence: 1-2-1-2
Phase sequence 1-2-1-2: lamps appear to move.
When all of the “1” bulbs are lit, the “2” bulbs are dark, and vice versa. With this
blinking sequence, there is no definite “motion” to the bulbs' light. Your eyes could
follow a “motion” from left to right just as easily as from right to left. Technically, the “1”
and “2” bulb blinking sequences are 180° out of phase (exactly opposite each other).
This is analogous to the single-phase AC motor, which can run just as easily in either
direction, but which cannot start on its own because its magnetic field alternation lacks
a definite “rotation.”
Now let's examine a string of lights where there are three sets of bulbs to be sequenced
instead of just two, and these three sets are equally out of phase with each other in
Figure below.
1293d4142a93idd4ae2g83d1é2 «3
all'1" bubs it @@@SCSCCOCCO
123 123 «12 3 «1 2 3
al 2 bibsit @@OCCOOCOOCS |...
123 123 12 3 «12 3
all’3" bubs it @@OSSCOCCOCO
123123 123 «1 2 3
all"1" bubs it @@@SCCCOCCCO
phase sequence = 1-2-3
bulbs appear to be "moving” from left to right
Phase sequence: 1-2-3: bulbs appear to move left to right.
If the lighting sequence is 1-2-3 (the sequence shown in (Figure above)), the bulbs will
appear to “move” from left to right. Now imagine this blinking string of bulbs arranged
into a circle as in Figure below.
2 3
® ®@
all "1" bulbs lit 1@-+—-—+ @1
e @
3 2
2 3
e @
The bulbs appear to
all "2" bulbs lit 1@ @ "move" in aclockwise
direction
e@ ®@
3 2
2 3
e @
all "3" bulbs lit 1@ Z @ i
e @
3 2
Circular arrangement; bulbs appear to rotate clockwise.
Now the lights in Figure above appear to be “moving” in a clockwise direction because
they are arranged around a circle instead of a straight line. It should come as no surprise
that the appearance of motion will reverse if the phase sequence of the bulbs is
reversed.
The blinking pattern will either appear to move clockwise or counter-clockwise
depending on the phase sequence. This is analogous to a three-phase AC motor with
three sets of windings energized by voltage sources of three different phase shifts in
Figure below.
Three-phase AC motor: A phase sequence of 1-2-3 spins the magnet clockwise, 3-2-1
spins the magnet counterclockwise.
With phase shifts of less than 180° we get true rotation of the magnetic field. With
single-phase motors, the rotating magnetic field necessary for self-starting must to be
created by way of capacitive phase shift. With polyphase motors, the necessary phase
shifts are there already. Plus, the direction of shaft rotation for polyphase motors is very
easily reversed: just swap any two “hot” wires going to the motor, and it will run in the
opposite direction!
¢ REVIEW:
e AC “induction” and “synchronous” motors work by having a rotating magnet follow
the alternating magnetic fields produced by stationary wire windings.
e Single-phase AC motors of this type need help to get started spinning in a particular
direction.
e By introducing a phase shift of less than 180° to the magnetic fields in such a motor,
a definite direction of shaft rotation can be established.
Single-phase induction motors often use an auxiliary winding connected in series
with a capacitor to create the necessary phase shift.
Polyphase motors don't need such measures; their direction of rotation is fixed by
the phase sequence of the voltage they're powered by.
Swapping any two “hot” wires on a polyphase AC motor will reverse its phase
sequence, thus reversing its shaft rotation.
Three-phase Y and Delta configurations
Initially we explored the idea of three-phase power systems by connecting three voltage
sources together in what is commonly known as the “Y” (or “star”) configuration. This
configuration of voltage sources is characterized by a common connection point joining
one side of each source. (Figure below)
Three-phase “Y” connection has three voltage sources connected to a common point.
If we draw a circuit showing each voltage source to be a coil of wire (alternator or
transformer winding) and do some slight rearranging, the “Y” configuration becomes
more obvious in Figure below.
"line"
"line"
"neutral"
"line"
Three-phase, four-wire “Y” connection uses a "common" fourth wire.
The three conductors leading away from the voltage sources (windings) toward a load
are typically called /ines, while the windings themselves are typically called phases. In a
Y-connected system, there may or may not (Figure below) be a neutral wire attached at
the junction point in the middle, although it certainly helps alleviate potential problems
should one element of a three-phase load fail open, as discussed earlier.
3-phase, 3-wire "Y" connection
"line"
”
"line"
(no "neutral" wire)
"line"
Three-phase, three-wire “Y” connection does not use the neutral wire.
When we measure voltage and current in three-phase systems, we need to be specific as
to where we're measuring. Line voltage refers to the amount of voltage measured
between any two line conductors in a balanced three-phase system. With the above
circuit, the line voltage is roughly 208 volts. Phase voltage refers to the voltage
measured across any one component (source winding or load impedance) in a balanced
three-phase source or load. For the circuit shown above, the phase voltage is 120 volts.
The terms /ine current and phase current follow the same logic: the former referring to
current through any one line conductor, and the latter to current through any one
component.
Y-connected sources and loads always have line voltages greater than phase voltages,
and line currents equal to phase currents. If the Y-connected source or load is balanced,
the line voltage will be equal to the phase voltage times the square root of 3:
For "Y" circuits:
Eiine = V 3 E phase
1
line — 1,, ase
However, the “Y” configuration is not the only valid one for connecting three-phase
voltage source or load elements together. Another configuration is known as the “Delta,”
for its geometric resemblance to the Greek letter of the same name (A). Take close notice
of the polarity for each winding in Figure below.
"line"
120V 20°
+ 2
"line"
"line"
Three-phase, three-wire A connection has no common.
At first glance it seems as though three voltage sources like this would create a short-
circuit, electrons flowing around the triangle with nothing but the internal impedance of
the windings to hold them back. Due to the phase angles of these three voltage sources,
however, this is not the case.
One quick check of this is to use Kirchhoff's Voltage Law to see if the three voltages
around the loop add up to zero. If they do, then there will be no voltage available to push
current around and around that loop, and consequently there will be no circulating
current. Starting with the top winding and progressing counter-clockwise, our KVL
expression looks something like this:
(120 V 2 0°) + (120 V Z 240°) + (120 V Z 120°)
Does it all equal 0?
Yes!
Indeed, if we add these three vector quantities together, they do add up to zero. Another
way to verify the fact that these three voltage sources can be connected together ina
loop without resulting in circulating currents is to open up the loop at one junction point
and calculate voltage across the break: (Figure below)
120V Z0°
+ 2
120 V
Z 240°
—+| —
Ex:ear SHOuld equal 0 V
Voltage across open A should be zero.
Starting with the right winding (120 V Z 120°) and progressing counter-clockwise, our
KVL equation looks like this:
(120 V Z 120°) + (120 20°) + (120 V Z 240°) + Exceni = 0
0+ Exreak = 9
E, 0)
break —
Sure enough, there will be zero voltage across the break, telling us that no current will
circulate within the triangular loop of windings when that connection is made complete.
Having established that a A-connected three-phase voltage source will not burn itself to
a crisp due to circulating currents, we turn to its practical use as a source of power in
three-phase circuits. Because each pair of line conductors is connected directly across a
single winding in a A circuit, the line voltage will be equal to the phase voltage.
Conversely, because each line conductor attaches at a node between two windings, the
line current will be the vector sum of the two joining phase currents. Not surprisingly, the
resulting equations for a A configuration are as follows:
For A ("delta") circuits:
tine = Ephase
line= V 3 L phase
Let's see how this works in an example circuit: (Figure below)
120V 20°
+ e
The load on the A source is wired ina A.
With each load resistance receiving 120 volts from its respective phase winding at the
source, the current in each phase of this circuit will be 83.33 amps:
i=2-
E
10 kW
120 V
1 = 83.33 A (for each load resistor and source winding)
line= V 3 Tnhase
line = V 3 (83.33 A)
line = 144.34 A
So each line current in this three-phase power system is equal to 144.34 amps, which is
substantially more than the line currents in the Y-connected system we looked at earlier.
One might wonder if we've lost all the advantages of three-phase power here, given the
fact that we have such greater conductor currents, necessitating thicker, more costly
wire. The answer is no. Although this circuit would require three number 1 gage copper
conductors (at 1000 feet of distance between source and load this equates to a little
over 750 pounds of copper for the whole system), it is still less than the 1000+ pounds
of copper required for a single-phase system delivering the same power (30 kW) at the
same voltage (120 volts conductor-to-conductor).
One distinct advantage of a A-connected system is its lack of a neutral wire. With a Y-
connected system, a neutral wire was needed in case one of the phase loads were to fail
open (or be turned off), in order to keep the phase voltages at the load from changing.
This is not necessary (or even possible!) in a A-connected circuit. With each load phase
element directly connected across a respective source phase winding, the phase voltage
will be constant regardless of open failures in the load elements.
Perhaps the greatest advantage of the A-connected source is its fault tolerance. It is
possible for one of the windings in a A-connected three-phase source to fail open (Figure
below) without affecting load voltage or current!
1200V 20
nd =
windin 7
falled open! L20V
Even with a source winding failure, the line voltage is still 120 V, and load phase voltage
is still 120 V. The only difference is extra current in the remaining functional source
windings.
The only consequence of a source winding failing open for a A-connected source is
increased phase current in the remaining windings. Compare this fault tolerance with a
Y-connected system suffering an open source winding in Figure below.
winding
failed open!
Open “Y” source winding halves the voltage on two loads of a A connected load.
With a A-connected load, two of the resistances suffer reduced voltage while one
remains at the original line voltage, 208. A Y-connected load suffers an even worse fate
(Figure below) with the same winding failure in a Y-connected source
winding
failed open!
Open source winding of a "Y-Y" system halves the voltage on two loads, and looses one
load entirely.
In this case, two load resistances suffer reduced voltage while the third loses supply
voltage completely! For this reason, A-connected sources are preferred for reliability.
However, if dual voltages are needed (e.g. 120/208) or preferred for lower line currents,
Y-connected systems are the configuration of choice.
¢ REVIEW:
e The conductors connected to the three points of a three-phase source or load are
called /ines.
e The three components comprising a three-phase source or load are called phases.
Line voltage is the voltage measured between any two lines in a three-phase circuit.
Phase voltage is the voltage measured across a single component in a three-phase
source or load.
e Line current is the current through any one line between a three-phase source and
load.
e Phase current is the current through any one component comprising a three-phase
source or load.
In balanced “Y” circuits, line voltage is equal to phase voltage times the square root
of 3, while line current is equal to phase current.
For "Y" circuits:
Eline = VY 3 Ephase
e Line = Lnnase
In balanced A circuits, line voltage is equal to phase voltage, while line current is
equal to phase current times the square root of 3.
For A ("delta") circuits:
Eiine = phase
o line= V 3 Lphase
A-connected three-phase voltage sources give greater reliability in the event of
winding failure than Y-connected sources. However, Y-connected sources can deliver
the same amount of power with less line current than A-connected sources.
Three-phase transformer circuits
Since three-phase is used so often for power distribution systems, it makes sense that we
would need three-phase transformers to be able to step voltages up or down. This is only
partially true, as regular single-phase transformers can be ganged together to transform
power between two three-phase systems in a variety of configurations, eliminating the
requirement for a special three-phase transformer. However, special three-phase
transformers are built for those tasks, and are able to perform with less material
requirement, less size, and less weight than their modular counterparts.
A three-phase transformer is made of three sets of primary and secondary windings,
each set wound around one leg of an iron core assembly. Essentially it looks like three
single-phase transformers sharing a joined core as in Figure below.
Three-phase transformer core
Three phase transformer core has three sets of windings.
Those sets of primary and secondary windings will be connected in either A or Y
configurations to form a complete unit. The various combinations of ways that these
windings can be connected together in will be the focus of this section.
Whether the winding sets share a common core assembly or each winding pair isa
separate transformer, the winding connection options are the same:
e« Primary - Secondary
2g
- Y -
e Y - A
» A - Y
« A - A
The reasons for choosing a Y or A configuration for transformer winding connections are
the same as for any other three-phase application: Y connections provide the
opportunity for multiple voltages, while A connections enjoy a higher level of reliability
(if one winding fails open, the other two can still maintain full line voltages to the load).
Probably the most important aspect of connecting three sets of primary and secondary
windings together to form a three-phase transformer bank is paying attention to proper
winding phasing (the dots used to denote “polarity” of windings). Remember the proper
phase relationships between the phase windings of A and Y: (Figure below)
(Y) The center point of the “Y” must tie either all the “-” or all the “+” winding points
together. (A) The winding polarities must stack together in a complementary manner ( +
to -).
Getting this phasing correct when the windings aren't shown in regular Y or A
configuration can be tricky. Let me illustrate, starting with Figure below.
A,
B,
C,
T, T, T;
A,
B,
C,
Inputs Aj, By, C; may be wired either “A” or “Y”, as may outputs A>, Bo, Co.
Three individual transformers are to be connected together to transform power from one
three-phase system to another. First, I'll show the wiring connections for a Y-Y
configuration: Figure below
Y-Y
Phase wiring for “Y-Y” transformer.
Note in Figure above how all the winding ends marked with dots are connected to their
respective phases A, B, and C, while the non-dot ends are connected together to form
the centers of each “Y”. Having both primary and secondary winding sets connected in
“Y” formations allows for the use of neutral conductors (N; and N>) in each power
system.
Now, we'll take a look at a Y-A configuration: (Figure below)
Y-A
Ag
B,
C,
Phase wiring for “Y-A” transformer.
Note how the secondary windings (bottom set, Figure above) are connected in a chain,
the “dot” side of one winding connected to the “non-dot” side of the next, forming the A
loop. At every connection point between pairs of windings, a connection is made to a line
of the second power system (A, B, and C).
Now, let's examine a A-Y system in Figure below.
A,
B,
C;
Ne
Ag
B,
C,
Phase wiring for “A-Y” transformer.
Such a configuration (Figure above) would allow for the provision of multiple voltages
(line-to-line or line-to-neutral) in the second power system, from a source power system
having no neutral.
And finally, we turn to the A-A configuration: (Figure below)
A-A
Az
Be
C2
Phase wiring for “A-A” transformer.
When there is no need for a neutral conductor in the secondary power system, A-A
connection schemes (Figure above) are preferred because of the inherent reliability of
the A configuration.
Considering that a A configuration can operate satisfactorily missing one winding, some
power system designers choose to create a three-phase transformer bank with only two
transformers, representing a A-A configuration with a missing winding in both the
primary and secondary sides: (Figure below)
"Open A"
A2
Bp
C2
“V” or “open-A” provides 2-g power with only two transformers.
This configuration is called “V” or “Open-A.” Of course, each of the two transformers
have to be oversized to handle the same amount of power as three in a standard A
configuration, but the overall size, weight, and cost advantages are often worth it. Bear
in mind, however, that with one winding set missing from the A shape, this system no
longer provides the fault tolerance of a normal A-A system. If one of the two transformers
were to fail, the load voltage and current would definitely be affected.
The following photograph (Figure below) shows a bank of step-up transformers at the
Grand Coulee hydroelectric dam in Washington state. Several transformers (green in
color) may be seen from this vantage point, and they are grouped in threes: three
transformers per hydroelectric generator, wired together in some form of three-phase
configuration. The photograph doesn't reveal the primary winding connections, but it
appears the secondaries are connected in a Y configuration, being that there is only one
large high-voltage insulator protruding from each transformer. This suggests the other
side of each transformer's secondary winding is at or near ground potential, which could
only be true in a Y system. The building to the left is the powerhouse, where the
generators and turbines are housed. On the right, the sloping concrete wall is the
downstream face of the dam:
Step-up transfromer bank at Grand Coulee hydroelectric dam, Washington state, USA.
In the chapter on mixed-frequency signals, we explored the concept of harmonics in AC
systems: frequencies that are integer multiples of the fundamental source frequency.
With AC power systems where the source voltage waveform coming from an AC
generator (alternator) is supposed to be a single-frequency sine wave, undistorted, there
should be no harmonic content . . . ideally.
This would be true were it not for nonlinear components. Nonlinear components draw
current disproportionately with respect to the source voltage, causing non-sinusoidal
current waveforms. Examples of nonlinear components include gas-discharge lamps,
semiconductor power-control devices (diodes, transistors, SCRs, TRIACs), transformers
(primary winding magnetization current is usually non-sinusoidal due to the B/H
saturation curve of the core), and electric motors (again, when magnetic fields within the
motor's core operate near saturation levels). Even incandescent lamps generate slightly
nonsinusoidal currents, as the filament resistance changes throughout the cycle due to
rapid fluctuations in temperature. As we learned in the mixed-frequency chapter, any
distortion of an otherwise sine-wave shaped waveform constitutes the presence of
harmonic frequencies.
When the nonsinusoidal waveform in question is symmetrical above and below its
average centerline, the harmonic frequencies will be odd integer multiples of the
fundamental source frequency only, with no even integer multiples. (Figure below) Most
nonlinear loads produce current waveforms like this, and so even-numbered harmonics
(2nd, 4th, 6th, 8th, 10th, 12th, etc.) are absent or only minimally present in most AC
power systems.
La NY ND
Pure sine wave =
1°" harmonic only
Examples of symmetrical waveforms -- odd harmonics only.
Examples of nonsymmetrical waveforms with even harmonics present are shown for
reference in Figure below.
Examples of nonsymmetrical waveforms -- even harmonics present.
Even though half of the possible harmonic frequencies are eliminated by the typically
symmetrical distortion of nonlinear loads, the odd harmonics can still cause problems.
Some of these problems are general to all power systems, single-phase or otherwise.
Transformer overheating due to eddy current losses, for example, can occur in any AC
power system where there is significant harmonic content. However, there are some
problems caused by harmonic currents that are specific to polyphase power systems,
and it is these problems to which this section is specifically devoted.
It is helpful to be able to simulate nonlinear loads in SPICE so as to avoid a lot of complex
mathematics and obtain a more intuitive understanding of harmonic effects. First, we'll
begin our simulation with a very simple AC circuit: a single sine-wave voltage source
with a purely linear load and all associated resistances: (Figure below)
R,
2] line 3
VW
LQ
Risuice 1Q
1 1kQ Road
Visiss 120 V
0 0
SPICE circuit with single sine-wave source.
The Reource ANd Riine resistances in this circuit do more than just mimic the real world:
they also provide convenient shunt resistances for measuring currents in the SPICE
simulation: by reading voltage across a 1 Q resistance, you obtain a direct indication of
current through it, since E = IR.
A SPICE simulation of this circuit (SPICE listing: “linear load simulation”) with Fourier
analysis on the voltage measured across Rjj,~ should show us the harmonic content of
this circuit's line current. Being completely linear in nature, we should expect no
harmonics other than the 1st (fundamental) of 60 Hz, assuming a 60 Hz source. See
SPICE output “Fourier components of transient response v(2,3)” and Figure below.
linear load simulation
vsource 1 0 sin(0 120 60 0 Q)
rsource 121
rline 231
rload 3 0 1k
-options itl5=0
.tran 0.5m 30m 0 lu
.plot tran v(2,3)
. four 60 v(2,3)
.end
Fourier components of transient response v(2,3)
dc component = 4.028E-12
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.000E+01 1.198E-01 1.000000 -72.000 0.000
1.200E+02 5.793E-12 0.000000 51.122 123.122
3 1.800E+02 7.407E-12 0.000000 -34.624 37.376
4 2.400E+02 9.056E-12 0.000000 4.267 76.267
5 3.Q00E+02 1.651E-11 0.000000 -83.461 -11.461
6 3.600E+02 3.931E-11 0.000000 36.399 108.399
7 4.200E+02 2.338E-11 0.000000 -41.343 30.657
8 4.800E+02 4.716E-11 0.000000 53.324 125.324
9 5.400E+02 3.453E-11 0.000000 21.691 93.691
total harmonic distortion = 0.000000 percent
Relative amplitude
0 1 2 3 4 5
Harmonic number
Frequency domain plot of single frequency component. See SPICE listing: “linear load
simulation”.
A .plot Command appears in the SPICE netlist, and normally this would result in a sine-
wave graph output. In this case, however, I've purposely omitted the waveform display
for brevity's sake -- the .plot command is in the netlist simply to satisfy a quirk of
SPICE's Fourier transform function.
No discrete Fourier transform is perfect, and so we see very small harmonic currents
indicated (in the pico-amp range!) for all frequencies up to the 9th harmonic (in the
table ), which is as far as SPICE goes in performing Fourier analysis. We show 0.1198
amps (1.198E-01) for the “Fourier component” of the 1st harmonic, or the fundamental
frequency, which is our expected load current: about 120 mA, given a source voltage of
120 volts and a load resistance of 1 kQ.
Next, I'd like to simulate a nonlinear load so as to generate harmonic currents. This can
be done in two fundamentally different ways. One way is to design a load using
nonlinear components such as diodes or other semiconductor devices which are easy to
simulate with SPICE. Another is to add some AC current sources in parallel with the load
resistor. The latter method is often preferred by engineers for simulating harmonics,
since current sources of known value lend themselves better to mathematical network
analysis than components with highly complex response characteristics. Since we're
letting SPICE do all the math work, the complexity of a semiconductor component would
cause no trouble for us, but since current sources can be fine-tuned to produce any
arbitrary amount of current (a convenient feature), I'll choose the latter approach shown
in Figure below and SPICE listing: “Nonlinear load simulation”.
2 Riine 3 3
50mA
180 Hz
0 0 0
SPICE circuit: 60 Hz source with 3rd harmonic added.
Nonlinear load simulation
vsource 1 0 sin(O 120 60 0 Q)
rsource 12 1
rline 231
rload 3 0 1k
i3har 3 0 sin(O 50m 180 0 0)
-options itl5=0
.tran 0.5m 30m 0 lu
.plot tran v(2,3)
.four 60 v(2,3)
.end
In this circuit, we have a current source of 50 mA magnitude and a frequency of 180 Hz,
which is three times the source frequency of 60 Hz. Connected in parallel with the 1 kO
load resistor, its current will add with the resistor's to make a nonsinusoidal total line
current. I'll show the waveform plot in Figure below just so you can see the effects of this
3rd-harmonic current on the total current, which would ordinarily be a plain sine wave.
SPICE time-domain plot showing sum of 60 Hz source and 3rd harmonic of 180 Hz.
Fourier components of transient response v(2,3)
dc component = 1.349E-11
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.Q000E+01 1.198E-01 1.000000 -72.000 0.000
1.200E+02 1.609E-11 0.000000 67.570 139.570
3 1.800E+02 4.990E-02 0.416667 144.000 216.000
4 2.400E+02 1.074E-10 0.000000 -169.546 -97.546
5 3.Q00E+02 3.871E-11 0.000000 169.582 241.582
6 3.600E+02 5.736E-11 0.000000 140.845 212.845
7 4.200E+02 8.407E-11 0.000000 177.071 249.071
8 4.800E+02 1.329E-10 0.000000 156.772 228.772
9 5.400E+02 2.619E-10 0.000000 160.498 232.498
total harmonic distortion = 41.666663 percent
0.12 -
2041 for using 03
0.1
0.08
0.06
0.04
Relative amplitude
0.02
0 1 2 3 4 5
Harmonic number
SPICE Fourier plot showing 60 Hz source and 3rd harmonic of 180 Hz.
In the Fourier analysis, (See Figure above and “Fourier components of transient response
v(2,3)”) the mixed frequencies are unmixed and presented separately. Here we see the
same 0.1198 amps of 60 Hz (fundamental) current as we did in the first simulation, but
appearing in the 3rd harmonic row we see 49.9 mA: our 50 mA, 180 Hz current source at
work. Why don't we see the entire 50 mA through the line? Because that current source
is connected across the 1 kQ load resistor, so some of its current is shunted through the
load and never goes through the line back to the source. It's an inevitable consequence
of this type of simulation, where one part of the load is “normal” (a resistor) and the
other part is imitated by a current source.
If we were to add more current sources to the “load,” we would see further distortion of
the line current waveform from the ideal sine-wave shape, and each of those harmonic
currents would appear in the Fourier analysis breakdown. See Figure below and SPICE
listing: “Nonlinear load simulation”.
Nonlinear load: 1st, 3rd, Sth, 7th, and 9th
harmonics present
2 Riine 3 3 3 3 3
Nonlinear load: 1st, 3rd, 5th, 7th, and 9th harmonics present.
Nonlinear load simulation
vsource 1 0 sin(O 120 60 0 Q)
rsource 12 1
rline 231
rload 3 0 1k
i3har 3 0 sin(@ 50m 180 0 0)
id5har 3 0 sin(@ 50m 300 0 0)
i7har 3 0 sin(O0 50m 420 0 Q)
i9har 3 0 sin(@ 50m 540 0 0)
-options itl5=0
.tran 0.5m 30m 0 lu
.plot tran v(2,3)
.four 60 v(2,3)
.end
Fourier components of transient response v(2,3)
dc component = 6.299E-11
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.0Q00E+01 1.198E-01 1.000000 -72.000 0.000
1.200E+02 1.900E-09 0.000000 -93.908 -21.908
3 1.800E+02 4.990E-02 0.416667 144.000 216.000
4 2.400E+02 5.469E-09 0.000000 -116.873 -44.873
5 3.Q00E+02 4.990E-02 0.416667 0.000 72.000
6 3.600E+02 6.271E-09 0.000000 85.062 157.062
7 4.200E+02 4.990E-02 0.416666 -144.000 -72.000
8 4.800E+02 2.742E-09 0.000000 -38.781 33.219
9 5.400E+02 4.990E-02 0.416666 72.000 144.000
total harmonic distortion = 83.333296 percent
"22044 for" using 033
0.08
0.06
0.04
Relative amplitude
0.02
0 123 4 5 6 7 8 9
Harmonic number
Fourier analysis: “Fourier components of transient response v(2,3)”.
As you can see from the Fourier analysis, (Figure above) every harmonic current source is
equally represented in the line current, at 49.9 mA each. So far, this is just a single-
phase power system simulation. Things get more interesting when we make it a three-
phase simulation. Two Fourier analyses will be performed: one for the voltage across a
line resistor, and one for the voltage across the neutral resistor. As before, reading
voltages across fixed resistances of 1 Q each gives direct indications of current through
those resistors. See Figure below and SPICE listing “Y-Y source/load 4-wire system with
harmonics”.
ta
- i wt rs oe + — i i ls
Lia Shas ) 4) @) (a) La SPias () (® ® ey)
f Soma Soma Soma Soma f Soma Soma Soma Soma
1 3 1BOH= 3OOH= 420H= S40H= 1BOH: SOOH: 420H: S40H=
180H: SOOH: 420H= S40H=
Soma 50m4 50m4 soma
@ @©@ © ®
6 110 SR,
Pew . Tio
“A
SPICE circuit: analysis of “line current” and “neutral current”, Y-Y source/load 4-wire
system with harmonics.
Y-Y source/load 4-wire system with harmonics
*
* phasel voltage source and r (120 v /_ 0 deg)
vsourcel 1 @ sin(0 120 60 0 0)
rsourcel 12 1
*
* phase2 voltage source and r (120 v /_ 120 deg)
vsource2 3 @ sin(0 120 60 5.55555m 0)
rsource2 341
*
* phase3 voltage source and r (120 v /_ 240 deg)
vsource3 5 @ sin(O 120 60 11.1111m 0)
rsource3 5 6 1
*
* Line and neutral wire resistances
rlinel 281
rline2 491
rline3 6 10 1
rneutral 071
*
* phase 1 of load
rloadl 8 7 1k
i3harl 8 7 sin(0 50m 180 © 0)
i5harl 8 7 sin(@ 50m 300 0 0)
i7harl 8 7 sin(0 50m 420 © 0)
i9harl 8 7 sin(@ 50m 540 0 0)
*
* phase 2 of load
rload2 9 7 1k
i3har2 9 7 sin(O 50m 180 5.55555m 0)
idhar2 9 7 sin(O 50m 300 5.55555m 0)
i7har2 9 7 sin(O 50m 420 5.55555m 0)
5.55555m Q)
i9har2 9 7 sin(0 50m 540
*
* phase 3 of load
rload3 10 7 1k
i3har3 10 7 sin(® 50m 180 11.1111m 0)
id5har3 10 7 sin(O 50m 300 11.1111m 0)
i7har3 10 7 sin(® 50m 420 11.1111m 0)
i9har3 10 7 sin(O 50m 540 11.1111m 0)
*
* analysis stuff
-options itl5=0
-tran 0.5m 100m 12m lu
.plot tran v(2,8)
.four 60 v(2,8)
.plot tran v(0,7)
.four 60 v(0,7)
.end
Fourier analysis of line current:
Fourier components of transient response v(2,8)
dc component = -6.404E-12
harmonic frequency Fourier normalized phase
no (hz) component component (deg)
1 6.000E+01 1.198E-01 1.000000 0.000
1.200E+02 2.218E-10 0.000000 172.985
3 1.800E+02 4.975E-02 0.415423 0.000
4 2.400E+02 4.236E-10 0.000000 166.990
5 3.000E+02 4.990E-02 0.416667 0.000
6 3.600E+02 1.877E-10 0.000000 -147.146
7 4.200E+02 4.990E-02 0.416666 0.000
8 4.800E+02 2.784E-10 0.000000 -148.811
9 5.400E+02 4.975E-02 0.415422 0.000
total harmonic distortion = 83.209009 percent
0.12 T -
"22045for using 03] —
© 0.1
no]
= |
= 0.08
a
E
ow 0.06
Ee
0.04
o
© 002
0 12 3 4 5 6 7 8 9
Harmonic number
Fourier analysis of line current in balanced Y-Y system
normalized
phase (deg)
0.000
172.985
0.000
166.990
0.000
-147.146
0.000
-148.811
0.000
Fourier analysis of neutral current:
Fourier components of transient response v(0,7)
dc component = 1.819E-10
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.Q00E+01 4.337E-07 1.000000 60.018 0.000
2 1.200E+02 1.869E-10 0.000431 91.206 31.188
3 1.800E+02 1.493E-01 344147.7638 -180.000 -240.018
4 2.400E+02 1.257E-09 0.002898 -21.103 -81.121
5 3.Q00E+02 9.023E-07 2.080596 119.981 59.963
6 3.600E+02 3.396E-10 0.000783 15.882 -44.136
7 4.200E+02 1.264E-06 2.913955 59.993 -0.025
8 4.800E+02 5.975E-10 0.001378 35.584 -24.434
9 5.400E+02 1.493E-01 344147.4889 -179.999 -240.017
0.16
‘usin 33: =!
0.14
0.12
0.1
0.08
0.06
0.04
0.02
Relative amplitude
0 12 3 4 5 6 7 8 9
Harmonic number
Fourier analysis of neutral current shows other than no harmonics! Compare to line
current in Figure above
This is a balanced Y-Y power system, each phase identical to the single-phase AC system
simulated earlier. Consequently, it should come as no surprise that the Fourier analysis
for line current in one phase of the 3-phase system is nearly identical to the Fourier
analysis for line current in the single-phase system: a fundamental (60 Hz) line current
of 0.1198 amps, and odd harmonic currents of approximately 50 mA each. See Figure
above and Fourier analysis: “Fourier components of transient response v(2,8)”
What should be surprising here is the analysis for the neutral conductor's current, as
determined by the voltage drop across the Ryeutray resistor between SPICE nodes 0 and 7.
(Figure above) In a balanced 3-phase Y load, we would expect the neutral current to be
zero. Each phase current -- which by itself would go through the neutral wire back to the
supplying phase on the source Y -- should cancel each other in regard to the neutral
conductor because they're all the same magnitude and all shifted 120° apart. Ina
system with no harmonic currents, this is what happens, leaving zero current through
the neutral conductor. However, we cannot say the same for harmonic currents in the
same system.
Note that the fundamental frequency (60 Hz, or the 1st harmonic) current is virtually
absent from the neutral conductor. Our Fourier analysis shows only 0.4337 UA of 1st
harmonic when reading voltage across Ryeuytral. Fhe Same may be said about the 5th and
7th harmonics, both of those currents having negligible magnitude. In contrast, the 3rd
and 9th harmonics are strongly represented within the neutral conductor, with 149.3 mA
(1.493E-01 volts across 1 QO) each! This is very nearly 150 mA, or three times the current
sources' values, individually. With three sources per harmonic frequency in the load, it
appears our 3rd and 9th harmonic currents in each phase are adding to form the neutral
current. See Fourier analysis: “Fourier components of transient response v(0,7) ”
This is exactly what's happening, though it might not be apparent why this is so. The key
to understanding this is made clear in a time-domain graph of phase currents. Examine
this plot of balanced phase currents over time, with a phase sequence of 1-2-3. (Figure
below)
phase sequence:
I= 2-2-1 2=3- 1-2-3
1 2 3
TIME —>
Phase sequence 1-2-3-1-2-3-1-2-3 of equally spaced waves.
With the three fundamental waveforms equally shifted across the time axis of the graph,
it is easy to see how they would cancel each other to give a resultant current of zero in
the neutral conductor. Let's consider, though, what a 3rd harmonic waveform for phase 1
would look like superimposed on the graph in Figure below.
1 2 3
TIME —>
Third harmonic waveform for phase-1 superimposed on three-phase fundamental
waveforms.
Observe how this harmonic waveform has the same phase relationship to the 2nd and
3rd fundamental waveforms as it does with the 1st: in each positive half-cycle of any of
the fundamental waveforms, you will find exactly two positive half-cycles and one
negative half-cycle of the harmonic waveform. What this means is that the 3rd-harmonic
waveforms of three 120° phase-shifted fundamental-frequency waveforms are actually jn
phase with each other. The phase shift figure of 120° generally assumed in three-phase
AC systems applies only to the fundamental frequencies, not to their harmonic
multiples!
If we were to plot all three 3rd-harmonic waveforms on the same graph, we would see
them precisely overlap and appear as a single, unified waveform (shown in bold in
(Figure below)
TIME —>
Third harmonics for phases 1, 2, 3 all coincide when superimposed on the fundamental
three-phase waveforms.
For the more mathematically inclined, this principle may be expressed symbolically.
Suppose that A represents one waveform and B another, both at the same frequency,
but shifted 120° from each other in terms of phase. Let's call the 3rd harmonic of each
waveform A’ and B', respectively. The phase shift between A’ and B' is not 120° (that is
the phase shift between A and B), but 3 times that, because the A’ and B' waveforms
alternate three times as fast as A and B. The shift between waveforms is only accurately
expressed in terms of phase angle when the same angular velocity is assumed. When
relating waveforms of different frequency, the most accurate way to represent phase
shift is in terms of time; and the time-shift between A' and B' is equivalent to 120° ata
frequency three times lower, or 360° at the frequency of A’ and B'. A phase shift of 360°
is the same as a phase shift of 0°, which is to say no phase shift at all. Thus, A’ and B'
must be in phase with each other:
Phase sequence = A-B-C
Fundamental
3rd harmonic
This characteristic of the 3rd harmonic in a three-phase system also holds true for any
integer multiples of the 3rd harmonic. So, not only are the 3rd harmonic waveforms of
each fundamental waveform in phase with each other, but so are the 6th harmonics, the
9th harmonics, the 12th harmonics, the 15th harmonics, the 18th harmonics, the 21st
harmonics, and so on. Since only odd harmonics appear in systems where waveform
distortion is symmetrical about the centerline -- and most nonlinear loads create
symmetrical distortion -- even-numbered multiples of the 3rd harmonic (6th, 12th, 18th,
etc.) are generally not significant, leaving only the odd-numbered multiples (3rd, 9th,
15th, 21st, etc.) to significantly contribute to neutral currents.
In polyphase power systems with some number of phases other than three, this effect
occurs with harmonics of the same multiple. For instance, the harmonic currents that
add in the neutral conductor of a star-connected 4-phase system where the phase shift
between fundamental waveforms is 90° would be the 4th, 8th, 12th, 16th, 20th, and so
on.
Due to their abundance and significance in three-phase power systems, the 3rd
harmonic and its multiples have their own special name: triplen harmonics. All triplen
harmonics add with each other in the neutral conductor of a 4-wire Y-connected load. In
power systems containing substantial nonlinear loading, the triplen harmonic currents
may be of great enough magnitude to cause neutral conductors to overheat. This is very
problematic, as other safety concerns prohibit neutral conductors from having
overcurrent protection, and thus there is no provision for automatic interruption of these
high currents.
The following illustration shows how triplen harmonic currents created at the load add
within the neutral conductor. The symbol “w” is used to represent angular velocity, and
is mathematically equivalent to 2nf. So, “w” represents the fundamental frequency, “3W
" represents the 3rd harmonic, “5w” represents the 5th harmonic, and so on: (Figure
below)
Source Load
line
—
® 30 So 70 90
© 30 50 70 Iw
“Y-Y’Triplen source/load: Harmonic currents add in neutral conductor.
In an effort to mitigate these additive triplen currents, one might be tempted to remove
the neutral wire entirely. If there is no neutral wire in which triplen currents can flow
together, then they won't, right? Unfortunately, doing so just causes a different problem:
the load's “Y” center-point will no longer be at the same potential as the source's,
meaning that each phase of the load will receive a different voltage than what is
produced by the source. We'll re-run the last SPICE simulation without the 1 Q Rpeutrat
resistor and see what happens:
Y-Y source/load (no neutral) with harmonics
*
* phasel voltage source and r (120 v /_ 0 deg)
vsourcel 1 @ sin(O 120 60 0 0)
rsourcel 12 1
*
* phase2 voltage source and r (120 v /_ 120 deg)
vsource2 3 @ sin(0 120 60 5.55555m 0)
rsource2 3 41
*
* phase3 voltage source and r (120 v /_ 240 deg)
vsource3 5 @ sin(0 120 60 11.1111m 0)
rsource3 561
*
* Line resistances
rlinel 2 81
rline2 491
rline3 6 10 1
*
* phase 1 of load
rloadl 8 7 1k
i3harl 8 7 sin(0 50m 180 © 0)
id5harl 8 7 sin(0 50m 300 © 0)
i7harl 8 7 sin(0 50m 420 © 0)
i9harl 8 7 sin(0 50m 540 © 0)
*
* phase 2 of load
rload2 9 7 1k
i3har2 9 7 sin(0 50m 180 5.55555m
i5har2 9 7 sin(0 50m 300 5.55555m
i7har2 9 7 sin(0 50m 420 5.55555m
i9har2 9 7 sin(0 50m 540 5.55555m
*
oooo
* phase 3 of load
rload3 10 7 1k
i3har3 10 7 sin(® 50m 180 11.1111m 0)
id5har3 10 7 sin(® 50m 300 11.1111m 0)
i7har3 10 7 sin(® 50m 420 11.1111m 0)
i9har3 10 7 sin(® 50m 540 11.1111m 0)
*
* analysis stuff
-options itl5=0
.tran 0.5m 100m 12m lu
.plot tran v(2,8)
.four 60 v(2,8)
.plot tran v(0,7)
.four 60 v(0,7)
.plot tran v(8,7)
.four 60 v(8,7)
.end
Fourier analysis of line current:
Fourier components of transient response v(2,8)
dc component = 5.423E-11
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.Q0Q00E+01 1.198E-01 1.000000 0.000 0.000
1.200E+02 2.388E-10 0.000000 158.016 158.016
3 1.800E+02 3.136E-07 0.000003 -90.009 -90.009
4 2.400E+02 5.963E-11 0.000000 -111.510 -111.510
5 3.Q00E+02 4.990E-02 0.416665 0.000 0.000
6 3.600E+02 8.606E-11 0.000000 -124.565 -124.565
7 4.200E+02 4.990E-02 0.416668 0.000 0.000
8 4.800E+02 8.126E-11 0.000000 -159.638 - 159.638
9 5.400E+02 9.406E-07 0.000008 -90.005 -90.005
total harmonic distortion = 58.925539 percent
Fourier analysis of voltage between the two “Y” center-points:
Fourier components of transient response v(0,7)
dc component = 6.093E-08
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.0Q00E+01 1.453E-04 1.000000 60.018 0.000
1.200E+02 6.263E-08 0.000431 91.206 31.188
3 1.800E+02 5.QQ0E+01 344147.7879 -180.000 -240.018
4 2.400E+02 4.210E-07 0.002898 -21.103 -81.121
5 3.000E+02 3.023E-04 2.080596 119.981 59.963
6 3.600E+02 1.138E-07 0.000783 15.882 -44,.136
7 4.200E+02 4.234E-04 2.913955 59.993 -0.025
8 4.800E+02 2.001E-07 0.001378 35.584 -24.434
9 5.400E+02 5.QQ0E+01 344147.4728 -179.999 -240.017
total harmonic distortion = ******#****** percent
Fourier analysis of load phase voltage:
Fourier components of transient response v(8,7)
dc component = 6.070E-08
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.Q0Q0E+01 1.198E+02 1.000000 0.000 0.000
1.200E+02 6.231E-08 0.000000 90.473 90.473
3 1.800E+02 5.QQ0E+01 0.417500 -180.000 - 180.000
4 2.400E+02 4.278E-07 0.000000 -19.747 -19.747
5 3.000E+02 9.995E-02 0.000835 179.850 179.850
6 3.600E+02 1.023E-07 0.000000 13.485 13.485
7 4.200E+02 9.959E-02 0.000832 179.790 179.789
8 4.800E+02 1.991E-07 0.000000 35.462 35.462
9 5.400E+02 5.000E+01 0.417499 -179.999 -179.999
total harmonic distortion = 59.043467 percent
Strange things are happening, indeed. First, we see that the triplen harmonic currents
(3rd and 9th) all but disappear in the lines connecting load to source. The 5th and 7th
harmonic currents are present at their normal levels (approximately 50 mA), but the 3rd
and 9th harmonic currents are of negligible magnitude. Second, we see that there is
substantial harmonic voltage between the two “Y” center-points, between which the
neutral conductor used to connect. According to SPICE, there is 50 volts of both 3rd and
9th harmonic frequency between these two points, which is definitely not normal ina
linear (no harmonics), balanced Y system. Finally, the voltage as measured across one of
the load's phases (between nodes 8 and 7 in the SPICE analysis) likewise shows strong
triplen harmonic voltages of 50 volts each.
Figure below is a graphical summary of the aforementioned effects.
Source Load
line
Three-wire “Y-Y” (no neutral) system: Triplen voltages appear between “Y” centers.
Triplen voltages appear across load phases. Non-triplen currents appear in line
conductors.
In summary, removal of the neutral conductor leads to a “hot” center-point on the load
“Y", and also to harmonic load phase voltages of equal magnitude, all comprised of
triplen frequencies. In the previous simulation where we had a 4-wire, Y-connected
system, the undesirable effect from harmonics was excessive neutral current, but at least
each phase of the load received voltage nearly free of harmonics.
Since removing the neutral wire didn't seem to work in eliminating the problems caused
by harmonics, perhaps switching to a A configuration will. Let's try a A source instead of
a Y, keeping the load in its present Y configuration, and see what happens. The
measured parameters will be line current (voltage across Rjj,e, Nodes O and 8), load
phase voltage (nodes 8 and 7), and source phase current (voltage across Reource, NOdes 1
and 2). (Figure below)
Rew
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ta
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4 . IV
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ta
Delta-Y source/load with harmonics
Delta-Y source/load with harmonics
*
* phasel voltage source and r (120 v /_ 0 deg)
vsourcel 1 0 sin(0 207.846 60 0 0)
rsourcel 121
*
* phase2 voltage source and r (120 v /_ 120 deg)
vsource2 3 2 sin(0 207.846 60 5.55555m 0)
rsource2 3 4 1
*
* phase3 voltage source and r (120 v /_ 240 deg)
vsource3 5 4 sin(0 207.846 60 11.1111m 0)
rsource3 5 0 1
*
* line resistances
rlinel 0 81
rline2 291
rline3 4 10 1
*
* phase 1 of load
rloadl 8 7 1k
i3harl 8 7 sin(0 50m 180 9.72222m 0)
id5harl 8 7 sin(0 50m 300 9.72222m Q)
i7harl 8 7 sin(0 50m 420 9.72222m Q)
i9harl 8 7 sin(0 50m 540 9.72222m Q)
*
* phase 2 of load
rload2 9 7 1k
i3har2 9 7 sin(0 50m 180 15.2777m Q)
id5har2 9 7 sin(O0 50m 300 15.2777m Q)
i7har2 9 7 sin(0 50m 420 15.2777m 0)
0)
i9har2 9 7 sin(O0 50m 540 15.2777m
*
* phase 3 of load
rload3 10 7 1k
i3har3 10 7 sin(O 50m 180 4.16666m 0)
i5har3 10 7 sin(O0 50m 300 4.16666m 0)
i7har3 10 7 sin(0 50m 420 4.16666m 0)
i9har3 10 7 sin(O0 50m 540 4.16666m 0)
*
* analysis stuff
-options itl5=0
.tran 0.5m 100m 16m lu
-plot tran v(0,8) v(8,7) v(1,2)
.four 60 v(0,8) v(8,7) v(1,2)
.end
Note: the following paragraph is for those curious readers who follow every detail of my
SPICE netlists. If you just want to find out what happens in the circuit, skip this
paragraph! When simulating circuits having AC sources of differing frequency and
differing phase, the only way to do it in SPICE is to set up the sources with a delay time
or phase offset specified in seconds. Thus, the 0° source has these five specifying
figures: “(0 207.846 60 0 0)”, which means 0 volts DC offset, 207.846 volts peak
amplitude (120 times the square root of three, to ensure the load phase voltages remain
at 120 volts each), 60 Hz, 0 time delay, and 0 damping factor. The 120° phase-shifted
source has these figures: “(0 207.846 60 5.55555m 0)”, all the same as the first except
for the time delay factor of 5.55555 milliseconds, or 1/3 of the full period of 16.6667
milliseconds for a 60 Hz waveform. The 240° source must be time-delayed twice that
amount, equivalent to a fraction of 240/360 of 16.6667 milliseconds, or 11.1111
milliseconds. This is for the A-connected source. The Y-connected load, on the other
hand, requires a different set of time-delay figures for its harmonic current sources,
because the phase voltages in a Y load are not in phase with the phase voltages of aA
source. If A source voltages Vac, Vga, and Vcp are referenced at 0°, 120°, and 240°,
respectively, then “Y” load voltages Va, Vp, and Vc will have phase angles of -30°, 90°,
and 210°, respectively. This is an intrinsic property of all A-Y circuits and not a quirk of
SPICE. Therefore, when | specified the delay times for the harmonic sources, | had to set
them at 15.2777 milliseconds (-30°, or +330°), 4.16666 milliseconds (90°), and 9.72222
milliseconds (210°). One final note: when delaying AC sources in SPICE, they don't “turn
on” until their delay time has elapsed, which means any mathematical analysis up to
that point in time will be in error. Consequently, | set the .tran transient analysis line to
hold off analysis until 16 milliseconds after start, which gives all sources in the netlist
time to engage before any analysis takes place.
The result of this analysis is almost as disappointing as the last. (Figure below) Line
currents remain unchanged (the only substantial harmonic content being the 5th and
7th harmonics), and load phase voltages remain unchanged as well, with a full 50 volts
of triplen harmonic (3rd and 9th) frequencies across each load component. Source phase
current is a fraction of the line current, which should come as no surprise. Both 5th and
7th harmonics are represented there, with negligible triplen harmonics:
Fourier analysis of line current:
Fourier components of transient response v(0,8)
dc component = -6.850E-11
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.000E+01 1.198E-01 1.000000 150.000 0.000
1.200E+02 2.491E-11 0.000000 159.723 9.722
3 1.800E+02 1.506E-06 0.000013 0.005 -149.996
4 2.400E+02 2.033E-11 0.000000 52.772 -97.228
5 3.000E+02 4.994E-02 0.416682 30.002 -119.998
6 3.600E+02 1.234E-11 0.000000 57.802 -92.198
7 4.200E+02 4.993E-02 0.416644 -29.998 -179.998
8 4.800E+02 8.024E-11 0.000000 -174.200 -324.200
9 5.400E+02 4.518E-06 0.000038 -179.995 -329.995
total harmonic distortion = 58.925038 percent
Fourier analysis of load phase voltage:
Fourier components of transient response v(8,7)
dc component = 1.259E-08
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.QQ00E+01 1.198E+02 1.000000 150.000 0.000
2 1.200E+02 1.941E-07 0.000000 49.693 - 100.307
3 1.800E+02 5.QQ00E+01 0.417222 -89.998 -239.998
4 2.400E+02 1.519E-07 0.000000 66.397 -83.603
5 3.QQ00E+02 6.466E-02 0.000540 -151.112 -301.112
6 3.600E+02 2.433E-07 0.000000 68.162 -81.838
7 4.200E+02 6.931E-02 0.000578 148.548 -1.453
8 4.800E+02 2.398E-07 0.000000 -174.897 -324.897
9 5.400E+02 5.QQ0E+01 0.417221 90.006 -59.995
total harmonic distortion = 59.004109 percent
Fourier analysis of source phase current:
Fourier components of transient response v(1,2)
dc component = 3.564E-11
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.QQ00E+01 6.906E-02 1.000000 -0.181 0.000
1.200E+02 1.525E-11 0.000000 -156.674 - 156.493
3 1.800E+02 1.422E-06 0.000021 -179.996 -179.815
4 2.400E+02 2.949E-11 0.000000 -110.570 -110.390
5 3.Q00E+02 2.883E-02 0.417440 -179.996 -179.815
6 3.600E+02 2.324E-11 0.000000 -91.926 -91.745
7 4.200E+02 2.883E-02 0.417398 -179.994 -179.813
8 4.800E+02 4.140E-11 0.000000 -39.875 -39.694
9 5.400E+02 4.267E-06 0.000062 0.006 0.186
total harmonic distortion = 59.031969 percent
Source line Load
“A-Y” source/load: Triplen voltages appear across load phases. Non-triplen currents
appear in line conductors and in source phase windings.
Really, the only advantage of the A-Y configuration from the standpoint of harmonics is
that there is no longer a center-point at the load posing a shock hazard. Otherwise, the
load components receive the same harmonically-rich voltages and the lines see the
same currents as in a three-wire Y system.
If we were to reconfigure the system into a A-A arrangement, (Figure below) that should
guarantee that each load component receives non-harmonic voltage, since each load
phase would be directly connected in parallel with each source phase. The complete lack
of any neutral wires or “center points” in a A-A system prevents strange voltages or
additive currents from occurring. It would seem to be the ideal solution. Let's simulate
and observe, analyzing line current, load phase voltage, and source phase current. See
SPICE listing: “Delta-Delta source/load with harmonics”, “Fourier analysis: Fourier
components of transient response v(0,6)”, and “Fourier components of transient
response v(2,1)”.
Rew
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5) TS 12
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reoms 5oma 50mA 50m4a
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wi es 1
Delta-Delta source/load with harmonics.
Delta-Delta source/load with harmonics
*
* phasel voltage source and r (120 v /_ © deg)
vsourcel 1 @ sin(0 120 60 0 0)
rsourcel 12 1
*
* phase2 voltage source and r (120 v /_ 120 deg)
vsource2 3 2 sin(@ 120 60 5.55555m 0)
rsource2 341
*
* phase3 voltage source and r (120 v /_ 240 deg)
vsource3 5 4 sin(0 120 60 11.1111m 0)
rsource3 501
*
* Line resistances
rlinel 061
rline2 271
rline3 481
*
* phase 1 of load
rloadl 7 6 1k
i3harl 7 6 sin(0 50m 180
id5harl 7 6 sin(0 50m 300
i7harl 7 6 sin(0 50m 420
i9harl 7 6 sin(0 50m 540
*
* phase 2 of load
rload2 8 7 1k
i3har2 8 7 sin(0 50m 180
id5har2 8 7 sin(0 50m 300
i7har2 8 7 sin(0 50m 420
i9har2 8 7 sin(0 50m 540
*
* phase 3 of load
rload3 6 8 1k
6 8 sin(® 50m 180
i5har3 6 8 sin(@ 50m 300
i 6 8 sin(0 50m 420
6 8 sin(® 50m 540
* analysis stuff
oooo
-55555m
-55555m
-55555m
-55555m
uMnuu
11.1111m
11.1111m
11.1111m
11.1111m
oooo
oooo
-options itl5=0
.tran 0.5m 100m 16m lu
.plot tran v(0,6) v(7,6) v(2,1) i(3har1)
.four 60 v(0,6) v(7,6) v(2,1)
.end
Fourier analysis of line current:
Fourier components of transient response v(0,6)
dc component = -6.007E-11
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.QQ00E+01 2.070E-01 1.000000 150.000 0.000
1.200E+02 5.480E-11 0.000000 156.666 6.666
3 1.800E+02 6.257E-07 0.000003 89.990 -60.010
4 2.400E+02 4.911E-11 0.000000 8.187 -141.813
5 3.Q00E+02 8.626E-02 0.416664 -149.999 -300.000
6 3.600E+02 1.089E-10 0.000000 -31.997 -181.997
7 4.200E+02 8.626E-02 0.416669 150.001 0.001
8 4.800E+02 1.578E-10 0.000000 -63.940 -213.940
9 5.400E+02 1.877E-06 0.000009 89.987 -60.013
total harmonic distortion = 58.925538 percent
Fourier analysis of load phase voltage:
Fourier components of transient response v(7,6)
dc component = -5.680E-10
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.Q00E+01 1.195E+02 1.000000 0.000 0.000
1.200E+02 1.039E-09 0.000000 144.749 144.749
3 1.800E+02 1.251E-06 0.000000 89.974 89.974
4 2.400E+02 4.215E-10 0.000000 36.127 36.127
5 3.Q00E+02 1.992E-01 0.001667 -180.000 - 180.000
6 3.600E+02 2.499E-09 0.000000 -4.760 -4.760
7 4.200E+02 1.992E-01 0.001667 -180.000 -180.000
8 4.800E+02 2.951E-09 0.000000 -151.385 -151.385
9 5.400E+02 3.752E-06 0.000000 89.905 89.905
total harmonic distortion = 0.235702 percent
Fourier analysis of source phase current:
Fourier components of transient response v(2,1)
dc component = -1.923E-12
harmonic frequency Fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.Q0Q00E+01 1.194E-01 1.000000 179.940 0.000
1.200E+02 2.569E-11 0.000000 133.491 -46.449
3 1.800E+02 3.129E-07 0.000003 89.985 -89.955
4 2.400E+02 2.657E-11 0.000000 23.368 -156.571
5 3.000E+02 4.980E-02 0.416918 -180.000 -359.939
6 3.600E+02 4.595E-11 0.000000 -22.475 -202.415
7 4.200E+02 4.980E-02 0.416921 -180.000 -359.939
8 4.800E+02 7.385E-11 0.000000 -63.759 - 243.699
9 5.400E+02 9.385E-07 0.000008 89.991 -89.949
total harmonic distortion = 58.961298 percent
As predicted earlier, the load phase voltage is almost a pure sine-wave, with negligible
harmonic content, thanks to the direct connection with the source phases in a A-A
system. But what happened to the triplen harmonics? The 3rd and 9th harmonic
frequencies don't appear in any substantial amount in the line current, nor in the load
phase voltage, nor in the source phase current! We know that triplen currents exist,
because the 3rd and 9th harmonic current sources are intentionally placed in the phases
of the load, but where did those currents go?
Remember that the triplen harmonics of 120° phase-shifted fundamental frequencies are
in phase with each other. Note the directions that the arrows of the current sources
within the load phases are pointing, and think about what would happen if the 3rd and
9th harmonic sources were DC sources instead. What we would have is current
circulating within the loop formed by the A-connected phases. This is where the triplen
harmonic currents have gone: they stay within the A of the load, never reaching the line
conductors or the windings of the source. These results may be graphically summarized
as such in Figure below.
Load
OIO50 7090
—>
Source
line
A-A source/load: Load phases receive undistorted sinewave voltages. Triplen currents
are confined to circulate within load phases. Non-triplen currents apprear in line
conductors and in source phase windings.
This is a major benefit of the A-A system configuration: triplen harmonic currents remain
confined in whatever set of components create them, and do not “spread” to other parts
of the system.
REVIEW:
Nonlinear components are those that draw a non-sinusoidal (non-sine-wave) current
waveform when energized by a sinusoidal (sine-wave) voltage. Since any distortion
of an originally pure sine-wave constitutes harmonic frequencies, we can say that
nonlinear components generate harmonic currents.
When the sine-wave distortion is symmetrical above and below the average
centerline of the waveform, the only harmonics present will be odd-numbered, not
even-numbered.
The 3rd harmonic, and integer multiples of it (6th, 9th, 12th, 15th) are known as
triplen harmonics. They are in phase with each other, despite the fact that their
respective fundamental waveforms are 120° out of phase with each other.
In a 4-wire Y-Y system, triplen harmonic currents add within the neutral conductor.
Triplen harmonic currents in a A-connected set of components circulate within the
loop formed by the A.
Harmonic phase sequences
In the last section, we saw how the 3rd harmonic and all of its integer multiples
(collectively called triplen harmonics) generated by 120° phase-shifted fundamental
waveforms are actually in phase with each other. In a 60 Hz three-phase power system,
where phases A, B, and C are 120° apart, the third-harmonic multiples of those
frequencies (180 Hz) fall perfectly into phase with each other. This can be thought of in
graphical terms, (Figure below) and/or in mathematical terms:
A B c
TIME —>
Harmonic currents of Phases A, B, C all coincide, that is, no rotation.
Phase sequence = A-B-C
Fundamental
3rd harmonic
If we extend the mathematical table to include higher odd-numbered harmonics, we will
notice an interesting pattern develop with regard to the rotation or sequence of the
harmonic frequencies:
Fundamental
3rd harmonic
5th harmonic
7th harmonic
9th harmonic
Harmonics such as the 7th, which “rotate” with the same sequence as the fundamental,
are called positive sequence. Harmonics such as the 5th, which “rotate” in the opposite
sequence as the fundamental, are called negative sequence. Triplen harmonics (3rd and
9th shown in this table) which don't “rotate” at all because they're in phase with each
other, are called zero sequence.
This pattern of positive-zero-negative-positive continues indefinitely for all odd-
numbered harmonics, lending itself to expression in a table like this:
Rotation sequences according
to harmonic number
19th} ~— Rotates with fundamental
|o [3rd {9th [15th] 21st} ~— Does not rotate
| - [5th [1 1th| 17th] 23rd —~— Rotates against fundamental
Sequence especially matters when we're dealing with AC motors, since the mechanical
rotation of the rotor depends on the torque produced by the sequential “rotation” of the
applied 3-phase power. Positive-sequence frequencies work to push the rotor in the
proper direction, whereas negative-sequence frequencies actually work against the
direction of the rotor's rotation. Zero-sequence frequencies neither contribute to nor
detract from the rotor's torque. An excess of negative-sequence harmonics (5th, 11th,
17th, and/or 23rd) in the power supplied to a three-phase AC motor will result in a
degradation of performance and possible overheating. Since the higher-order harmonics
tend to be attenuated more by system inductances and magnetic core losses, and
generally originate with less amplitude anyway, the primary harmonic of concern is the
5th, which is 300 Hz in 60 Hz power systems and 250 Hz in 50 Hz power systems.
Contributors
Contributors to this chapter are listed in chronological order of their contributions, from
most recent to first. See Appendix 2 (Contributor List) for dates and contact information.
Ed Beroset (May 6, 2002): Suggested better ways to illustrate the meaning of the
prefix “poly-”.
Jason Starck (June 2000): HTML document formatting, which led to a much better-
looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt, under the terms
and conditions of the Design Science License.
—| [+4]
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 11
POWER FACTOR
Power in resistive and reactive AC circuits
e Calculating power factor
e Practical power factor correction
e Contributors
Power in resistive and reactive AC
circuits
Consider a circuit for a single-phase AC power system, where
a 120 volt, 60 Hz AC voltage source is delivering power toa
resistive load: (Figure below)
120 V
60 Hz (“) 60 2
Ac source drives a purely resistive load.
Zp =60+j0Q or 60220
E
[=
Z
In this example, the current to the load would be 2 amps,
RMS. The power dissipated at the load would be 240 watts.
Because this load is purely resistive (no reactance), the
current is in phase with the voltage, and calculations look
similar to that in an equivalent DC circuit. If we were to plot
the voltage, current, and power waveforms for this circuit, it
would look like Figure below.
Time -—>
Current is in phase with voltage in a resistive circuit.
Note that the waveform for power is always positive, never
negative for this resistive circuit. This means that power is
always being dissipated by the resistive load, and never
returned to the source as it is with reactive loads. If the
source were a mechanical generator, it would take 240 watts
worth of mechanical energy (about 1/3 horsepower) to turn
the shaft.
Also note that the waveform for power is not at the same
frequency as the voltage or current! Rather, its frequency is
double that of either the voltage or current waveforms. This
different frequency prohibits our expression of power in an
AC circuit using the same complex (rectangular or polar)
notation as used for voltage, current, and impedance,
because this form of mathematical symbolism implies
unchanging phase relationships. When frequencies are not
the same, phase relationships constantly change.
As strange as it may seem, the best way to proceed with AC
power calculations is to use sca/ar notation, and to handle
any relevant phase relationships with trigonometry.
For comparison, let's consider a simple AC circuit with a
purely reactive load in Figure below.
120 V
60 Hz 160 mH
AC circuit with a purely reactive (inductive) load.
X, = 60.319 Q
Z, =0+j60.319Q or 60.3192 290°
l= —
Z
120 V
60.319 Q
1= L989 A
Power is not dissipated in a purely reactive load. Though it Is
alternately absorbed from and returned to the source.
Note that the power alternates equally between cycles of
positive and negative. (Figure above) This means that power
is being alternately absorbed from and returned to the
source. If the source were a mechanical generator, it would
take (practically) no net mechanical energy to turn the
shaft, because no power would be used by the load. The
generator shaft would be easy to spin, and the inductor
would not become warm as a resistor would.
Now, let's consider an AC circuit with a load consisting of
both inductance and resistance in Figure below.
Load
120 V
60 Hz
AC circuit with both reactance and resistance.
X, = 60.319 Q
Z, =0+j60.319Q or 60.3192 290°
Zp=604+j0Q2 or 60220°
Zrota = 60 + j60.319Q or 85.078Q 2 45.152°
E
l= —
Zz
120 V
85.078 Q
1=1410A
At a frequency of 60 Hz, the 160 millihenrys of inductance
gives us 60.319 Q of inductive reactance. This reactance
combines with the 60 QO of resistance to form a total load
impedance of 60 + j60.319 O, or 85.078 O Z 45.152°. If
we're not concerned with phase angles (which we're not at
this point), we may calculate current in the circuit by taking
the polar magnitude of the voltage source (120 volts) and
dividing it by the polar magnitude of the impedance (85.078
Q). With a power supply voltage of 120 volts RMS, our load
current is 1.410 amps. This is the figure an RMS ammeter
would indicate if connected in series with the resistor and
inductor.
We already know that reactive components dissipate zero
power, as they equally absorb power from, and return power
to, the rest of the circuit. Therefore, any inductive reactance
in this load will likewise dissipate zero power. The only thing
left to dissipate power here is the resistive portion of the
load impedance. If we look at the waveform plot of voltage,
current, and total power for this circuit, we see how this
combination works in Figure below.
A combined resistive/reactive circuit dissipates more power
than it returns to the source. The reactance dissipates no
power; though, the resistor does.
As with any reactive circuit, the power alternates between
positive and negative instantaneous values over time. In a
purely reactive circuit that alternation between positive and
negative power is equally divided, resulting in a net power
dissipation of zero. However, in circuits with mixed
resistance and reactance like this one, the power waveform
will still alternate between positive and negative, but the
amount of positive power will exceed the amount of
negative power. In other words, the combined
inductive/resistive load will consume more power than it
returns back to the source.
Looking at the waveform plot for power, it should be evident
that the wave spends more time on the positive side of the
center line than on the negative, indicating that there is
more power absorbed by the load than there is returned to
the circuit. What little returning of power that occurs is due
to the reactance; the imbalance of positive versus negative
power is due to the resistance as it dissipates energy outside
of the circuit (usually in the form of heat). If the source were
a mechanical generator, the amount of mechanical energy
needed to turn the shaft would be the amount of power
averaged between the positive and negative power cycles.
Mathematically representing power in an AC circuit is a
challenge, because the power wave isn't at the same
frequency as voltage or current. Furthermore, the phase
angle for power means something quite different from the
phase angle for either voltage or current. Whereas the angle
for voltage or current represents a relative shift in timing
between two waves, the phase angle for power represents a
ratio between power dissipated and power returned.
Because of this way in which AC power differs from AC
voltage or current, it is actually easier to arrive at figures for
power by calculating with sca/ar quantities of voltage,
Current, resistance, and reactance than it is to try to derive it
from vector, or complex quantities of voltage, current, and
impedance that we've worked with so far.
¢ REVIEW:
e In a purely resistive circuit, all circuit power is dissipated
by the resistor(s). Voltage and current are in phase with
each other.
In a purely reactive circuit, no circuit power is dissipated
by the load(s). Rather, power is alternately absorbed
from and returned to the AC source. Voltage and current
are 90° out of phase with each other.
e In a circuit consisting of resistance and reactance mixed,
there will be more power dissipated by the load(s) than
returned, but some power will definitely be dissipated
and some will merely be absorbed and returned. Voltage
and current in such a circuit will be out of phase by a
value somewhere between 0° and 90°.
We know that reactive loads such as inductors and
Capacitors dissipate zero power, yet the fact that they drop
voltage and draw current gives the deceptive impression
that they actually do dissipate power. This “phantom power”
is called reactive power, and it is measured in a unit called
Volt-Amps-Reactive (VAR), rather than watts. The
mathematical symbol for reactive power is (unfortunately)
the capital letter Q. The actual amount of power being used,
or dissipated, in a circuit is called true power, and it is
measured in watts (symbolized by the capital letter P, as
always). The combination of reactive power and true power
is called apparent power, and it is the product of a circuit's
voltage and current, without reference to phase angle.
Apparent power is measured in the unit of Vo/t-Amps (VA)
and is symbolized by the capital letter S.
As arule, true power is a function of a circuit's dissipative
elements, usually resistances (R). Reactive power is a
function of a circuit's reactance (X). Apparent power is a
function of a circuit's total impedance (Z). Since we're
dealing with scalar quantities for power calculation, any
complex starting quantities such as voltage, current, and
impedance must be represented by their po/ar magnitudes,
not by real or imaginary rectangular components. For
instance, if I'm calculating true power from current and
resistance, | must use the polar magnitude for current, and
not merely the “real” or “imaginary” portion of the current. If
I'm calculating apparent power from voltage and
impedance, both of these formerly complex quantities must
be reduced to their polar magnitudes for the scalar
arithmetic.
There are several power equations relating the three types
of power to resistance, reactance, and impedance (all using
scalar quantities):
5
P = true power =TR BS
Measured in units of Watts
Q=reactive power Q=I1X Q= ~
Measured in units of Volt-Amps-Reactive (VAR)
S=apparentpower S=IZ S= =. S=l1E
Measured in units of Volt-Amps (VA)
Please note that there are two equations each for the
calculation of true and reactive power. There are three
equations available for the calculation of apparent power,
P=IE being useful on/y for that purpose. Examine the
following circuits and see how these three types of power
interrelate for: a purely resistive load in Figure below, a
purely reactive load in Figure below, and a resistive/reactive
load in Figure below.
Resistive load only:
1=2A
120 V
60 Hz WY) reactance
P = true power =1’R = 240 W
Q =reactive power = 1X =O VAR
S = apparent power = 1'Z = 240 VA
True power, reactive power, and apparent power for a purely
resistive load.
Reactive load only:
1= 1.989 A
no
120 V resistance 7
160 mH
60 Hz
X, = 60.3192
P = true power = 1’R =0 W
Q = reactive power = 1X = 238.73 VAR
S = apparent power = °Z = 238.73 VA
True power, reactive power, and apparent power for a purely
reactive load.
Resistive/reactive load:
Load
1=1410A
=; 160 mH
X, = 60.319 Q
120 V
60 Hz
60 Q
P = true power = I°R = 119.365 W
Q = reactive power = IX = 119.998 VAR
S = apparent power = 1°Z = 169.256 VA
True power, reactive power, and apparent power for a
resistive/reactive load.
These three types of power -- true, reactive, and apparent --
relate to one another in trigonometric form. We call this the
power triangle: (Figure below).
The "Power Triangle"
Apparent power (S)
measured in VA Reactive power ©)
measured in VA
Impedance
phase angle
True power (P)
measured in Watts
Power triangle relating appearant power to true power and
reactive power.
Using the laws of trigonometry, we can solve for the length
of any side (amount of any type of power), given the lengths
of the other two sides, or the length of one side and an
angle.
e REVIEW:
e Power dissipated by a load is referred to as true power.
True power is symbolized by the letter P and is measured
in the unit of Watts (W).
e Power merely absorbed and returned in load due to its
reactive properties is referred to as reactive power.
Reactive power is symbolized by the letter Q and is
measured in the unit of Volt-Amps-Reactive (VAR).
Total power in an AC circuit, both dissipated and
absorbed/returned is referred to as apparent power.
Apparent power is symbolized by the letter S and is
measured in the unit of Volt-Amps (VA).
e These three types of power are trigonometrically related
to one another. In a right triangle, P = adjacent length,
Q = opposite length, and S = hypotenuse length. The
opposite angle is equal to the circuit's impedance (Z)
phase angle.
Calculating power factor
As was mentioned before, the angle of this “power triangle”
graphically indicates the ratio between the amount of
dissipated (or consumed) power and the amount of
absorbed/returned power. It also happens to be the same
angle as that of the circuit's impedance in polar form. When
expressed as a fraction, this ratio between true power and
apparent power is called the power factor for this circuit.
Because true power and apparent power form the adjacent
and hypotenuse sides of a right triangle, respectively, the
power factor ratio is also equal to the cosine of that phase
angle. Using values from the last example circuit:
Power factor = __True power
Apparent power
Power factor = Pee
169.256 VA
Power factor = 0.705
cos 45.152° = 0.705
It should be noted that power factor, like all ratio
measurements, iS a unitless quantity.
For the purely resistive circuit, the power factor is 1
(perfect), because the reactive power equals zero. Here, the
power triangle would look like a horizontal line, because the
opposite (reactive power) side would have zero length.
For the purely inductive circuit, the power factor is zero,
because true power equals zero. Here, the power triangle
would look like a vertical line, because the adjacent (true
power) side would have zero length.
The same could be said for a purely capacitive circuit. If
there are no dissipative (resistive) components in the circuit,
then the true power must be equal to zero, making any
power in the circuit purely reactive. The power triangle for a
purely capacitive circuit would again be a vertical line
(pointing down instead of up as it was for the purely
inductive circuit).
Power factor can be an important aspect to consider in an
AC circuit, because any power factor less than 1 means that
the circuit's wiring has to carry more current than what
would be necessary with zero reactance in the circuit to
deliver the same amount of (true) power to the resistive
load. If our last example circuit had been purely resistive, we
would have been able to deliver a full 169.256 watts to the
load with the same 1.410 amps of current, rather than the
mere 119.365 watts that it is presently dissipating with that
same current quantity. The poor power factor makes for an
inefficient power delivery system.
Poor power factor can be corrected, paradoxically, by adding
another load to the circuit drawing an equal and opposite
amount of reactive power, to cancel out the effects of the
load's inductive reactance. Inductive reactance can only be
canceled by capacitive reactance, so we have to add a
capacitor in parallel to our example circuit as the additional
load. The effect of these two opposing reactances in parallel
is to bring the circuit's total impedance equal to its total
resistance (to make the impedance phase angle equal, or at
least closer, to zero).
Since we know that the (uncorrected) reactive power is
119.998 VAR (inductive), we need to calculate the correct
Capacitor size to produce the same quantity of (capacitive)
reactive power. Since this capacitor will be directly in
parallel with the source (of Known voltage), we'll use the
power formula which starts from voltage and reactance:
E-
a
... solving forX...
x. E
2nfC
20 Vy" ;
meek ... Solving forC...
119.998 VAR
C=
X = 120.002 2 2nfX
7 l
2n(60 Hz)( 120.002 Q)
C = 22.105 pF
Let's use a rounded capacitor value of 22 uF and see what
happens to our circuit: (Figure below)
lor = 994.716 mA Load
l.= Tioag = 141 A
995.257
mA | = 160 mH
load
120 V X, = 60.319 Q
60 Hz
60 Q
Parallel capacitor corrects lagging power factor of inductive
load. V2 and node numbers: 0, 1, 2, and 3 are SPICE related,
and may be ignored for the moment.
LZrotal = Ze // (Z, -- ZR)
Ziota = (120.57 Q Z -90°) // (60.319 Q 290° -- 60.2 Z 0°)
Zrota = 120.64 - j573.58m Q or 120.642 20.2724°
P = true power =1°R = 119.365 W
S = apparent power = 1°Z = 119.366 VA
The power factor for the circuit, overall, has been
substantially improved. The main current has been
decreased from 1.41 amps to 994.7 milliamps, while the
power dissipated at the load resistor remains unchanged at
119.365 watts. The power factor is much closer to being 1:
True power
Power factor =} ——*———__
Apparent power
Power factor = 119.365 W_
119.366 VA
Power factor = 0.9999887
Impedance (polar) angle = 0.272°
cos 0.272° = 0.9999887
Since the impedance angle is still a positive number, we
know that the circuit, overall, is still more inductive than it is
Capacitive. If our power factor correction efforts had been
perfectly on-target, we would have arrived at an impedance
angle of exactly zero, or purely resistive. If we had added too
large of a capacitor in parallel, we would have ended up with
an impedance angle that was negative, indicating that the
circuit was more capacitive than inductive.
A SPICE simulation of the circuit of (Figure above) shows
total voltage and total current are nearly in phase. The
SPICE circuit file has a zero volt voltage-source (V2) in series
with the capacitor so that the capacitor current may be
measured. The start time of 200 msec ( instead of 0) in the
transient analysis statement allows the DC conditions to
stabilize before collecting data. See SPICE listing “pf.cir
power factor”.
pf.cir power factor
V1 10 sin(@ 170 60)
C1 13 22uF
v2 0 0
L1 1 2 160mH
R1 2 0 60
# resolution stop start
.tran 1m 200m 160m
.end
The Nutmeg plot of the various currents with respect to the
applied voltage V;o¢4) is shown in (Figure below). The
reference iS Viota, to which all other measurements are
compared. This is because the applied voltage, Viota),
appears across the parallel branches of the circuit. There is
no single current common to all components. We can
compare those currents to Viota)-
~~
L1#branch)
ee
Units I{v2#branch) = ¥(1) 100 I
Zero phase angle due to in-phase Vio44; aNd Iigtas - The
lagging I, with respect to Vio; 's corrected by a leading Ic.
Note that the total current (l:¢q)) is in phase with the applied
voltage (Viota), indicating a phase angle of near zero. This is
no coincidence. Note that the lagging current, |, of the
inductor would have caused the total current to have a
lagging phase somewhere between (lita) and |,. However,
the leading capacitor current, Ic, compensates for the
lagging inductor current. The result is a total current phase-
angle somewhere between the inductor and capacitor
currents. Moreover, that total current (lio¢a;) was forced to be
in-phase with the total applied voltage (V;,;4)), by the
calculation of an appropriate capacitor value.
Since the total voltage and current are in phase, the product
of these two waveforms, power, will always be positive
throughout a 60 Hz cycle, real power as in Figure above. Had
the phase-angle not been corrected to zero (PF=1), the
product would have been negative where positive portions
of one waveform overlapped negative portions of the other
as in Figure above. Negative power is fed back to the
generator. It cannot be sold; though, it does waste power in
the resistance of electric lines between load and generator.
The parallel capacitor corrects this problem.
Note that reduction of line losses applies to the lines from
the generator to the point where the power factor correction
Capacitor is applied. In other words, there is still circulating
current between the capacitor and the inductive load. This is
not normally a problem because the power factor correction
iS applied close to the offending load, like an induction
motor.
It should be noted that too much capacitance in an AC
circuit will result in a low power factor just as well as too
much inductance. You must be careful not to over-correct
when adding capacitance to an AC circuit. You must also be
very careful to use the proper capacitors for the job (rated
adequately for power system voltages and the occasional
voltage spike from lightning strikes, for continuous AC
service, and capable of handling the expected levels of
current).
If a circuit is predominantly inductive, we say that its power
factor is Jagging (because the current wave for the circuit
lags behind the applied voltage wave). Conversely, if a
circuit is predominantly capacitive, we say that its power
factor is /eading. Thus, our example circuit started out with a
power factor of 0.705 lagging, and was corrected to a power
factor of 0.999 lagging.
e REVIEW:
e Poor power factor in an AC circuit may be “corrected”, or
re-established at a value close to 1, by adding a parallel
reactance opposite the effect of the load's reactance. If
the load's reactance is inductive in nature (which is
almost always will be), parallel capacitance is what is
needed to correct poor power factor.
Practical power factor correction
When the need arises to correct for poor power factor in an
AC power system, you probably won't have the luxury of
knowing the load's exact inductance in henrys to use for
your calculations. You may be fortunate enough to have an
instrument called a power factor meter to tell you what the
power factor is (a number between 0 and 1), and the
apparent power (which can be figured by taking a voltmeter
reading in volts and multiplying by an ammeter reading in
amps). In less favorable circumstances you may have to use
an oscilloscope to compare voltage and current waveforms,
measuring phase shift in degrees and calculating power
factor by the cosine of that phase shift.
Most likely, you will have access to a wattmeter for
measuring true power, whose reading you can compare
against a calculation of apparent power (from multiplying
total voltage and total current measurements). From the
values of true and apparent power, you can determine
reactive power and power factor. Let's do an example
problem to see how this works: (Figure below)
wattmeter ammeter
PP}
240 V
RMS Load
60 Hz
Wattmeter reading = 1.5 kW
Ammeter reading = 9.615 A RMS
Wattmeter reads true power; product of voltmeter and
ammeter readings yields appearant power.
First, we need to calculate the apparent power in kVA. We
can do this by multiplying load voltage by load current:
S=l1E
S = (9.615 A)(240 V)
S =2.308kVA
As we can see, 2.308 kVA is a much larger figure than 1.5
kW, which tells us that the power factor in this circuit is
rather poor (substantially less than 1). Now, we figure the
power factor of this load by dividing the true power by the
apparent power:
P
Power factor = —
L5kW
Power factor = ————————
2.308 kVA
Power factor= 0.65
Using this value for power factor, we can draw a power
triangle, and from that determine the reactive power of this
load: (Figure below)
Apparent power (S)
2.308 kVA
Reactive power (Q)
22?
True power (P)
1.5 kW
Reactive power may be calculated from true power and
appearant power.
To determine the unknown (reactive power) triangle
quantity, we use the Pythagorean Theorem “backwards,”
given the length of the hypotenuse (apparent power) and
the length of the adjacent side (true power):
Reactive power = (Apparent power)’ - (True power)*
Q=1.754kVAR
If this load is an electric motor, or most any other industrial
AC load, it will have a lagging (inductive) power factor,
which means that we'll have to correct for it with a capacitor
of appropriate size, wired in parallel. Now that we know the
amount of reactive power (1.754 kVAR), we can calculate
the size of capacitor needed to counteract its effects:
E
Qe
... solving forX...
x. E .
Q <= —_—
2mfC
JAN
oe ... Solving forC...
1.754 kVAR
C= l
X = 32.8450 OnEX,
- l
2m(60 Hz)(32.845 Q)
C = 80.761 pF
Rounding this answer off to 80 UF, we can place that size of
Capacitor in the circuit and calculate the results: (Figure
below)
wattmeter ammeter
240 V
RMS
60 Hz
Parallel capacitor corrects lagging (inductive) load.
An 80 uF capacitor will have a capacitive reactance of
33.157 QO, giving a current of 7.238 amps, anda
corresponding reactive power of 1.737 kVAR (for the
Capacitor only). Since the capacitor's current is 180° out of
phase from the the load's inductive contribution to current
draw, the capacitor's reactive power will directly subtract
from the load's reactive power, resulting in:
Inductive kV AR - Capacitive kVAR = Total kVAR
1.754 kKVAR - 1.737 KVAR = 16.519 VAR
This correction, of course, will not change the amount of true
power consumed by the load, but it will result in a
substantial reduction of apparent power, and of the total
current drawn from the 240 Volt source: (Figure below)
Power triangle for uncorrected (original) circuit
Apparent power (S)
2.308 kVA Reactive power (Q)
1L.754kVAR
(inductive)
True power (P)
LIkW
1.737kVAR
(capacitive)
Power triangle after adding capacitor
Apparent power (S) Reactive power (Q)
16.519 VAR
True power (P)
LIkW
Power triangle before and after capacitor correction.
The new apparent power can be found from the true and
new reactive power values, using the standard form of the
Pythagorean Theorem:
Apparent power = (Reactive power)” + (True power)°
Apparent power = 1.50009kVA
This gives a corrected power factor of (1.5kW / 1.5009 kVA),
or 0.99994, and a new total current of (1.50009 kVA / 240
Volts), or 6.25 amps, a substantial improvement over the
uncorrected value of 9.615 amps! This lower total current
will translate to less heat losses in the circuit wiring,
meaning greater system efficiency (less power wasted).
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4]
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 12
AC METERING CIRCUITS
AC voltmeters and ammeters
Frequency and phase measurement
Power measurement
Power quality measurement
AC bridge circuits
AC instrumentation transducers
Contributors
Bibliography
AC voltmeters and ammeters
AC electromechanical meter movements come in two basic
arrangements: those based on DC movement designs, and
those engineered specifically for AC use. Permanent-magnet
moving coil (PMMC) meter movements will not work
correctly if directly connected to alternating current,
because the direction of needle movement will change with
each half-cycle of the AC. (Figure below) Permanent-magnet
meter movements, like permanent-magnet motors, are
devices whose motion depends on the polarity of the
applied voltage (or, you can think of it in terms of the
direction of the current).
magnet
N\
\
wire coil
Passing AC through this D'Arsonval meter movement causes
useless flutter of the needle.
In order to use a DC-style meter movement such as the
D'Arsonval design, the alternating current must be rectified
into DC. This is most easily accomplished through the use of
devices called diodes. We saw diodes used in an example
circuit demonstrating the creation of harmonic frequencies
from a distorted (or rectified) sine wave. Without going into
elaborate detail over how and why diodes work as they do,
just remember that they each act like a one-way valve for
electrons to flow: acting as a conductor for one polarity and
an insulator for another. Oddly enough, the arrowhead in
each diode symbol points against the permitted direction of
electron flow rather than with it as one might expect.
Arranged in a bridge, four diodes will serve to steer AC
through the meter movement in a constant direction
throughout all portions of the AC cycle: (Figure below)
Meter movement needle
will always be driven in
the proper direction.
.
Bridge
rectifier
AC
source
>
Passing AC through this Rectified AC meter movement will
drive it in one direction.
Another strategy for a practical AC meter movement is to
redesign the movement without the inherent polarity
sensitivity of the DC types. This means avoiding the use of
permanent magnets. Probably the simplest design is to use
a nonmagnetized iron vane to move the needle against
spring tension, the vane being attracted toward a stationary
coil of wire energized by the AC quantity to be measured as
in Figure below.
wire coil
_iron vane
po
lron-vane electromechanical meter movement.
Electrostatic attraction between two metal plates separated
by an air gap is an alternative mechanism for generating a
needle-moving force proportional to applied voltage. This
works just as well for AC as it does for DC, or should | say,
just as poorly! The forces involved are very small, much
smaller than the magnetic attraction between an energized
coil and an iron vane, and as such these “electrostatic”
meter movements tend to be fragile and easily disturbed by
physical movement. But, for some high-voltage AC
applications, the electrostatic movement is an elegant
technology. If nothing else, this technology possesses the
advantage of extremely high input impedance, meaning
that no current need be drawn from the circuit under test.
Also, electrostatic meter movements are capable of
measuring very high voltages without need for range
resistors or other, external apparatus.
When a sensitive meter movement needs to be re-ranged to
function as an AC voltmeter, series-connected “multiplier”
resistors and/or resistive voltage dividers may be employed
just as in DC meter design: (Figure below)
AC voltmeter
AC voltmeter
Sensitive
Voltage Sensitive faa meter movement
to be meter movement measured
measured
R
multiplier
R
multiplier
(a) (0)
Multiplier resistor (a) or resistive divider (b) scales the range
of the basic meter movement.
Capacitors may be used instead of resistors, though, to
make voltmeter divider circuits. This strategy has the
advantage of being non-dissipative (no true power
consumed and no heat produced): (Figure below)
Sensitive
meter movement
Rou tiplier
Voltage
to be
measured
AC voltmeter with capacitive divider.
If the meter movement is electrostatic, and thus inherently
Capacitive in nature, a single “multiplier” capacitor may be
connected in series to give it a greater voltage measuring
range, just as a series-connected multiplier resistor gives a
moving-coil (inherently resistive) meter movement a greater
voltage range: (Figure below)
Electrostatic
meter movement
C
multiplier
Voltage
to be
measured
An electrostatic meter movement may use a Capacitive
multiplier to multiply the scale of the basic meter
movement...
The Cathode Ray Tube (CRT) mentioned in the DC metering
chapter is ideally suited for measuring AC voltages,
especially if the electron beam is swept side-to-side across
the screen of the tube while the measured AC voltage drives
the beam up and down. A graphical representation of the AC
wave shape and not just a measurement of magnitude can
easily be had with such a device. However, CRT's have the
disadvantages of weight, size, significant power
consumption, and fragility (being made of evacuated glass)
working against them. For these reasons, electromechanical
AC meter movements still have a place in practical usage.
With some of the advantages and disadvantages of these
meter movement technologies having been discussed
already, there is another factor crucially important for the
designer and user of AC metering instruments to be aware
of. This is the issue of RMS measurement. As we already
know, AC measurements are often cast in a scale of DC
power equivalence, called RMS (Root-Mean-Square) for the
sake of meaningful comparisons with DC and with other AC
waveforms of varying shape. None of the meter movement
technologies so far discussed inherently measure the RMS
value of an AC quantity. Meter movements relying on the
motion of a mechanical needle (“rectified” D'Arsonval, iron-
vane, and electrostatic) all tend to mechanically average the
instantaneous values into an overall average value for the
waveform. This average value is not necessarily the same as
RMS, although many times it is mistaken as such. Average
and RMS values rate against each other as such for these
three common waveform shapes: (Figure below)
RMS = 0.707 (Peak)
AVG = 0.637 (Peak)
P-P = 2 (Peak)
RMS = Peak
AVG = Peak
P-P = 2 (Peak)
RMS = 0.577 (Peak)
AVG = 0.5 (Peak)
P-P = 2 (Peak)
RMS, Average, and Peak-to-Peak values for sine, square, and
triangle waves.
Since RMS seems to be the kind of measurement most
people are interested in obtaining with an instrument, and
electromechanical meter movements naturally deliver
average measurements rather than RMS, what are AC meter
designers to do? Cheat, of course! Typically the assumption
is made that the waveform shape to be measured is going to
be sine (by far the most common, especially for power
systems), and then the meter movement scale is altered by
the appropriate multiplication factor. For sine waves we see
that RMS is equal to 0.707 times the peak value while
Average is 0.637 times the peak, so we can divide one figure
by the other to obtain an average-to-RMS conversion factor
of 1.109:
0.707
0.637
= 1.1099
In other words, the meter movement will be calibrated to
indicate approximately 1.11 times higher than it would
ordinarily (naturally) indicate with no special
accommodations. It must be stressed that this “cheat” only
works well when the meter is used to measure pure sine
wave sources. Note that for triangle waves, the ratio
between RMS and Average is not the same as for sine waves:
0.577
0.5
= 1.154
With square waves, the RMS and Average values are
identical! An AC meter calibrated to accurately read RMS
voltage or current on a pure sine wave will not give the
proper value while indicating the magnitude of anything
other than a perfect sine wave. This includes triangle waves,
square waves, or any kind of distorted sine wave. With
harmonics becoming an ever-present phenomenon in large
AC power systems, this matter of accurate RMS
measurement is no small matter.
The astute reader will note that | have omitted the CRT
“movement” from the RMS/Average discussion. This is
because a CRT with its practically weightless electron beam
“movement” displays the Peak (or Peak-to-Peak if you wish)
of an AC waveform rather than Average or RMS. Still, a
similar problem arises: how do you determine the RMS value
of a waveform from it? Conversion factors between Peak and
RMS only hold so long as the waveform falls neatly into a
known category of shape (sine, triangle, and square are the
only examples with Peak/RMS/Average conversion factors
given here!).
One answer is to design the meter movement around the
very definition of RMS: the effective heating value of an AC
voltage/current as it powers a resistive load. Suppose that
the AC source to be measured is connected across a resistor
of known value, and the heat output of that resistor is
measured with a device like a thermocouple. This would
provide a far more direct measurement means of RMS than
any conversion factor could, for it will work with ANY
waveform shape whatsoever: (Figure below)
sensitive
meter
movement
thermocouple bonded
with resistive heating
element
AC voltage to
be measured
Direct reading thermal RMS voltmeter accommodates any
wave shape.
While the device shown above is somewhat crude and would
suffer from unique engineering problems of its own, the
concept illustrated is very sound. The resistor converts the
AC voltage or current quantity into a thermal (heat)
quantity, effectively squaring the values in real-time. The
system's mass works to average these values by the
principle of thermal inertia, and then the meter scale itself is
calibrated to give an indication based on the square-root of
the thermal measurement: perfect Root-Mean-Square
indication all in one device! In fact, one major instrument
manufacturer has implemented this technique into its high-
end line of handheld electronic multimeters for “true-RMS”
Capability.
Calibrating AC voltmeters and ammeters for different full-
scale ranges of operation is much the same as with DC
instruments: series “multiplier” resistors are used to give
voltmeter movements higher range, and parallel “shunt”
resistors are used to allow ammeter movements to measure
currents beyond their natural range. However, we are not
limited to these techniques as we were with DC: because we
can use transformers with AC, meter ranges can be
electromagnetically rather than resistively “stepped up” or
“stepped down,” sometimes far beyond what resistors would
have practically allowed for. Potential Transformers (PT's)
and Current Transformers (CT's) are precision instrument
devices manufactured to produce very precise ratios of
transformation between primary and secondary windings.
They can allow small, simple AC meter movements to
indicate extremely high voltages and currents in power
systems with accuracy and complete electrical isolation
(something multiplier and shunt resistors could never do):
(Figure below)
0-5 A AC movement range
A)
199) precision
oT step-up
high-voltage load
power source
precision
step-down
7 ratio
0-120 V AC movement range
(CT) Current transformer scales current down. (PT) Potential
transformer scales voltage down.
Shown here is a voltage and current meter panel from a
three-phase AC system. The three “donut” current
transformers (CT's) can be seen in the rear of the panel.
Three AC ammeters (rated 5 amps full-scale deflection each)
on the front of the panel indicate current through each
conductor going through a CT. As this panel has been
removed from service, there are no current-carrying
conductors threaded through the center of the CT “donuts”
anymore: (Figure below)
Toroidal current transformers scale high current levels down
for application to 5 A full-scale AC ammeters.
Because of the expense (and often large size) of instrument
transformers, they are not used to scale AC meters for any
applications other than high voltage and high current. For
scaling a milliamp or microamp movement to a range of 120
volts or 5 amps, normal precision resistors (multipliers and
shunts) are used, just as with DC.
e REVIEW:
e Polarized (DC) meter movements must use devices
called diodes to be able to indicate AC quantities.
e Electromechanical meter movements, whether
electromagnetic or electrostatic, naturally provide the
average value of a measured AC quantity. These
instruments may be ranged to indicate RMS value, but
only if the shape of the AC waveform is precisely known
beforehand!
e So-called true RMS meters use different technology to
provide indications representing the actual RMS (rather
than skewed average or peak) of an AC waveform.
Frequency and phase measurement
An important electrical quantity with no equivalent in DC
circuits is frequency. Frequency measurement is very
important in many applications of alternating current,
especially in AC power systems designed to run efficiently at
one frequency and one frequency only. If the AC is being
generated by an electromechanical alternator, the
frequency will be directly proportional to the shaft speed of
the machine, and frequency could be measured simply by
measuring the speed of the shaft. If frequency needs to be
measured at some distance from the alternator, though,
other means of measurement will be necessary.
One simple but crude method of frequency measurement in
power systems utilizes the principle of mechanical
resonance. Every physical object possessing the property of
elasticity (Springiness) has an inherent frequency at which it
will prefer to vibrate. The tuning fork is a great example of
this: strike it once and it will continue to vibrate at a tone
specific to its length. Longer tuning forks have lower
resonant frequencies: their tones will be lower on the
musical scale than shorter forks.
Imagine a row of progressively-sized tuning forks arranged
side-by-side. They are all mounted on a common base, and
that base is vibrated at the frequency of the measured AC
voltage (or current) by means of an electromagnet.
Whichever tuning fork is closest in resonant frequency to
the frequency of that vibration will tend to shake the most
(or the loudest). If the forks' tines were flimsy enough, we
could see the relative motion of each by the length of the
blur we would see as we inspected each one from an end-
view perspective. Well, make a collection of “tuning forks”
out of a strip of sheet metal cut in a pattern akin to a rake,
and you have the vibrating reed frequency meter: (Figure
below)
=
sheet metal reeds. to AC voltage
shaken by magnetic
field from thecoil a
Vibrating reed frequency meter diagram.
The user of this meter views the ends of all those unequal
length reeds as they are collectively shaken at the
frequency of the applied AC voltage to the coil. The one
closest in resonant frequency to the applied AC will vibrate
the most, looking something like Figure below.
Frequency Meter
52 54 56 58 60 62 64 66 68
noopboo0a
120 Volts AC
Vibrating reed frequency meter front panel.
Vibrating reed meters, obviously, are not precision
instruments, but they are very simple and therefore easy to
manufacture to be rugged. They are often found on small
engine-driven generator sets for the purpose of setting
engine speed so that the frequency is somewhat close to 60
(50 in Europe) Hertz.
While reed-type meters are imprecise, their operational
principle is not. In lieu of mechanical resonance, we may
substitute electrical resonance and design a frequency
meter using an inductor and capacitor in the form of a tank
circuit (parallel inductor and capacitor). See Figure below.
One or both components are made adjustable, and a meter
is placed in the circuit to indicate maximum amplitude of
voltage across the two components. The adjustment knob(s)
are calibrated to show resonant frequency for any given
setting, and the frequency is read from them after the
device has been adjusted for maximum indication on the
meter. Essentially, this is a tunable filter circuit which is
adjusted and then read in a manner similar to a bridge
circuit (which must be balanced for a “null” condition and
then read).
Sensitive AC
meter movement
variable capacitor with
adjustment knob calibrated
in Hertz.
Resonant frequency meter “peaks” as L-C resonant
frequency is tuned to test frequency.
This technique is a popular one for amateur radio operators
(or at least it was before the advent of inexpensive digital
frequency instruments called counters), especially because
it doesn't require direct connection to the circuit. So long as
the inductor and/or capacitor can intercept enough stray
field (magnetic or electric, respectively) from the circuit
under test to cause the meter to indicate, it will work.
In frequency as in other types of electrical measurement, the
most accurate means of measurement are usually those
where an unknown quantity is compared against a known
standard, the basic instrument doing nothing more than
indicating when the two quantities are equal to each other.
This is the basic principle behind the DC (Wheatstone)
bridge circuit and it is a sound metrological principle applied
throughout the sciences. If we have access to an accurate
frequency standard (a source of AC voltage holding very
precisely to a single frequency), then measurement of any
unknown frequency by comparison should be relatively easy.
For that frequency standard, we turn our attention back to
the tuning fork, or at least a more modern variation of it
called the quartz crystal. Quartz is a naturally occurring
mineral possessing a very interesting property called
piezoelectricity. Piezoelectric materials produce a voltage
across their length when physically stressed, and will
physically deform when an external voltage is applied across
their lengths. This deformation is very, very slight in most
cases, but it does exist.
Quartz rock is elastic (Springy) within that small range of
bending which an external voltage would produce, which
means that it will have a mechanical resonant frequency of
its own capable of being manifested as an electrical voltage
signal. In other words, if a chip of quartz is struck, it will
“ring” with its own unique frequency determined by the
length of the chip, and that resonant oscillation will produce
an equivalent voltage across multiple points of the quartz
chip which can be tapped into by wires fixed to the surface
of the chip. In reciprocal manner, the quartz chip will tend to
vibrate most when it is “excited” by an applied AC voltage
at precisely the right frequency, just like the reeds ona
vibrating-reed frequency meter.
Chips of quartz rock can be precisely cut for desired
resonant frequencies, and that chip mounted securely inside
a protective shell with wires extending for connection to an
external electric circuit. When packaged as such, the
resulting device is simply called a crysta/ (or sometimes
“xtal”). The schematic symbol is shown in Figure below.
crystal or xtal
il
Cc)
7
Crystal (frequency determing element) schematic symbol.
Electrically, that quartz chip is equivalent to a series LC
resonant circuit. (Figure below) The dielectric properties of
quartz contribute an additional capacitive element to the
equivalent circuit.
C
capacitance. .C characteristics
caused by wire of the quartz
connections
across quartz L
Quartz crystal equivalent circuit.
The “capacitance” and “inductance” shown in series are
merely electrical equivalents of the quartz's mechanical
resonance properties: they do not exist as discrete
components within the crystal. The capacitance shown in
parallel due to the wire connections across the dielectric
(insulating) quartz body is real, and it has an effect on the
resonant response of the whole system. A full discussion on
crystal dynamics is not necessary here, but what needs to be
understood about crystals is this resonant circuit
equivalence and how it can be exploited within an oscillator
circuit to achieve an output voltage with a stable, known
frequency.
Crystals, as resonant elements, typically have much higher
“Q” (quality) values than tank circuits built from inductors
and capacitors, principally due to the relative absence of
stray resistance, making their resonant frequencies very
definite and precise. Because the resonant frequency Is
solely dependent on the physical properties of quartz (a
very stable substance, mechanically), the resonant
frequency variation over time with a quartz crystal is very,
very low. This is how quartz movement watches obtain their
high accuracy: by means of an electronic oscillator stabilized
by the resonant action of a quartz crystal.
For laboratory applications, though, even greater frequency
stability may be desired. To achieve this, the crystal in
question may be placed in a temperature stabilized
environment (usually an oven), thus eliminating frequency
errors due to thermal expansion and contraction of the
quartz.
For the ultimate in a frequency standard though, nothing
discovered thus far surpasses the accuracy of a single
resonating atom. This is the principle of the so-called atomic
clock, which uses an atom of mercury (or cesium) suspended
in a vacuum, excited by outside energy to resonate at its
own unique frequency. The resulting frequency is detected
as a radio-wave signal and that forms the basis for the most
accurate clocks known to humanity. National standards
laboratories around the world maintain a few of these hyper-
accurate clocks, and broadcast frequency signals based on
those atoms' vibrations for scientists and technicians to tune
in and use for frequency calibration purposes.
Now we get to the practical part: once we have a source of
accurate frequency, how do we compare that against an
unknown frequency to obtain a measurement? One way is to
use a CRT as a frequency-comparison device. Cathode Ray
Tubes typically have means of deflecting the electron beam
in the horizontal as well as the vertical axis. If metal plates
are used to electrostatically deflect the electrons, there will
be a pair of plates to the left and right of the beam as well as
a pair of plates above and below the beam as in Figure
below.
horizontal
deflection
electron "gun" plates
view-
(vacuum) screen
_ electrons
vertical
deflection
light
plates ~—
Cathode ray tube (CRT) with vertical and horizontal
deflection plates.
If we allow one AC signal to deflect the beam up and down
(connect that AC voltage source to the “vertical” deflection
plates) and another AC signal to deflect the beam left and
right (using the other pair of deflection plates), patterns will
be produced on the screen of the CRT indicative of the ratio
of these two AC frequencies. These patterns are called
Lissajous figures and are a common means of comparative
frequency measurement in electronics.
If the two frequencies are the same, we will obtain a simple
figure on the screen of the CRT, the shape of that figure
being dependent upon the phase shift between the two AC
signals. Here is a sampling of Lissajous figures for two sine-
wave signals of equal frequency, shown as they would
appear on the face of an oscilloscope (an AC voltage-
measuring instrument using a CRT as its “movement”). The
first picture is of the Lissajous figure formed by two AC
voltages perfectly in phase with each other: (Figure below)
OSCILLOSCOPE
vertical
¥
©
DC GND AC
|
Vidiv
timebase
Xx
— DC GND 4c
sidiv Cc
Lissajous figure: same frequency, zero degrees phase shift.
If the two AC voltages are not in phase with each other, a
straight line will not be formed. Rather, the Lissajous figure
will take on the appearance of an oval, becoming perfectly
circular if the phase shift is exactly 90° between the two
signals, and if their amplitudes are equal: (Figure below)
OSCILLOSCOPE
vertical
; ¥
©
Dc GND Ac
| a
Vidiv
timebase
; X
— DC GND AC
sidiv |
Lissajous figure: same frequency, 90 or 270 degrees phase
shift.
Finally, if the two AC signals are directly opposing one
another in phase (180° shift), we will end up with a line
again, only this time it will be oriented in the opposite
direction: (Figure below)
OSCILLOSCOPE
vertical
\ Y
=i DC GND Ac
Vidiv ——I
timebase
Lissajous figure: same frequency, 180 degrees phase shift.
When we are faced with signal frequencies that are not the
same, Lissajous figures get quite a bit more complex.
Consider the following examples and their given
vertical/horizontal frequency ratios: (Figure below)
OSCILLOSCOPE
vertical
Y
—— DC GND Ac
Vidiv | |
timebase
Lissajous figure: Horizontal frequency is twice that of
vertical.
The more complex the ratio between horizontal and vertical
frequencies, the more complex the Lissajous figure. Consider
the following illustration of a 3:1 frequency ratio between
horizontal and vertical: (Figure below)
OSCILLOSCOPE
vertical
¥
i DC GND Ac
Vidiv —
Lissajous figure: Horizontal frequency is three times that of
vertical.
...and a 3:2 frequency ratio (horizontal = 3, vertical = 2) in
Figure below.
OSCILLOSCOPE
vertical
Y
©
pall DC_GND Ac
Vidiv —
trigger © |
timebase
Lissajous figure: Horizontal/Vertical frequency ratio is 3:2
Lissajous figure: Horizontal/vertical frequency ratio is 3:2.
In cases where the frequencies of the two AC signals are not
exactly a simple ratio of each other (but close), the Lissajous
figure will appear to “move,” slowly changing orientation as
the phase angle between the two waveforms rolls between
0° and 180°. If the two frequencies are locked in an exact
integer ratio between each other, the Lissajous figure will be
stable on the viewscreen of the CRT.
The physics of Lissajous figures limits their usefulness as a
frequency-comparison technique to cases where the
frequency ratios are simple integer values (1:1, 1:2, 1:3, 2:3,
3:4, etc.). Despite this limitation, Lissajous figures are a
popular means of frequency comparison wherever an
accessible frequency standard (signal generator) exists.
e REVIEW:
e Some frequency meters work on the principle of
mechanical resonance, indicating frequency by relative
oscillation among a set of uniquely tuned “reeds”
shaken at the measured frequency.
Other frequency meters use electric resonant circuits (LC
tank circuits, usually) to indicate frequency. One or both
components is made to be adjustable, with an
accurately calibrated adjustment knob, and a sensitive
meter is read for maximum voltage or current at the
point of resonance.
e Frequency can be measured in a comparative fashion, as
is the case when using a CRT to generate Lissajous
figures. Reference frequency signals can be made with a
high degree of accuracy by oscillator circuits using
quartz crystals as resonant devices. For ultra precision,
atomic clock signal standards (based on the resonant
frequencies of individual atoms) can be used.
Power measurement
Power measurement in AC circuits can be quite a bit more
complex than with DC circuits for the simple reason that
phase shift complicates the matter beyond multiplying
voltage by current figures obtained with meters. What is
needed is an instrument able to determine the product
(multiplication) of instantaneous voltage and current.
Fortunately, the common electrodynamometer movement
with its stationary and moving coil does a fine job of this.
Three phase power measurement can be accomplished
using two dynamometer movements with a common shaft
linking the two moving coils together so that a single
pointer registers power on a meter movement scale. This,
obviously, makes for a rather expensive and complex
movement mechanism, but it is a workable solution.
An ingenious method of deriving an electronic power meter
(one that generates an electric signal representing power in
the system rather than merely move a pointer) is based on
the Hall effect. The Hall effect is an unusual effect first
noticed by E. H. Hall in 1879, whereby a voltage is
generated along the width of a current-carrying conductor
exposed to a perpendicular magnetic field: (Figure below)
voltage
x
__» current
Hall effect: Voltage is proportional to current and strength of
the perpendicular magnetic field.
The voltage generated across the width of the flat,
rectangular conductor is directly proportional to both the
magnitude of the current through it and the strength of the
magnetic field. Mathematically, it is a product
(multiplication) of these two variables. The amount of “Hall
Voltage” produced for any given set of conditions also
depends on the type of material used for the flat,
rectangular conductor. It has been found that specially
prepared “semiconductor” materials produce a greater Hall
voltage than do metals, and so modern Hall Effect devices
are made of these.
It makes sense then that if we were to build a device using a
Hall-effect sensor where the current through the conductor
was pushed by AC voltage from an external circuit and the
magnetic field was set up by a pair or wire coils energized
by the current of the AC power circuit, the Hall voltage
would be in direct proportion to the multiple of circuit
current and voltage. Having no mass to move (unlike an
electromechanical movement), this device is able to provide
instantaneous power measurement: (Figure below)
voltage
x
R
multiplier
source
Hall effect power sensor measures instantaneous power.
Not only will the output voltage of the Hall effect device be
the representation of instantaneous power at any point in
time, but it will also be a DC signal! This is because the Hall
voltage polarity is dependent upon both the polarity of the
magnetic field and the direction of current through the
conductor. If both current direction and magnetic field
polarity reverses -- as it would ever half-cycle of the AC
power -- the output voltage polarity will stay the same.
If voltage and current in the power circuit are 90° out of
phase (a power factor of zero, meaning no real power
delivered to the load), the alternate peaks of Hall device
current and magnetic field will never coincide with each
other: when one is at its peak, the other will be zero. At
those points in time, the Hall output voltage will likewise be
zero, being the product (multiplication) of current and
magnetic field strength. Between those points in time, the
Hall output voltage will fluctuate equally between positive
and negative, generating a signal corresponding to the
instantaneous absorption and release of power through the
reactive load. The net DC output voltage will be zero,
indicating zero true power in the circuit.
Any phase shift between voltage and current in the power
circuit less than 90° will result in a Hall output voltage that
oscillates between positive and negative, but soends more
time positive than negative. Consequently there will bea
net DC output voltage. Conditioned through a low-pass filter
circuit, this net DC voltage can be separated from the AC
mixed with it, the final output signal registered ona
sensitive DC meter movement.
Often it is useful to have a meter to totalize power usage
over a period of time rather than instantaneously. The
output of such a meter can be set in units of Joules, or total
energy consumed, since power is a measure of work being
done per unit time. Or, more commonly, the output of the
meter can be set in units of Watt-Hours.
Mechanical means for measuring Watt-Hours are usually
centered around the concept of the motor: build an AC
motor that spins at a rate of soeed proportional to the
instantaneous power in a circuit, then have that motor turn
an “odometer” style counting mechanism to keep a running
total of energy consumed. The “motor” used in these meters
has a rotor made of a thin aluminum disk, with the rotating
magnetic field established by sets of coils energized by line
voltage and load current so that the rotational speed of the
disk is dependent on both voltage and current.
Power quality measurement
It used to be with large AC power systems that “power
quality” was an unheard-of concept, aside from power factor.
Almost all loads were of the “linear” variety, meaning that
they did not distort the shape of the voltage sine wave, or
cause non-sinusoidal currents to flow in the circuit. This is
not true anymore. Loads controlled by “nonlinear” electronic
components are becoming more prevalent in both home and
industry, meaning that the voltages and currents in the
power system(s) feeding these loads are rich in harmonics:
what should be nice, clean sine-wave voltages and currents
are becoming highly distorted, which is equivalent to the
presence of an infinite series of high-frequency sine waves
at multiples of the fundamental power line frequency.
Excessive harmonics in an AC power system can overheat
transformers, cause exceedingly high neutral conductor
currents in three-phase systems, create electromagnetic
“noise” in the form of radio emissions that can interfere with
sensitive electronic equipment, reduce electric motor
horsepower output, and can be difficult to pinpoint. With
problems like these plaguing power systems, engineers and
technicians require ways to precisely detect and measure
these conditions.
Power Quality is the general term given to represent an AC
power system's freedom from harmonic content. A “power
quality” meter is one that gives some form of harmonic
content indication.
A simple way for a technician to determine power quality in
their system without sophisticated equipment is to compare
voltage readings between two accurate voltmeters
measuring the same system voltage: one meter being an
“averaging” type of unit (Such as an electromechanical
movement meter) and the other being a “true-RMS” type of
unit (Such as a high-quality digital meter). Remember that
“averaging” type meters are calibrated so that their scales
indicate volts RMS, based on the assumption that the AC
voltage being measured Is sinusoidal. \f the voltage is
anything but sinewave-shaped, the averaging meter will not
register the proper value, whereas the true-RMS meter
always will, regardless of waveshape. The rule of thumb here
is this: the greater the disparity between the two meters, the
worse the power quality is, and the greater its harmonic
content. A power system with good quality power should
generate equal voltage readings between the two meters, to
within the rated error tolerance of the two instruments.
Another qualitative measurement of power quality is the
oscilloscope test: connect an oscilloscope (CRT) to the AC
voltage and observe the shape of the wave. Anything other
than a clean sine wave could be an indication of trouble:
(Figure below)
OSCILLOSCOPE
vertical
Y
©
DC_GND AC
——
Vidiv
timebase
X
©
— DC GND 4c
sidiv sd
This is a moderately ugly “sine” wave. Definite harmonic
content here!
Still, if quantitative analysis (definite, numerical figures) is
necessary, there is no substitute for an instrument
specifically designed for that purpose. Such an instrument is
called a power quality meter and is sometimes better known
in electronic circles as a low-frequency spectrum analyzer.
What this instrument does is provide a graphical
representation on a CRT or digital display screen of the AC
voltage's frequency “spectrum.” Just as a prism splits a
beam of white light into its constituent color components
(how much red, orange, yellow, green, and blue is in that
light), the spectrum analyzer splits a mixed-frequency signal
into its constituent frequencies, and displays the result in
the form of a histogram: (Figure below)
135 7 § 1113
Total distortion = 43.7 %
Power Quality Meter
Power quality meter is a low frequency spectrum analyzer.
Each number on the horizontal scale of this meter
represents a harmonic of the fundamental frequency. For
American power systems, the “1” represents 60 Hz (the 1st
harmonic, or fundamental), the “3” for 180 Hz (the 3rd
harmonic), the “5” for 300 Hz (the 5th harmonic), and so on.
The black rectangles represent the relative magnitudes of
each of these harmonic components in the measured AC
voltage. A pure, 60 Hz sine wave would show only a tall
black bar over the “1” with no black bars showing at all over
the other frequency markers on the scale, because a pure
sine wave has no harmonic content.
Power quality meters such as this might be better referred to
as overtone meters, because they are designed to display
only those frequencies known to be generated by the power
system. In three-phase AC power systems (predominant for
large power applications), even-numbered harmonics tend
to be canceled out, and so only harmonics existing in
significant measure are the odd-numbered.
Meters like these are very useful in the hands of a skilled
technician, because different types of nonlinear loads tend
to generate different spectrum “signatures” which can clue
the troubleshooter to the source of the problem. These
meters work by very quickly sampling the AC voltage at
many different points along the waveform shape, digitizing
those points of information, and using a microprocessor
(small computer) to perform numerical Fourier analysis (the
Fast Fourier Transform or “FFT” algorithm) on those data
points to arrive at harmonic frequency magnitudes. The
process is not much unlike what the SPICE program tells a
computer to do when performing a Fourier analysis on a
simulated circuit voltage or current waveform.
AC bridge circuits
As we saw with DC measurement circuits, the circuit
configuration known as a bridge can be a very useful way to
measure unknown values of resistance. This is true with AC
as well, and we can apply the very same principle to the
accurate measurement of unknown impedances.
To review, the bridge circuit works as a pair of two-
component voltage dividers connected across the same
source voltage, with a nu//-detector meter movement
connected between them to indicate a condition of
“balance” at zero volts: (Figure below)
A balanced bridge shows a “null”, or minimum reading, on
the indicator.
Any one of the four resistors in the above bridge can be the
resistor of unknown value, and its value can be determined
by a ratio of the other three, which are “calibrated,” or
whose resistances are known to a precise degree. When the
bridge is in a balanced condition (zero voltage as indicated
by the null detector), the ratio works out to be this:
In a condition of balance:
R, RR;
J
R, R,
One of the advantages of using a bridge circuit to measure
resistance is that the voltage of the power source Is
irrelevant. Practically speaking, the higher the supply
voltage, the easier it is to detect a condition of imbalance
between the four resistors with the null detector, and thus
the more sensitive it will be. A greater supply voltage leads
to the possibility of increased measurement precision.
However, there will be no fundamental error introduced as a
result of a lesser or greater power supply voltage unlike
other types of resistance measurement schemes.
Impedance bridges work the same, only the balance
equation is with complex quantities, as both magnitude and
phase across the components of the two dividers must be
equal in order for the null detector to indicate “zero.” The
null detector, of course, must be a device capable of
detecting very small AC voltages. An oscilloscope is often
used for this, although very sensitive electromechanical
meter movements and even headphones (small speakers)
may be used if the source frequency is within audio range.
One way to maximize the effectiveness of audio headphones
as a null detector is to connect them to the signal source
through an impedance-matching transformer. Headphone
speakers are typically low-impedance units (8 Q), requiring
substantial current to drive, and so a step-down transformer
helps “match” low-current signals to the impedance of the
headphone speakers. An audio output transformer works
well for this purpose: (Figure below)
Null detector for AC bridge
made from audio headphones
Headphones
Test
leads 1kQ
“Modern” low-Ohm headphones require an impedance
matching transformer for use as a sensitive null detector.
Using a pair of headphones that completely surround the
ears (the “closed-cup” type), I've been able to detect
currents of less than 0.1 YA with this simple detector circuit.
Roughly equal performance was obtained using two different
step-down transformers: a small power transformer (120/6
volt ratio), and an audio output transformer (1000:8 ohm
impedance ratio). With the pushbutton switch in place to
interrupt current, this circuit is usable for detecting signals
from DC to over 2 MHz: even if the frequency is far above or
below the audio range, a “click” will be heard from the
headphones each time the switch is pressed and released.
Connected to a resistive bridge, the whole circuit looks like
Figure below.
Headphones
Bridge with sensitive AC null detector.
Listening to the headphones as one or more of the resistor
“arms” of the bridge is adjusted, a condition of balance will
be realized when the headphones fail to produce “clicks” (or
tones, if the bridge's power source frequency is within audio
range) as the switch is actuated.
When describing general AC bridges, where impedances and
not just resistances must be in proper ratio for balance, it is
sometimes helpful to draw the respective bridge legs in the
form of box-shaped components, each one with a certain
impedance: (Figure below)
LPS.
ES,
Generalized AC impedance bridge: Z = nonspecific complex
impedance.
For this general form of AC bridge to balance, the impedance
ratios of each branch must be equal:
2 . 2
a
Zz “Ze
Again, it must be stressed that the impedance quantities in
the above equation must be complex, accounting for both
magnitude and phase angle. It is insufficient that the
impedance magnitudes alone be balanced; without phase
angles in balance as well, there will still be voltage across
the terminals of the null detector and the bridge will not be
balanced.
Bridge circuits can be constructed to measure just about any
device value desired, be it capacitance, inductance,
resistance, or even “Q.” As always in bridge measurement
circuits, the unknown quantity is always “balanced” against
a known standard, obtained from a high-quality, calibrated
component that can be adjusted in value until the null
detector device indicates a condition of balance. Depending
on how the bridge is set up, the unknown component's
value may be determined directly from the setting of the
calibrated standard, or derived from that standard through a
mathematical formula.
A couple of simple bridge circuits are shown below, one for
inductance (Figure below) and one for capacitance: (Figure
below)
_unknown
inductance
standard
#4, inductance
Symmetrical bridge measures unknown inductor by
comparison to a standard inductor.
unknown
capacitance
standard
capacitance
Symmetrical bridge measures unknown capacitor by
comparison to a standard capacitor.
Simple “symmetrical” bridges such as these are so named
because they exhibit symmetry (mirror-image similarity)
from left to right. The two bridge circuits shown above are
balanced by adjusting the calibrated reactive component (L,
or C,). They are a bit simplified from their real-life
counterparts, as practical symmetrical bridge circuits often
have a calibrated, variable resistor in series or parallel with
the reactive component to balance out stray resistance in
the unknown component. But, in the hypothetical world of
perfect components, these simple bridge circuits do just fine
to illustrate the basic concept.
An example of a little extra complexity added to
compensate for real-world effects can be found in the so-
called Wien bridge, which uses a parallel capacitor-resistor
standard impedance to balance out an unknown series
Capacitor-resistor combination. (Figure below) All capacitors
have some amount of internal resistance, be it literal or
equivalent (in the form of dielectric heating losses) which
tend to spoil their otherwise perfectly reactive natures. This
internal resistance may be of interest to measure, and so the
Wien bridge attempts to do so by providing a balancing
impedance that isn't “pure” either:
Wein Bridge measures both capacitive C,, and resistive R,,
components of “real” capacitor.
Being that there are two standard components to be
adjusted (a resistor and a capacitor) this bridge will take a
little more time to balance than the others we've seen so far.
The combined effect of R, and C, is to alter the magnitude
and phase angle until the bridge achieves a condition of
balance. Once that balance is achieved, the settings of R,
and C, can be read from their calibrated knobs, the parallel
impedance of the two determined mathematically, and the
unknown capacitance and resistance determined
mathematically from the balance equation (Z4/Z5 = Z3/Z,).
It is assumed in the operation of the Wien bridge that the
standard capacitor has negligible internal resistance, or at
least that resistance is already known so that it can be
factored into the balance equation. Wien bridges are useful
for determining the values of “lossy” capacitor designs like
electrolytics, where the internal resistance is relatively high.
They are also used as frequency meters, because the
balance of the bridge is frequency-dependent. When used in
this fashion, the capacitors are made fixed (and usually of
equal value) and the top two resistors are made variable and
are adjusted by means of the same knob.
An interesting variation on this theme is found in the next
bridge circuit, used to precisely measure inductances.
Maxwell-Wein bridge measures an inductor in terms of a
capacitor standard.
This ingenious bridge circuit is Known as the Maxwell-Wien
bridge (sometimes known plainly as the Maxwell bridge),
and is used to measure unknown inductances in terms of
calibrated resistance and capacitance. (Figure above)
Calibration-grade inductors are more difficult to manufacture
than capacitors of similar precision, and so the use of a
simple “symmetrical” inductance bridge is not always
practical. Because the phase shifts of inductors and
Capacitors are exactly opposite each other, a capacitive
impedance can balance out an inductive impedance if they
are located in opposite legs of a bridge, as they are here.
Another advantage of using a Maxwell bridge to measure
inductance rather than a symmetrical inductance bridge is
the elimination of measurement error due to mutual
inductance between two inductors. Magnetic fields can be
difficult to shield, and even a small amount of coupling
between coils in a bridge can introduce substantial errors in
certain conditions. With no second inductor to react with in
the Maxwell bridge, this problem is eliminated.
For easiest operation, the standard capacitor (C,) and the
resistor in parallel with it (R,) are made variable, and both
must be adjusted to achieve balance. However, the bridge
can be made to work if the capacitor is fixed (non-variable)
and more than one resistor made variable (at least the
resistor in parallel with the capacitor, and one of the other
two). However, in the latter configuration it takes more trial-
and-error adjustment to achieve balance, as the different
variable resistors interact in balancing magnitude and
phase.
Unlike the plain Wien bridge, the balance of the Maxwell-
Wien bridge is independent of source frequency, and in
some cases this bridge can be made to balance in the
presence of mixed frequencies from the AC voltage source,
the limiting factor being the inductor's stability over a wide
frequency range.
There are more variations beyond these designs, but a full
discussion is not warranted here. General-purpose
impedance bridge circuits are manufactured which can be
switched into more than one configuration for maximum
flexibility of use.
A potential problem in sensitive AC bridge circuits is that of
stray capacitance between either end of the null detector
unit and ground (earth) potential. Because capacitances can
“conduct” alternating current by charging and discharging,
they form stray current paths to the AC voltage source which
may affect bridge balance: (Figure below)
Stray capacitance to ground may introduce errors into the
bridge.
While reed-type meters are imprecise, their operational
principle is not. In lieu of mechanical resonance, we may
substitute electrical resonance and design a frequency
meter using an inductor and capacitor in the form of a tank
circuit (parallel inductor and capacitor). One or both
components are made adjustable, and a meter is placed in
the circuit to indicate maximum amplitude of voltage across
the two components. The adjustment knob(s) are calibrated
to show resonant frequency for any given setting, and the
frequency is read from them after the device has been
adjusted for maximum indication on the meter. Essentially,
this is a tunable filter circuit which is adjusted and then read
in a manner similar to a bridge circuit (which must be
balanced for a “null” condition and then read). The problem
is worsened if the AC voltage source is firmly grounded at
one end, the total stray impedance for leakage currents
made far less and any leakage currents through these stray
Capacitances made greater as a result: (Figure below)
Stray capacitance errors are more severe if one side of the
AC supply is grounded.
One way of greatly reducing this effect is to keep the null
detector at ground potential, so there will be no AC voltage
between it and the ground, and thus no current through
stray capacitances. However, directly connecting the null
detector to ground is not an option, as it would create a
direct current path for stray currents, which would be worse
than any capacitive path. Instead, a special voltage divider
circuit called a Wagner ground or Wagner earth may be used
to maintain the null detector at ground potential without the
need for a direct connection to the null detector. (Figure
below)
Wagner
earth
Wagner ground for AC supply minimizes the effects of stray
capacitance to ground on the bridge.
The Wagner earth circuit is nothing more than a voltage
divider, designed to have the voltage ratio and phase shift
as each side of the bridge. Because the midpoint of the
Wagner divider is directly grounded, any other divider
circuit (including either side of the bridge) having the same
voltage proportions and phases as the Wagner divider, and
powered by the same AC voltage source, will be at ground
potential as well. Thus, the Wagner earth divider forces the
null detector to be at ground potential, without a direct
connection between the detector and ground.
There is often a provision made in the null detector
connection to confirm proper setting of the Wagner earth
divider circuit: a two-position switch, (Figure below) so that
one end of the null detector may be connected to either the
bridge or the Wagner earth. When the null detector registers
zero signal in both switch positions, the bridge is not only
guaranteed to be balanced, but the null detector is also
guaranteed to be at zero potential with respect to ground,
thus eliminating any errors due to leakage currents through
stray detector-to-ground capacitances:
I
i
(
stra.
Switch-up position allows adjustment of the Wagner ground.
e REVIEW:
e AC bridge circuits work on the same basic principle as
DC bridge circuits: that a balanced ratio of impedances
(rather than resistances) will result in a “balanced”
condition as indicated by the null-detector device.
e Null detectors for AC bridges may be sensitive
electromechanical meter movements, oscilloscopes
(CRT's), headphones (amplified or unamplified), or any
other device capable of registering very small AC
voltage levels. Like DC null detectors, its only required
point of calibration accuracy is at zero.
e AC bridge circuits can be of the “symmetrical” type
where an unknown impedance is balanced by a standard
impedance of similar type on the same side (top or
bottom) of the bridge. Or, they can be
“nonsymmetrical,” using parallel impedances to balance
series impedances, or even capacitances balancing out
inductances.
e AC bridge circuits often have more than one adjustment,
since both impedance magnitude and phase angle must
be properly matched to balance.
Some impedance bridge circuits are frequency-sensitive
while others are not. The frequency-sensitive types may
be used as frequency measurement devices if all
component values are accurately known.
A Wagner earth or Wagner ground is a voltage divider
circuit added to AC bridges to help reduce errors due to
stray capacitance coupling the null detector to ground.
AC instrumentation transducers
Just as devices have been made to measure certain physical
quantities and repeat that information in the form of DC
electrical signals (thermocouples, strain gauges, pH probes,
etc.), soecial devices have been made that do the same with
AC.
It is often necessary to be able to detect and transmit the
physical position of mechanical parts via electrical signals.
This is especially true in the fields of automated machine
tool control and robotics. A simple and easy way to do this is
with a potentiometer: (Figure below)
potentiometer shaft moved
by physical motion of an object
+ voltmeter indicates
position of that object
Potentiometer tap voltage indicates position of an object
slaved to the shaft.
However, potentiometers have their own unique problems.
For one, they rely on physical contact between the “wiper”
and the resistance strip, which means they suffer the effects
of physical wear over time. As potentiometers wear, their
proportional output versus shaft position becomes less and
less certain. You might have already experienced this effect
when adjusting the volume control on an old radio: when
twisting the knob, you might hear “scratching” sounds
coming out of the speakers. Those noises are the result of
poor wiper contact in the volume control potentiometer.
Also, this physical contact between wiper and strip creates
the possibility of arcing (Sparking) between the two as the
wiper is moved. With most potentiometer circuits, the
Current is so low that wiper arcing is negligible, but itisa
possibility to be considered. If the potentiometer is to be
operated in an environment where combustible vapor or
dust is present, this potential for arcing translates into a
potential for an explosion!
Using AC instead of DC, we are able to completely avoid
Sliding contact between parts if we use a variable
transformer instead of a potentiometer. Devices made for
this purpose are called LVDT's, which stands for Linear
Variable Differential Transformers. The design of an LVDT
looks like this: (Figure below)
AC output
voltage
AC "excitation"
voltage
v1" > so vl ov
movable core
AC output of linear variable differential transformer (LVDT)
indicates core position.
Obviously, this device is a transformer. it has a single
primary winding powered by an external source of AC
voltage, and two secondary windings connected in series-
bucking fashion. It is variable because the core is free to
move between the windings. It is differential because of the
way the two secondary windings are connected. Being
arranged to oppose each other (180° out of phase) means
that the output of this device will be the difference between
the voltage output of the two secondary windings. When the
core is centered and both windings are outputting the same
voltage, the net result at the output terminals will be zero
volts. It is called /inear because the core's freedom of motion
is straight-line.
The AC voltage output by an LVDT indicates the position of
the movable core. Zero volts means that the core is
centered. The further away the core is from center position,
the greater percentage of input (“excitation”) voltage will be
seen at the output. The phase of the output voltage relative
to the excitation voltage indicates which direction from
center the core is offset.
The primary advantage of an LVDT over a potentiometer for
position sensing is the absence of physical contact between
the moving and stationary parts. The core does not contact
the wire windings, but slides in and out within a
nonconducting tube. Thus, the LVDT does not “wear” like a
potentiometer, nor is there the possibility of creating an arc.
Excitation of the LVDT is typically 10 volts RMS or less, at
frequencies ranging from power line to the high audio (20
kHz) range. One potential disadvantage of the LVDT is its
response time, which is mostly dependent on the frequency
of the AC voltage source. If very quick response times are
desired, the frequency must be higher to allow whatever
voltage-sensing circuits enough cycles of AC to determine
voltage level as the core is moved. To illustrate the potential
problem here, imagine this exaggerated scenario: an LVDT
powered by a 60 Hz voltage source, with the core being
moved in and out hundreds of times per second. The output
of this LVDT wouldn't even look like a sine wave because the
core would be moved throughout its range of motion before
the AC source voltage could complete a single cycle! It
would be almost impossible to determine instantaneous core
position if it moves faster than the instantaneous source
voltage does.
A variation on the LVDT is the RVDT, or Rotary Variable
Differential Transformer. This device works on almost the
same principle, except that the core revolves on a shaft
instead of moving in a straight line. RVDT's can be
constructed for limited motion of 360° (full-circle) motion.
Continuing with this principle, we have what is known asa
Synchro or Selsyn, which is a device constructed a lot like a
wound-rotor polyphase AC motor or generator. The rotor is
free to revolve a full 360°, just like a motor. On the rotor is a
single winding connected to a source of AC voltage, much
like the primary winding of an LVDT. The stator windings are
usually in the form of a three-phase Y, although synchros
with more than three phases have been built. (Figure below)
A device with a two-phase stator is known as a reso/ver. A
resolver produces sine and cosine outputs which indicate
Shaft position.
Resolver
x
Synchro (a.k.a "Selsyn")
(V) AC voltage ,
source 4
rotor 4 three-phase
winding stator winding rotor wo-phase |
winding stator winding
stator rotor . stator rotor |
connections connections connections connections
modern schematic symbol
A synchro is wound with a three-phase stator winding, and a
rotating field. A resolver has a two-phase stator.
Voltages induced in the stator windings from the rotor's AC
excitation are not phase-shifted by 120° as in a real three-
phase generator. If the rotor were energized with DC current
rather than AC and the shaft spun continuously, then the
voltages would be true three-phase. But this is not how a
synchro is designed to be operated. Rather, this isa
position-sensing device much like an RVDT, except that its
output signal is much more definite. With the rotor
energized by AC, the stator winding voltages will be
proportional in magnitude to the angular position of the
rotor, phase either 0° or 180° shifted, like a regular LVDT or
RVDT. You could think of it as a transformer with one primary
winding and three secondary windings, each secondary
winding oriented at a unique angle. As the rotor is slowly
turned, each winding in turn will line up directly with the
rotor, producing full voltage, while the other windings will
produce something less than full voltage.
Synchros are often used in pairs. With their rotors connected
in parallel and energized by the same AC voltage source,
their shafts will match position to a high degree of accuracy:
(Figure below)
Synchro "transmitter" Synchro "receiver"
The receiver rotor will turn to match position withthe —
transmitter rotor so long as the two rotors remain energized.
Synchro shafts are slaved to each other. Rotating one moves
the other.
Such “transmitter/receiver” pairs have been used on ships
to relay rudder position, or to relay navigational gyro
position over fairly long distances. The only difference
between the “transmitter” and the “receiver” is which one
gets turned by an outside force. The “receiver” can just as
easily be used as the “transmitter” by forcing its shaft to
turn and letting the synchro on the left match position.
If the receiver's rotor is left unpowered, it will act as a
position-error detector, generating an AC voltage at the rotor
if the shaft is anything other than 90° or 270° shifted from
the shaft position of the transmitter. The receiver rotor will
no longer generate any torque and consequently will no
longer automatically match position with the transmitter's:
(Figure below)
Synchro "transmitter" Synchro "receiver"
AC voltmeter
AC voltmeter registers voltage if the receiver rotor is not
rotated exactly 90 or 270 degrees from the transmitter
rotor.
This can be thought of almost as a sort of bridge circuit that
achieves balance only if the receiver shaft is brought to one
of two (matching) positions with the transmitter shaft.
One rather ingenious application of the synchro is in the
creation of a phase-shifting device, provided that the stator
is energized by three-phase AC: (Figure below)
three-phase AC voltage
source (can be Y or Delta)
Synchro
- i
Full rotation of the rotor will smoothly shift the phase from
0° all the way to 360° (back to 0°).
As the synchro's rotor is turned, the rotor coil will
progressively align with each stator coil, their respective
magnetic fields being 120° phase-shifted from one another.
In between those positions, these phase-shifted fields will
mix to produce a rotor voltage somewhere between 0°, 120°,
or 240° shift. The practical result is a device capable of
providing an infinitely variable-phase AC voltage with the
twist of a knob (attached to the rotor shaft).
A synchro or a resolver may measure linear motion if geared
with a rack and pinion mechanism. A linear movement of a
few inches (or cm) resulting in multiple revolutions of the
synchro (resolver) generates a train of sinewaves. An
Inductosyn® is a linear version of the resolver. It outputs
signals like a resolver; though, it bears slight resemblance.
The Inductosyn consists of two parts: a fixed serpentine
winding having a 0.1 in or 2 mm pitch, and a movable
winding known as a s/ider. (Figure below) The slider has a
pair of windings having the same pitch as the fixed winding.
The slider windings are offset by a quarter pitch so both sine
and cosine waves are produced by movement. One slider
winding is adequate for counting pulses, but provides no
direction information. The 2-phase windings provide
direction information in the phasing of the sine and cosine
waves. Movement by one pitch produces a cycle of sine and
cosine waves; multiple pitches produce a train of waves.
sin(@) cos(6)
(a) (b)
Inductosyn: (a) Fixed serpentine winding, (b) movable slider
2-phase windings. Adapted from Figure 6.16 [WAK]
When we Say sine and cosine waves are produces as a
function of linear movement, we really mean a high
frequency carrier is amplitude modulated as the slider
moves. The two slider AC signals must be measured to
determine position within a pitch, the fine position. How
many pitches has the slider moved? The sine and cosine
signals' relationship does not reveal that. However, the
number of pitches (number of waves) may be counted from
a known starting point yielding coarse position. This is an
incremental encoder. |f absolute position must be known
regardless of the starting point, an auxiliary resolver geared
for one revolution per length gives a coarse position. This
constitutes an absolute encoder.
A linear Inductosyn has a transformer ratio of 100:1.
Compare this to the 1:1 ratio for a resolver. A few volts AC
excitation into an Inductosyn yields a few millivolts out. This
low signal level is converted to to a 12-bit digital format by a
resolver to digital converter (RDC). Resolution of 25
microinches is achievable.
There is alSo a rotary version of the Inductosyn having 360
pattern pitches per revolution. When used with a 12-bit
resolver to digital converter, better that 1 arc second
resolution is achievable. This is an incremental encoder.
Counting of pitches from a known starting point is necessary
to determine absolute position. Alternatively, a resolver may
determine coarse absolute position. [WAK]
So far the transducers discussed have all been of the
inductive variety. However, it is possible to make
transducers which operate on variable capacitance as well,
AC being used to sense the change in capacitance and
generate a variable output voltage.
Remember that the capacitance between two conductive
surfaces varies with three major factors: the overlapping
area of those two surfaces, the distance between them, and
the dielectric constant of the material in between the
surfaces. If two out of three of these variables can be fixed
(stabilized) and the third allowed to vary, then any
measurement of capacitance between the surfaces will be
solely indicative of changes in that third variable.
Medical researchers have long made use of capacitive
sensing to detect physiological changes in living bodies. As
early as 1907, a German researcher named H. Cremer placed
two metal plates on either side of a beating frog heart and
measured the capacitance changes resulting from the heart
alternately filling and emptying itself of blood. Similar
measurements have been performed on human beings with
metal plates placed on the chest and back, recording
respiratory and cardiac action by means of capacitance
changes. For more precise capacitive measurements of
organ activity, metal probes have been inserted into organs
(especially the heart) on the tips of catheter tubes,
Capacitance being measured between the metal probe and
the body of the subject. With a sufficiently high AC
excitation frequency and sensitive enough voltage detector,
not just the pumping action but also the sounds of the
active heart may be readily interpreted.
Like inductive transducers, capacitive transducers can also
be made to be self-contained units, unlike the direct
physiological examples described above. Some transducers
work by making one of the capacitor plates movable, either
in such a way as to vary the overlapping area or the distance
between the plates. Other transducers work by moving a
dielectric material in and out between two fixed plates:
(Figure below)
(a) (b) (c)
Variable capacitive transducer varies; (a) area of overlap,
(b) distance between plates, (c) amount of dielectric
between plates.
Transducers with greater sensitivity and immunity to
changes in other variables can be obtained by way of
differential design, much like the concept behind the LVDT
(Linear Variable Differential Transformer). Here are a few
examples of differential capacitive transducers: (Figure
below)
ED EF I
Differential capacitive transducer varies capacitance ratio
by changing: (a) area of overlap, (b) distance between
plates, (c) dielectric between plates.
As you Can see, all of the differential devices shown in the
above illustration have three wire connections rather than
two: one wire for each of the “end” plates and one for the
“common” plate. As the capacitance between one of the
“end” plates and the “common” plate changes, the
Capacitance between the other “end” plate and the
“common” plate is such to change in the opposite direction.
This kind of transducer lends itself very well to
implementation in a bridge circuit: (Figure below)
Pictoral diagram
capacitive
sensor
~—— —pP»
Schematic diagram
| | b few
$s 06 8 se
Differential capacitive transducer bridge measurement
circuit.
Capacitive transducers provide relatively small capacitances
for a measurement circuit to operate with, typically in the
picofarad range. Because of this, high power supply
frequencies (in the megahertz range!) are usually required
to reduce these capacitive reactances to reasonable levels.
Given the small capacitances provided by typical capacitive
transducers, stray capacitances have the potential of being
major sources of measurement error. Good conductor
shielding is essentia/ for reliable and accurate capacitive
transducer circuitry!
The bridge circuit is not the only way to effectively interpret
the differential capacitance output of such a transducer, but
it is one of the simplest to implement and understand. As
with the LVDT, the voltage output of the bridge is
proportional to the displacement of the transducer action
from its center position, and the direction of offset will be
indicated by phase shift. This kind of bridge circuit is similar
in function to the kind used with strain gauges: it is not
intended to be in a “balanced” condition all the time, but
rather the degree of imbalance represents the magnitude of
the quantity being measured.
An interesting alternative to the bridge circuit for
interpreting differential capacitance is the twin-T. It requires
the use of diodes, those “one-way valves” for electric current
mentioned earlier in the chapter: (Figure below)
Differential capacitive transducer “Twin-T” measurement
circuit.
This circuit might be better understood if re-drawn to
resemble more of a bridge configuration: (Figure below)
Differential capacitor transducer “Twin-T” measurement
circuit redrawn as a bridge.Output Is across Rigag.-
Capacitor C, is charged by the AC voltage source during
every positive half-cycle (positive as measured in reference
to the ground point), while C, is charged during every
negative half-cycle. While one capacitor is being charged,
the other capacitor discharges (at a slower rate than it was
charged) through the three-resistor network. As a
consequence, C,; maintains a positive DC voltage with
respect to ground, and C, a negative DC voltage with
respect to ground.
If the capacitive transducer is displaced from center
position, one capacitor will increase in capacitance while the
other will decrease. This has little effect on the peak voltage
charge of each capacitor, as there is negligible resistance in
the charging current path from source to capacitor, resulting
in a very short time constant (Tt). However, when it comes
time to discharge through the resistors, the capacitor with
the greater capacitance value will hold its charge longer,
resulting in a greater average DC voltage over time than the
lesser-value Capacitor.
The load resistor (Rjgag), Connected at one end to the point
between the two equal-value resistors (R) and at the other
end to ground, will drop no DC voltage if the two capacitors'
DC voltage charges are equal in magnitude. If, on the other
hand, one capacitor maintains a greater DC voltage charge
than the other due to a difference in capacitance, the load
resistor will drop a voltage proportional to the difference
between these voltages. Thus, differential capacitance is
translated into a DC voltage across the load resistor.
Across the load resistor, there is both AC and DC voltage
present, with only the DC voltage being significant to the
difference in capacitance. If desired, a low-pass filter may be
added to the output of this circuit to block the AC, leaving
only a DC signal to be interpreted by measurement circuitry:
(Figure below)
R Low-pass
filter
Addition of low-pass filter to “twin-T” feeds pure DC to
measurement indicator.
As a measurement circuit for differential capacitive sensors,
the twin-T configuration enjoys many advantages over the
standard bridge configuration. First and foremost,
transducer displacement is indicated by a simple DC
voltage, not an AC voltage whose magnitude and phase
must be interpreted to tell which capacitance is greater.
Furthermore, given the proper component values and power
supply output, this DC output signal may be strong enough
to directly drive an electromechanical meter movement,
eliminating the need for an amplifier circuit. Another
important advantage is that all important circuit elements
have one terminal directly connected to ground: the source,
the load resistor, and both capacitors are all ground-
referenced. This helps minimize the ill effects of stray
Capacitance commonly plaguing bridge measurement
circuits, likewise eliminating the need for compensatory
measures such as the Wagner earth.
This circuit is also easy to specify parts for. Normally, a
measurement circuit incorporating complementary diodes
requires the selection of “matched” diodes for good
accuracy. Not so with this circuit! So long as the power
supply voltage is significantly greater than the deviation in
voltage drop between the two diodes, the effects of
mismatch are minimal and contribute little to measurement
error. Furthermore, supply frequency variations have a
relatively low impact on gain (how much output voltage is
developed for a given amount of transducer displacement),
and square-wave supply voltage works as well as sine-wave,
assuming a 50% duty cycle (equal positive and negative
half-cycles), of course.
Personal experience with using this circuit has confirmed its
impressive performance. Not only is it easy to prototype and
test, but its relative insensitivity to stray capacitance and its
high output voltage as compared to traditional bridge
circuits makes it a very robust alternative.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jason Starck (June 2000): HTML document formatting,
which led to a much better-looking second edition.
Bibliography
1. [WAK]Walt Kester, “Position and Motion Sensors”, Analog
Devices. https://www.analog.com/media/en/training-
seminars/design-handbooks/Practical-Design-
Techniques-Sensor-Signal/Section6.PDF
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] 4]\—
—/ | 4]
Lessons In Electric Circuits
-- Volume Il
Chapter 13
AC MOTORS
Introduction
o Hysteresis and Eddy Current
e Synchronous Motors
e Synchronous condenser
e Reluctance motor
o Synchronous reluctance
o Switched reluctance
o Electronic driven variable reluctance motor
Stepper motors
o Characteristics
o Variable reluctance stepper
o Permanent magnet stepper
» Wave drive
» Full step drive
» Half step drive
= Construction
o Hybrid stepper motor
Brushless DC motor
Tesla polyphase induction motors
o Construction
o Theory of operation
» Motor speed
» Torque
» NEMA design classes
=» Power factor
» Efficiency
o Nola power factor corrector
o Induction motor alternator
o Motor starting and speed control
» Running 3-phase motors on 1-phase
» Multiple fields
» Variable voltage
» Electronic speed control
o Linear induction motor
e Wound rotor induction motors
o Speed control
o Doubly-fed induction generator
e Single-phase induction motors
o Permanent-split capacitor motor
o Capacitor-start induction motor
o Capacitor-run motor induction motor
o Resistance split-phase induction motor
o Nola power factor corrector
e Other specialized motors
o Shaded pole induction motor
o 2-phase servo motor
o Hysteresis motor
o Eddy current clutch
o Transmitter - receiver
o Differential transmitter - receiver
» Addition vs subtraction
o Control transformer
o Resolver
e AC commutator motors
o Single phase series motor
o Compensated series motor
Universal motor
-Repulsion motor
-Repulsion start induction motor
e Bibliography
O°
{e)
O°
Original author: Dennis Crunkilton
Conductors of squirrel cage induction motor removed from
rotor.
Introduction
After the introduction of the DC electrical distribution
system by Edison in the United States, a gradual transition
to the more economical AC system commenced. Lighting
worked as well on AC as on DC. Transmission of electrical
energy covered longer distances at lower loss with
alternating current. However, motors were a problem with
alternating current. Initially, AC motors were constructed like
DC motors. Numerous problems were encountered due to
changing magnetic fields, as compared to the static fields in
DC motor field coils.
Electric motor family tree
Squirel Perma nent lit W ; Synchronous
split ound Variable
cage = tor phase rotor reluctance reluctance
Wound Capacitor Shaded PM Switched
rotor start pok se reluctance
, Capacitor Variable Synchronaus ,
AC electric motor family diagram.
Charles P. Steinmetz contributed to solving these problems
with his investigation of hysteresis losses in iron armatures.
Nikola Tesla envisioned an entirely new type of motor when
he visualized a spinning turbine, not soun by water or
steam, but by a rotating magnetic field. His new type of
motor, the AC induction motor, is the workhorse of industry
to this day. Its ruggedness and simplicity (Figure above)
make for long life, high reliability, and low maintenance. Yet
small brushed AC motors, similar to the DC variety, persist in
small appliances along with small Tesla induction motors.
Above one horsepower (750 W), the Tesla motor reigns
supreme.
Modern solid state electronic circuits drive brushless DC
motors with AC waveforms generated from a DC source. The
brushless DC motor, actually an AC motor, is replacing the
conventional brushed DC motor in many applications. And,
the stepper motor, a digital version of motor, is driven by
alternating current square waves, again, generated by solid
state circuitry Figure above shows the family tree of the AC
motors described in this chapter.
Cruise ships and other large vessels replace reduction
geared drive shafts with large multi-megawatt generators
and motors. Such has been the case with diesel-electric
locomotives on a smaller scale for many years.
Mechanical enegy
Electrical energy
Heat
Motor system level diagram.
At the system level, (Figure above) a motor takes in
electrical energy in terms of a potential difference and a
current flow, converting it to mechanical work. Alas, electric
motors are not 100% efficient. Some of the electric energy is
lost to heat, another form of energy, due to I7R losses in the
motor windings. The heat is an undesired byproduct of the
conversion. It must be removed from the motor and may
adversely affect longevity. Thus, one goal is to maximize
motor efficiency, reducing the heat loss. AC motors also
have some losses not encountered by DC motors: hysteresis
and eddy currents.
_Hysteresis and Eddy Current
Early designers of AC motors encountered problems traced
to losses unique to alternating current magnetics. These
problems were encountered when adapting DC motors to AC
operation. Though few AC motors today bear any
resemblance to DC motors, these problems had to be solved
before AC motors of any type could be properly designed
before they were built.
Both rotor and stator cores of AC motors are composed of a
stack of insulated laminations. The laminations are coated
with insulating varnish before stacking and bolting into the
final form. Eddy currents are minimized by breaking the
potential conductive loop into smaller less lossy segments.
(Figure below) The current loops look like shorted
transformer secondary turns. The thin isolated laminations
break these loops. Also, the silicon (a semiconductor) added
to the alloy used in the laminations increases electrical
resistance which decreases the magnitude of eddy currents.
solid core laminated core
Eddy currents in iron cores.
If the laminations are made of silicon alloy grain oriented
steel, hysteresis losses are minimized. Magnetic hysteresis is
a lagging behind of magnetic field strength as compared to
magnetizing force. If a soft iron nail is temporarily
magnetized by a solenoid, one would expect the nail to lose
the magnetic field once the solenoid is de-energized.
However, a small amount of residual magnetization, B, due
to hysteresis remains. (Figure below) An alternating current
has to expend energy, -H, the coercive force, in overcoming
this residual magnetization before it can magnetize the core
back to zero, let alone in the opposite direction. Hysteresis
loss is encountered each time the polarity of the AC
reverses. The loss is proportional to the area enclosed by the
hysteresis loop on the B-H curve. “Soft” iron alloys have
lower losses than “hard” high carbon steel alloys. Silicon
grain oriented steel, 4% silicon, rolled to preferentially orient
the grain or crystalline structure, has still lower losses.
low hysteresis loss high loss
Hysteresis curves for low and high loss alloys.
Once Steinmetz's Laws of hysteresis could predict iron core
losses, it was possible to design AC motors which performed
as designed. This was akin to being able to design a bridge
ahead of time that would not collapse once it was actually
built. This knowledge of eddy current and hysteresis was
first applied to building AC commutator motors similar to
their DC counterparts. Today this is but a minor category of
AC motors. Others invented new types of AC motors bearing
little resemblance to their DC kin.
_Synchronous Motors
Single phase synchronous motors are available in small sizes
for applications requiring precise timing such as time
keeping, (clocks) and tape players. Though battery powered
quartz regulated clocks are widely available, the AC line
operated variety has better long term accuracy-- over a
period of months. This is due to power plant operators
purposely maintaining the long term accuracy of the
frequency of the AC distribution system. If it falls behind by
a few cycles, they will make up the lost cycles of AC so that
clocks lose no time.
Above 10 Horsepower (10 kW) the higher efficiency and
leading powerfactor make large synchronous motors useful
in industry. Large synchronous motors are a few percent
more efficient than the more common induction motors.
Though, the synchronous motor is more complex.
Since motors and generators are similar in construction, it
should be possible to use a generator as a motor,
conversely, use a motor as a generator. A synchronous motor
is similar to an alternator with a rotating field. The figure
below shows small alternators with a permanent magnet
rotating field. This figure below could either be two
paralleled and synchronized alternators driven by a
mechanical energy source, or an alternator driving a
synchronous motor. Or, it could be two motors, if an external
power source were connected. The point is that in either
case the rotors must run at the same nominal frequency,
and be in phase with each other. That is, they must be
synchronized. The procedure for synchronizing two
alternators is to (1) open the switch, (2) drive both
alternators at the same rotational rate, (3) advance or retard
the phase of one unit until both AC outputs are in phase, (4)
close the switch before they drift out of phase. Once
synchronized, the alternators will be locked to each other,
requiring considerable torque to break one unit loose (out of
synchronization) from the other.
Synchronous motor running in step with alternator.
If more torque in the direction of rotation is applied to the
rotor of one of the above rotating alternators, the angle of
the rotor will advance (opposite of (3)) with respect to the
magnetic field in the stator coils while still synchronized and
the rotor will deliver energy to the AC line like an alternator.
The rotor will also be advanced with respect to the rotor in
the other alternator. If a load such as a brake is applied to
one of the above units, the angle of the rotor will lag the
stator field as at (3), extracting energy from the AC line, like
a motor. If excessive torque or drag is applied, the rotor will
exceed the maximum torque angle advancing or lagging so
much that synchronization is lost. Torque is developed only
when synchronization of the motor is maintained.
In the case of a small synchronous motor in place of the
alternator Figure above right, it is not necessary to go
through the elaborate synchronization procedure for
alternators. However, the synchronous motor is not self
starting and must still be brought up to the approximate
alternator electrical speed before it will lock (synchronize) to
the generator rotational rate. Once up to speed, the
synchronous motor will maintain synchronism with the AC
power source and develop torque.
Sinewave drives synchronous motor.
Assuming that the motor is up to synchronous speed, as the
sine wave Changes to positive in Figure above (1), the lower
north coil pushes the north rotor pole, while the upper south
coil attracts that rotor north pole. In a similar manner the
rotor south pole is repelled by the upper south coil and
attracted to the lower north coil. By the time that the sine
wave reaches a peak at (2), the torque holding the north
pole of the rotor up is at a maximum. This torque decreases
as the sine wave decreases to 0 Vpc at (3) with the torque at
a minimum.
As the sine wave changes to negative between (3&4), the
lower south coil pushes the south rotor pole, while attracting
rotor north rotor pole. In a similar manner the rotor north
pole is repelled by the upper north coil and attracted to the
lower south coil. At (4) the sinewave reaches a negative
peak with holding torque again at a maximum. As the sine
wave changes from negative to 0 Vpc to positive, The
process repeats for a new cycle of sine wave.
Note, the above figure illustrates the rotor position for a no-
load condition (a=0°). In actual practice, loading the rotor
will cause the rotor to lag the positions shown by angle a.
This angle increases with loading until the maximum motor
torque is reached at a=90° electrical. Synchronization and
torque are lost beyond this angle.
The current in the coils of a single phase synchronous motor
pulsates while alternating polarity. If the permanent magnet
rotor speed is close to the frequency of this alternation, it
synchronizes to this alternation. Since the coil field pulsates
and does not rotate, it is necessary to bring the permanent
magnet rotor up to speed with an auxiliary motor. This is a
small induction motor similar to those in the next section.
Addition of field poles decreases speed.
A 2-pole (pair of N-S poles) alternator will generate a 60 Hz
sine wave when rotated at 3600 rpm (revolutions per
minute). The 3600 rpm corresponds to 60 revolutions per
second. A similar 2-pole permanent magnet synchronous
motor will also rotate at 3600 rpm. A lower speed motor may
be constructed by adding more pole pairs. A 4-pole motor
would rotate at 1800 rpm, a 12-pole motor at 600 rpm. The
style of construction shown (Figure above)) is for illustration.
Higher efficiency higher torque multi-pole stator
synchronous motors actually have multiple poles in the
rotor.
One-winding 12-pole synchronous motor.
Rather than wind 12-coils for a 12-pole motor, wind a single
coil with twelve interdigitated steel poles pieces as shown in
Figure above. Though the polarity of the coil alternates due
to the appplied AC, assume that the top is temporarily north,
the bottom south. Pole pieces route the south flux from the
bottom and outside of the coil to the top. These 6-souths are
interleaved with 6-north tabs bent up from the top of the
steel pole piece of the coil. Thus, a permanent magnet rotor
bar will encounter 6-pole pairs corresponding to 6-cycles of
AC in one physical rotation of the bar magnet. The rotation
speed will be 1/6 of the electrical soeed of the AC. Rotor
speed will be 1/6 of that experienced with a 2-pole
synchronous motor. Example: 60 Hz would rotate a 2-pole
motor at 3600 rpm, or 600 rpm for a 12-pole motor.
’ Ce)
| at q
sagt
Usfeattn
ea
Reprinted by permission of Westclox History at
www.clockHistory.com
The stator (Figure above) shows a 12-pole Westclox
synchronous clock motor. Construction is similar to the
previous figure with a single coil. The one coil style of
construction is economical for low torque motors. This 600
rom motor drives reduction gears moving clock hands.
If the Westclox motor were to run at 600 rpm from a 50 Hz
power source, how many poles would be required? A 10-pole
motor would have 5-pairs of N-S poles. It would rotate at
50/5 = 10 rotations per second or 600 rpm (10 s? x 60
s/minute.)
Reprinted by permission of Westclox History at
www.clockHistory.com
The rotor (Figure above) consists of a permanent magnet bar
and a steel induction motor cup. The synchronous motor bar
rotating within the pole tabs keeps accurate time. The
induction motor cup outside of the bar magnet fits outside
and over the tabs for self starting. At one time non-self-
starting motors without the induction motor cup were
manufactured.
A 3-phase synchronous motor as shown in Figure below
generates an electrically rotating field in the stator. Such
motors are not self starting if started from a fixed frequency
power source such as 50 or 60 Hz as found in an industrial
setting. Furthermore, the rotor is not a permanent magnet as
shown below for the multi-horsepower (multi-kilowatt)
motors used in industry, but an electromagnet. Large
industrial synchronous motors are more efficient than
induction motors. They are used when constant speed is
required. Having a leading power factor, they can correct the
AC line for a lagging power factor.
The three phases of stator excitation add vectorially to
produce a single resultant magnetic field which rotates f/2n
times per second, where f is the power line frequency, 50 or
60 Hz for industrial power line operated motors. The number
of poles is n. For rotor speed in rom, multiply by 60.
S = f120/n
where: S = rotor speed in rpm
f = AC line frequency
n = number of poles per phase
The 3-phase 4-pole (per phase) synchronous motor (Figure
below) will rotate at 1800 rpm with 60 Hz power or 1500
rom with 50 Hz power. If the coils are energized one ata
time in the sequence qg-1, @-2, o-3, the rotor should point to
the corresponding poles in turn. Since the sine waves
actually overlap, the resultant field will rotate, not in steps,
but smoothly. For example, when the g-1 and @-2 sinewaves
coincide, the field will be at a peak pointing between these
poles. The bar magnet rotor shown is only appropriate for
small motors. The rotor with multiple magnet poles (below
right) is used in any efficient motor driving a substantial
load. These will be slip ring fed electromagnets in large
industrial motors. Large industrial synchronous motors are
self started by embedded squirrel cage conductors in the
armature, acting like an induction motor. The
electromagnetic armature is only energized after the rotor is
brought up to near synchronous speed.
Three phase, 4-pole synchronous motor
Small multi-phase synchronous motors (Figure above) may
be started by ramping the drive frequency from zero to the
final running frequency. The multi-phase drive signals are
generated by electronic circuits, and will be square waves in
all but the most demanding applications. Such motors are
known as brushless DC motors. True synchronous motors are
driven by sine waveforms. Two or three phase drive may be
used by supplying the appropriate number of windings in
the stator. Only 3-phase is shown above.
ol torque
output
waveform
gen &
power
Electronic synchronous motor
The block diagram (Figure above) shows the drive
electronics associated with a low voltage (12 Vp )
synchronous motor. These motors have a position sensor
integrated within the motor, which provides a low level
signal with a frequency proportional to the speed of rotation
of the motor. The position sensor could be as simple as solid
state magnetic field sensors such as Hal/ effect devices
providing commutation (armature current direction) timing
to the drive electronics. The position sensor could be a high
resolution angular sensor such as a resolver, an inductosyn
(magnetic encoder), or an optical encoder.
If constant and accurate speed of rotation is required, (as for
a disk drive) a tachometer and phase locked loop may be
included. (Figure below) This tachometer signal, a pulse
train proportional to motor speed, is fed back to a phase
locked loop, which compares the tachometer frequency and
phase to a stable reference frequency source such as a
crystal oscillator.
| torque
waveform
output
gen & 2
oe
drive >
position sensor
tachometer
reference
frequency
Phase locked loop controls synchronous motor speed.
A motor driven by square waves of current, as provided by
simple Hall effect sensors, is known as a brushless DC motor.
This type of motor has higher ripple torque torque variation
through a shaft revolution than a sine wave driven motor.
This is not a problem for many applications. Though, we are
primarily interested in synchronous motors in this section.
Ripple torque mechanical analog
Motor ripple torque and mechanical analog.
Ripple torque, or cogging is caused by magnetic attraction
of the rotor poles to the stator pole pieces. (Figure above)
Note that there are no stator coils, not even a motor. The PM
rotor may be rotated by hand but will encounter attraction
to the pole pieces when near them. This is analogous to the
mechanical situation. Would ripple torque be a problem for a
motor used in a tape player? Yes, we do not want the motor
to alternately speed and slow as it moves audio tape past a
tape playback head. Would ripple torque be a problem for a
fan motor? No.
a0 op
phi phi
Single phase belt
Windings distributed in a belt produce a more sinusoidal
field.
If a motor is driven by sinewaves of current synchronous
with the motor back emf, it is classified as a synchronous AC
motor, regardless of whether the drive waveforms are
generated by electronic means. A synchronous motor will
generate a sinusoidal back emf if the stator magnetic field
has a sinusoidal distribution. It will be more sinusoidal if pole
windings are distributed in a belt (Figure above) across
many slots instead of concentrated on one large pole (as
drawn in most of our simplified illustrations). This
arrangement cancels many of the stator field odd harmonics.
Slots having fewer windings at the edge of the phase
winding may share the space with other phases. Winding
belts may take on an alternate concentric form as shown in
Figure below.
Concentric belts.
For a 2-phase motor, driven by a sinewave, the torque is
constant throughout a revolution by the trigonometric
identity:
sin28 + cos’e = 1
The generation and synchronization of the drive waveform
requires a more precise rotor position indication than
provided by the Hall effect sensors used in brushless DC
motors. A resolver, or optical or magnetic encoder provides
resolution of hundreds to thousands of parts (pulses) per
revolution. A resolver provides analog angular position
signals in the form of signals proportional to the sine and
cosine of shaft angle. Encoders provide a digital angular
position indication in either serial or parallel format. The sine
wave drive may actually be from a PWM, Pulse Width
Modulator, a high efficiency method of approximating a
sinewave with a digital waveform. (Figure below) Each phase
requires drive electronics for this wave form phase-shifted
by the appropriate amount per phase.
UU UUTUUUOUUU
PWM
PWM approximates a sinewave.
Synchronous motor efficiency is higher than that of
induction motors. The synchronous motor can also be
smaller, especially if high energy permanent magnets are
used in the rotor. The advent of modern solid state
electronics makes it possible to drive these motors at
variable speed. Induction motors are mostly used in railway
traction. However, a small synchronous motor, which mounts
inside a drive wheel, makes it attractive for such
applications. The high temperature superconducting version
of this motor is one fifth to one third the weight of a copper
wound motor.[1] The largest experimental superconducting
synchronous motor is capable of driving a naval destroyer
class ship. In all these applications the electronic variable
speed drive is essential.
The variable speed drive must also reduce the drive voltage
at low speed due to decreased inductive reactance at lower
frequency. To develop maximum torque, the rotor needs to
lag the stator field direction by 90°. Any more, it loses
synchronization. Much less results in reduced torque. Thus,
the position of the rotor needs to be known accurately. And
the position of the rotor with respect to the stator field needs
to be calculated, and controlled. This type of control is
known as vector phase control. It is implemented with a fast
microprocessor driving a pulse width modulator for the
stator phases.
The stator of a synchronous motor is the same as that of the
more popular induction motor. As a result the industrial
grade electronic speed control used with induction motors is
also applicable to large industrial synchronous motors.
If the rotor and stator of a conventional rotary synchronous
motor are unrolled, a synchronous linear motor results. This
type of motor is applied to precise high speed linear
positioning.[2]
A larger version of the linear synchronous motor with a
movable carriage containing high energy NdBFe permanent
magnets is being developed to launch aircraft from naval
aircraft carriers.[3]
Synchronous condenser
Synchronous motors load the power line with a leading
power factor. This is often useful in cancelling out the more
commonly encountered lagging power factor caused by
induction motors and other inductive loads. Originally, large
industrial synchronous motors came into wide use because
of this ability to correct the lagging power factor of induction
motors.
This leading power factor can be exaggerated by removing
the mechanical load and over exciting the field of the
synchronous motor. Such a device is known as a
synchronous condenser. Furthermore, the leading power
factor can be adjusted by varying the field excitation. This
makes it possible to nearly cancel an arbitrary lagging
power factor to unity by paralleling the lagging load with a
synchronous motor. A synchronous condenser is operated in
a borderline condition between a motor and a generator with
no mechanical load to fulfill this function. It can compensate
either a leading or lagging power factor, by absorbing or
supplying reactive power to the line. This enhances power
line voltage regulation.
Since a synchronous condenser does not supply a torque,
the output shaft may be dispensed with and the unit easily
enclosed in a gas tight shell. The synchronous condenser
may then be filled with hydrogen to aid cooling and reduce
windage losses. Since the density of hydrogen is 7% of that
of air, the windage loss for a hydrogen filled unit is 7% of
that encountered in air. Furthermore, the thermal
conductivity of hydrogen is ten times that of air. Thus, heat
removal is ten times more efficient. As a result, a hydrogen
filled synchronous condenser can be driven harder than an
air cooled unit, or it may be physically smaller for a given
Capacity. There is no explosion hazard as long as the
hydrogen concentration is maintained above 70%, typically
above 91%.
The efficiency of long power transmission lines may be
increased by placing synchronous condensers along the line
to compensate lagging currents caused by line inductance.
More real power may be transmitted through a fixed size line
if the power factor is brought closer to unity by synchronous
condensers absorbing reactive power.
The ability of synchronous condensers to absorb or produce
reactive power on a transient basis stabilizes the power grid
against short circuits and other transient fault conditions.
Transient sags and dips of milliseconds duration are
stabilized. This supplements longer response times of quick
acting voltage regulation and excitation of generating
equipment. The synchronous condenser aids voltage
regulation by drawing leading current when the line voltage
sags, which increases generator excitation thereby restoring
line voltage. (Figure below) A capacitor bank does not have
this ability.
20% 40% 60% 80% 100%
Line current
Synchronous condenser improves power line voltage
regulation.
The capacity of a synchronous condenser can be increased
by replacing the copper wound iron field rotor with an
ironless rotor of high temperature superconducting wire,
which must be cooled to the liquid nitrogen boiling point of
77°K (-196°C). The superconducting wire carries 160 times
the current of comparable copper wire, while producing a
flux density of 3 Teslas or higher. An iron core would saturate
at 2 Teslas in the rotor air gap. Thus, an iron core,
approximate YW,=1000, is of no more use than air, or any
other material with a relative permeability u,=1, in the rotor.
Such a machine is said to have considerable additional
transient ability to supply reactive power to troublesome
loads like metal melting arc furnaces. The manufacturer
describes it as being a “reactive power shock absorber”.
Such a synchronous condenser has a higher power density
(smaller physically) than a switched capacitor bank. The
ability to absorb or produce reactive power on a transient
basis stabilizes the overall power grid against fault
conditions.
Reluctance motor
The variable reluctance motor is based on the principle that
an unrestrained piece of iron will move to complete a
magnetic flux path with minimum re/uctance, the magnetic
analog of electrical resistance. (Figure below)
Synchronous reluctance
If the rotating field of a large synchronous motor with salient
poles is de-energized, it will still develop 10 or 15% of
synchronous torque. This is due to variable reluctance
throughout a rotor revolution. There is no practical
application for a large synchronous reluctance motor.
However, it is practical in small sizes.
If slots are cut into the conductorless rotor of an induction
motor, corresponding to the stator slots, a synchronous
reluctance motor results. It starts like an induction motor but
runs with a small amount of synchronous torque. The
synchronous torque is due to changes in reluctance of the
magnetic path from the stator through the rotor as the slots
align. This motor is an inexpensive means of developing a
moderate synchronous torque. Low power factor, low pull-
out torque, and low efficiency are characteristics of the
direct power line driven variable reluctance motor. Such was
the status of the variable reluctance motor for a century
before the development of semiconductor power control.
Switched reluctance
If an iron rotor with poles, but without any conductors, is
fitted to a multi-phase stator, a switched reluctance motor,
capable of synchronizing with the stator field results. When
a stator coil pole pair is energized, the rotor will move to the
lowest magnetic reluctance path. (Figure below) A switched
reluctance motor is also known as a variable reluctance
motor. The reluctance of the rotor to stator flux path varies
with the position of the rotor.
high reluctance low reluctance
Reluctance Is a function of rotor position in a variable
reluctance motor.
Sequential switching (Figure below) of the stator phases
moves the rotor from one position to the next. The mangetic
flux seeks the path of least reluctance, the magnetic analog
of electric resistance. This is an over simplified rotor and
waveforms to illustrate operation.
CL hl
Variable reluctance motor, over-simplified operation.
If one end of each 3-phase winding of the switched
reluctance motor is brought out via a common lead wire, we
can explain operation as if it were a stepper motor. (Figure
above) The other coil connections are successively pulled to
ground, one at a time, in a wave drive pattern. This attracts
the rotor to the clockwise rotating magnetic field in 60°
increments.
Various waveforms may drive variable reluctance motors.
(Figure below) Wave drive (a) is simple, requiring only a
single ended unipolar switch. That is, one which only
switches in one direction. More torque is provided by the
bipolar drive (b), but requires a bipolar switch. The power
driver must pull alternately high and low. Waveforms (a & b)
are applicable to the stepper motor version of the variable
reluctance motor. For smooth vibration free operation the 6-
step approximation of a sine wave (c) is desirable and easy
to generate. Sine wave drive (d) may be generated by a
pulse width modulator (PWM), or drawn from the power line.
Variable reluctance motor drive waveforms: (a) unipolar
wave drive, (b) bipolar full step (c) sinewave (d) bipolar 6-
step.
Doubling the number of stator poles decreases the rotating
speed and increases torque. This might eliminate a gear
reduction drive. A variable reluctance motor intended to
move in discrete steps, stop, and start is a variable
reluctance stepper motor, covered in another section. If
smooth rotation is the goal, there is an electronic driven
version of the switched reluctance motor. Variable
reluctance motors or steppers actually use rotors like those
in Figure below.
Electronic driven variable reluctance motor
Variable reluctance motors are poor performers when direct
power line driven. However, microprocessors and solid state
power drive makes this motor an economical high
performance solution in some high volume applications.
Though difficult to control, this motor is easy to spin.
Sequential switching of the field coils creates a rotating
magnetic field which drags the irregularly shaped rotor
around with it as it seeks out the lowest magnetic reluctance
path. The relationship between torque and stator current is
highly nonlinear- difficult to control.
Electronic driven variable reluctance motor.
An electronic driven variable reluctance motor (Figure
below) resembles a brushless DC motor without a permanent
magnet rotor. This makes the motor simple and inexpensive.
However, this is offset by the cost of the electronic control,
which is not nearly as simple as that for a brushless DC
motor.
While the variable reluctance motor is simple, even more so
than an induction motor, it is difficult to control. Electronic
control solves this problem and makes it practical to drive
the motor well above and below the power line frequency. A
variable reluctance motor driven by a servo, an electronic
feedback system, controls torque and speed, minimizing
ripple torque. Figure below
variable
reluctance
uiprocessor
control
Electronic driven variable reluctance motor.
stator current
rotor position
This is the opposite of the high ripple torque desired in
stepper motors. Rather than a stepper, a variable reluctance
motor is optimized for continuous high speed rotation with
minimum ripple torque. It is necessary to measure the rotor
position with a rotary position sensor like an optical or
magnetic encoder, or derive this from monitoring the stator
back EMF. A microprocessor performs complex calculations
for switching the windings at the proper time with solid state
devices. This must be done precisely to minimize audible
noise and ripple torque. For lowest ripple torque, winding
current must be monitored and controlled. The strict drive
requirements make this motor only practical for high volume
applications like energy efficient vacuum cleaner motors,
fan motors, or pump motors. One such vacuum cleaner uses
a compact high efficiency electronic driven 100,000 rpm fan
motor. The simplicity of the motor compensates for the drive
electronics cost. No brushes, no commutator, no rotor
windings, no permanent magnets, simplifies motor
manufacture. The efficiency of this electronic driven motor
can be high. But, it requires considerable optimization,
using specialized design techniques, which is only justified
for large manufacturing volumes.
Advantages
e Simple construction- no brushes, commutator, or
permanent magnets, no Cu or Al in the rotor.
e High efficiency and reliability compared to conventional
AC or DC motors.
e High starting torque.
e Cost effective compared to bushless DC motor in high
volumes.
e Adaptable to very high ambient temperature.
e Low cost accurate speed control possible if volume is
high enough.
Disadvantages
e Current versus torque is highly nonlinear
e Phase switching must be precise to minimize ripple
torque
e Phase current must be controlled to minimize ripple
torque
e Acoustic and electrical noise
e Not applicable to low volumes due to complex control
issues
Stepper motors
A stepper motor is a “digital” version of the electric motor.
The rotor moves in discrete steps as commanded, rather
than rotating continuously like a conventional motor. When
stopped but energized, a stepper (short for stepper motor)
holds its load steady with a holding torque. Wide spread
acceptance of the stepper motor within the last two decades
was driven by the ascendancy of digital electronics. Modern
solid state driver electronics was a key to its success. And,
microprocessors readily interface to stepper motor driver
circuits.
Application wise, the predecessor of the stepper motor was
the servo motor. Today this is a higher cost solution to high
performance motion control applications. The expense and
complexity of a servomotor is due to the additional system
components: position sensor and error amplifier. (Figure
below) It is still the way to position heavy loads beyond the
grasp of lower power steppers. High acceleration or
unusually high accuracy still requires a servo motor.
Otherwise, the default is the stepper due to low cost, simple
drive electronics, good accuracy, good torque, moderate
speed, and low cost.
ez
command
, command
se a /
servo motor load positior
‘nae “error “
stepper motor load
sensor
Stepper motor vs servo motor.
A stepper motor positions the read-write heads in a floppy
drive. They were once used for the same purpose in
harddrives. However, the high speed and accuracy required
of modern harddrive head positioning dictates the use of a
linear servomotor (voice coil).
The servo amplifier is a linear amplifier with some difficult to
integrate discrete components. A considerable design effort
is required to optimize the servo amplifier gain vs phase
response to the mechanical components. The stepper motor
drivers are less complex solid state switches, being either
“on” or “off”. Thus, a stepper motor controller is less complex
and costly than a servo motor controller.
Slo-syn synchronous motors can run from AC line voltage
like a single-phase permanent-capacitor induction motor.
The capacitor generates a 90° second phase. With the direct
line voltage, we have a 2-phase drive. Drive waveforms of
bipolar (+) square waves of 2-24V are more common these
days. The bipolar magnetic fields may also be generated
from unipolar (one polarity) voltages applied to alternate
ends of a center tapped winding. (Figure below) In other
words, DC can be switched to the motor so that it sees AC.
As the windings are energized in sequence, the rotor
synchronizes with the consequent stator magnetic field.
Thus, we treat stepper motors as a class of AC synchronous
motor.
erase Sa il i
Vy Vy 1
(a) bipolar (b) unipolar
Unipolar drive of center tapped coil at (b), emulates AC
current in single coil at (a).
Characteristics
Stepper motors are rugged and inexpensive because the
rotor contains no winding slip rings, or commutator. The
rotor is a cylindrical solid, which may also have either salient
poles or fine teeth. More often than not the rotor is a
permanent magnet. Determine that the rotor is a permanent
magnet by unpowered hand rotation showing detent torque,
torque pulsations. Stepper motor coils are wound within a
laminated stator, except for can stack construction. There
may be as few as two winding phases or as many as five.
These phases are frequently split into pairs. Thus, a 4-pole
stepper motor may have two phases composed of in-line
pairs of poles spaced 90° apart. There may also be multiple
pole pairs per phase. For example a 12-pole stepper has 6-
pairs of poles, three pairs per phase.
Since stepper motors do not necessarily rotate continuously,
there is no horsepower rating. If they do rotate continuously,
they do not even approach a sub-fractional hp rated
capability. They are truly small low power devices compared
to other motors. They have torque ratings to a thousand in-
oz (inch-ounces) or ten n-m (newton-meters) for a 4 kg size
unit. A small “dime” size stepper has a torque of a
hundredth of a newton-meter or a few inch-ounces. Most
steppers are a few inches in diameter with a fraction of a n-
m or a few in-oz torque. The torque available is a function of
motor speed, load inertia, load torque, and drive electronics
as illustrated on the speed vs torque curve. (Figure below)
An energized, holding stepper has a relatively high holding
torque rating. There is less torque available for a running
motor, decreasing to zero at some high speed. This speed is
frequently not attainable due to mechanical resonance of
the motor load combination.
maximum speed
olding torque
cutott speed
Speed
Stepper speed characteristics.
Stepper motors move one step at a time, the step angle,
when the drive waveforms are changed. The step angle is
related to motor construction details: number of coils,
number of poles, number of teeth. It can be from 90° to
0.75°, corresponding to 4 to 500 steps per revolution. Drive
electronics may halve the step angle by moving the rotor in
half-steps.
Steppers cannot achieve the speeds on the speed torque
curve instantaneously. The maximum start frequency is the
highest rate at which a stopped and unloaded stepper can
be started. Any load will make this parameter unattainable.
In practice, the step rate is ramped up during starting from
well below the maximum start frequency. When stopping a
stepper motor, the step rate may be decreased before
stopping.
The maximum torque at which a stepper can start and stop
is the pull-in torque. This torque load on the stepper is due
to frictional (brake) and inertial (flywheel) loads on the
motor shaft. Once the motor is up to speed, pull-out torque
is the maximum sustainable torque without losing steps.
There are three types of stepper motors in order of
increasing complexity: variable reluctance, permanent
magnet, and hybrid. The variable reluctance stepper has s
solid soft steel rotor with salient poles. The permanent
magnet stepper has a cylindrical permanent magnet rotor.
The hybrid stepper has soft steel teeth added to the
permanent magnet rotor for a smaller step angle.
Variable reluctance stepper
A variable reluctance stepper motor relies upon magnetic
flux seeking the lowest reluctance path through a magnetic
circuit. This means that an irregularly shaped soft magnetic
rotor will move to complete a magnetic circuit, minimizing
the length of any high reluctance air gap. The stator
typically has three windings distributed between pole pairs ,
the rotor four salient poles, yielding a 30° step angle.(Figure
below) A de-energized stepper with no detent torque when
hand rotated is identifiable as a variable reluctance type
stepper.
30° step
Three phase and four phase variable reluctance stepper
motors.
The drive waveforms for the 3-@ stepper can be seen in the
“Reluctance motor” section. The drive for a 4-@ stepper is
shown in Figure below. Sequentially switching the stator
phases produces a rotating magnetic field which the rotor
follows. However, due to the lesser number of rotor poles,
the rotor moves less than the stator angle for each step. For
a variable reluctance stepper motor, the step angle is given
by:
O05 = 360°/No
Op = 360°/Np
Ocp = Op - Os
where: O, = stator angle, Op = Rotor angle,
0s; = step angle
Ns = number stator poles, Np = number rotor
poles
3 counterclockwise 15° step reverse step, clockwise
Stepping sequence for variable reluctance stepper.
In Figure above, moving from q@, to @>, etc., the stator
magnetic field rotates clockwise. The rotor moves
counterclockwise (CCW). Note what does not happen! The
dotted rotor tooth does not move to the next stator tooth.
Instead, the @> stator field attracts a different tooth in
moving the rotor CCW, which is a smaller angle (15°) than
the stator angle of 30°. The rotor tooth angle of 45° enters
into the calculation by the above equation. The rotor moved
CCW to the next rotor tooth at 45°, but it aligns with a CW
by 30° stator tooth. Thus, the actual step angle is the
difference between a stator angle of 45° and a rotor angle of
30° . How far would the stepper rotate if the rotor and stator
had the same number of teeth? Zero- no rotation.
Starting at rest with phase @, energized, three pulses are
required (@>, 3, @4) to align the “dotted” rotor tooth to the
next CCW stator tooth, which is 45°. With 3-pulses per stator
tooth, and 8-stator teeth, 24-pulses or steps move the rotor
through 360°.
By reversing the sequence of pulses, the direction of rotation
is reversed above right. The direction, step rate, and number
of steps are controlled by a stepper motor controller feeding
a driver or amplifier. This could be combined into a single
circuit board. The controller could be a microprocessor or a
specialized integrated circuit. The driver is not a linear
amplifier, but a simple on-off switch capable of high enough
current to energize the stepper. In principle, the driver could
be a relay or even a toggle switch for each phase. In
practice, the driver is either discrete transistor switches or
an integrated circuit. Both driver and controller may be
combined into a single integrated circuit accepting a
direction command and step pulse. It outputs current to the
proper phases in sequence.
Variable reluctance stepper motor.
Disassemble a reluctance stepper to view the internal
components. Otherwise, we show the internal construction
of a variable reluctance stepper motor in Figure above. The
rotor has protruding poles so that they may be attracted to
the rotating stator field as it is switched. An actual motor, is
much longer than our simplified illustration.
optical —knife edge
senna e ae
s_
guide rails
Variable reluctance stepper drives lead screw.
The shaft is frequently fitted with a drive screw. (Figure
above) This may move the heads of a floppy drive upon
command by the floppy drive controller.
Variable reluctance stepper motors are applied when only a
moderate level of torque is required and a coarse step angle
is adequate. A screw drive, as used in a floppy disk drive is
such an application. When the controller powers-up, it does
not know the position of the carriage. However, it can drive
the carriage toward the optical interrupter, calibrating the
position at which the knife edge cuts the interrupter as
“home”. The controller counts step pulses from this position.
As long as the load torque does not exceed the motor
torque, the controller will know the carriage position.
Summary: variable reluctance stepper motor
e The rotor is a soft iron cylinder with salient (protruding)
poles.
This is the least complex, most inexpensive stepper
motor.
The only type stepper with no detent torque in hand
rotation of a de-energized motor shaft.
Large step angle
A lead screw is often mounted to the shaft for linear
stepping motion.
Permanent magnet stepper
A permanent magnet stepper motor has a cylindrical
permanent magnet rotor. The stator usually has two
windings. The windings could be center tapped to allow fora
unipolar driver circuit where the polarity of the magnetic
field is changed by switching a voltage from one end to the
other of the winding. A bipo/ar drive of alternating polarity is
required to power windings without the center tap. A pure
permanent magnet stepper usually has a large step angle.
Rotation of the shaft of a de-energized motor exhibits detent
torque. If the detent angle is large, say 7.5° to 90°, it is likely
a permanent magnet stepper rather than a hybrid stepper
(next subsection).
Permanent magnet stepper motors require phased
alternating currents applied to the two (or more) windings.
In practice, this is almost always square waves generated
from DC by solid state electronics. Bipo/ar drive is square
waves alternating between (+) and (-) polarities, say, +2.5 V
to -2.5 V. Unipolar drive supplies a (+) and (-) alternating
magnetic flux to the coils developed from a pair of positive
square waves applied to opposite ends of a center tapped
coil. The timing of the bipolar or unipolar wave is wave
drive, full step, or half step.
Wave drive
Wave drive
PM wave drive sequence (a) 9,+, (b) @o+, (Cc) Q1-, (d)
Q2-.
Conceptually, the simplest drive is wave drive. (Figure
above) The rotation sequence left to right is positive @-1
points rotor north pole up, (+) @-2 points rotor north right,
negative g-1 attracts rotor north down, (-) @-2 points rotor
left. The wave drive waveforms below show that only one
coil is energized at a time. While simple, this does not
produce as much torque as other drive techniques.
4
pl 1
2 qe ;
PAM OO
¥
2
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Waveforms: bipolar wave drive.
r
The waveforms (Figure above) are bipolar because both
polarities , (+) and (-) drive the stepper. The coil magnetic
field reverses because the polarity of the drive current
reverses.
e TLS Le tay? 2
e —F LH Pe
ee ee vs
e Co FL
6-wire pr
Waveforms: unipolar wave drive.
The (Figure above) waveforms are unipolar because only one
polarity is required. This simplifies the drive electronics, but
requires twice as many drivers. There are twice as many
waveforms because a pair of (+) waves is required to
produce an alternating magnetic field by application to
opposite ends of a center tapped coil. The motor requires
alternating magnetic fields. These may be produced by
either unipolar or bipolar waves. However, motor coils must
have center taps for unipolar drive.
Permanent magnet stepper motors are manufactured with
various lead-wire configurations. (Figure below)
. 2 Paz
$2 $2
4-wire O -wire ; 5-wire os wire O
bipolar unipolar unipolar or unipolar
Stepper motor wiring diagrams.
The 4-wire motor can only be driven by bipolar waveforms.
The 6-wire motor, the most common arrangement, is
intended for unipolar drive because of the center taps.
Though, it may be driven by bipolar waves if the center taps
are ignored. The 5-wire motor can only be driven by unipolar
waves, as the common center tap interferes if both windings
are energized simultaneously. The 8-wire configuration is
rare, but provides maximum flexibility. It may be wired for
unipolar drive as for the 6-wire or 5-wire motor. A pair of
coils may be connected in series for high voltage bipolar low
Current drive, or in parallel for low voltage high current
drive.
A bifilar winding is produced by winding the coils with two
wires in parallel, often a red and green enamelled wire. This
method produces exact 1:1 turns ratios for center tapped
windings. This winding method is applicable to all but the 4-
wire arrangement above.
Full step drive
Full step drive provides more torque than wave drive
because both coils are energized at the same time. This
attracts the rotor poles midway between the two field poles.
(Figure below)
Full step, bipolar drive.
Full step bipolar drive as shown in Figure above has the
same step angle as wave drive. Unipolar drive (not shown)
would require a pair of unipolar waveforms for each of the
above bipolar waveforms applied to the ends of a center
tapped winding. Unipolar drive uses a less complex, less
expensive driver circuit. The additional cost of bipolar drive
is justified when more torque is required.
Half step drive
The step angle for a given stepper motor geometry is cut in
half with ha/f step drive. This corresponds to twice as many
step pulses per revolution. (Figure below) Half stepping
provides greater resolution in positioning of the motor shaft.
For example, half stepping the motor moving the print head
across the paper of an inkjet printer would double the dot
density.
Half step
Half step, bipolar drive.
Half step drive is a combination of wave drive and full step
drive with one winding energized, followed by both windings
energized, yielding twice as many steps. The unipolar
waveforms for half step drive are shown above. The rotor
aligns with the field poles as for wave drive and between the
poles as for full step drive.
Microstepping is possible with specialized controllers. By
varying the currents to the windings sinusoidally many
microsteps can be interpolated between the normal
positions.
Construction
The contruction of a permanent magnet stepper motor is
considerably different from the drawings above. It is
desirable to increase the number of poles beyond that
illustrated to produce a smaller step angle. It is also
desirable to reduce the number of windings, or at least not
increase the number of windings for ease of manufacture.
ceramic permanent magnet
rotor
b-1 coil -2 coil
Permanent magnet stepper motor, 24-pole can-stack
construction.
The permanent magnet stepper (Figure above) only has two
windings, yet has 24-poles in each of two phases. This style
of construction is known as can stack. A phase winding is
wrapped with a mild steel shell, with fingers brought to the
center. One phase, on a transient basis, will have a north
side and a south side. Each side wraps around to the center
of the doughnut with twelve interdigitated fingers for a total
of 24 poles. These alternating north-south fingers will attract
the permanent magnet rotor. If the polarity of the phase
were reversed, the rotor would jump 3609/24 = 15°. We do
not know which direction, which is not useful. However, if we
energize @-1 followed by g-2, the rotor will move 7.5°
because the @-2 is offset (rotated) by 7.5° from @-1. See
below for offset. And, it will rotate in a reproducible direction
if the phases are alternated. Application of any of the above
waveforms will rotate the permanent magnet rotor.
Note that the rotor is a gray ferrite ceramic cylinder
magnetized in the 24-pole pattern shown. This can be
viewed with magnet viewer film or iron filings applied to a
paper wrapping. Though, the colors will be green for both
north and south poles with the film.
Can stack permanent magnet stepper
(a) External view of can stack, (b) field offset detail.
Can-stack style construction of a PM stepper is distinctive
and easy to identify by the stacked “cans”. (Figure above)
Note the rotational offset between the two phase sections.
This is key to making the rotor follow the switching of the
fields between the two phases.
Summary: permanent magnet stepper motor
e The rotor is a permanent magnet, often a ferrite sleeve
magnetized with numerous poles.
e Can-stack construction provides numerous poles from a
single coil with interleaved fingers of soft iron.
e Large to moderate step angle.
e Often used in computer printers to advance paper.
Hybrid stepper motor
The hybrid stepper motor combines features of both the
variable reluctance stepper and the permanent magnet
stepper to produce a smaller step angle. The rotor is a
cylindrical permanent magnet, magnetized along the axis
with radial soft iron teeth (Figure below). The stator coils are
wound on alternating poles with corresponding teeth. There
are typically two winding phases distributed between pole
pairs. This winding may be center tapped for unipolar drive.
The center tap is achieved by a bifilar winding, a pair of
wires wound physically in parallel, but wired in series. The
north-south poles of a phase swap polarity when the phase
drive current is reversed. Bipolar drive is required for un-
tapped windings.
rotor pole detail
S
permanent magnet
rotor, 96-pole
8-pole stator
Hybrid stepper motor.
Note that the 48-teeth on one rotor section are offset by half
a pitch from the other. See rotor pole detail above. This rotor
tooth offset is also shown below. Due to this offset, the rotor
effectively has 96 interleaved poles of opposite polarity. This
offset allows for rotation in 1/96 th of a revolution steps by
reversing the field polarity of one phase. Two phase windings
are common as shown above and below. Though, there could
be as many as five phases.
The stator teeth on the 8-poles correspond to the 48-rotor
teeth, except for missing teeth in the space between the
poles. Thus, one pole of the rotor, say the south pole, may
align with the stator in 48 distinct positions. However, the
teeth of the south pole are offset from the north teeth by
half a tooth. Therefore, the rotor may align with the stator in
96 distinct positions. This half tooth offset shows in the rotor
pole detail above, or Figure below.
As if this were not complicated enough, the stator main
poles are divided into two phases (9-1, @-2). These stator
phases are offset from one another by one-quarter of a
tooth. This detail is only discernable on the schematic
diagrams below. The result is that the rotor moves in steps of
a quarter of a tooth when the phases are alternately
energized. In other words, the rotor moves in 2x96=192
steps per revolution for the above stepper.
The above drawing is representative of an actual hybrid
stepper motor. However, we provide a simplified pictorial
and schematic representation (Figure below) to illustrate
details not obvious above. Note the reduced number of coils
and teeth in rotor and stator for simplicity. In the next two
figures, we attempt to illustrate the quarter tooth rotation
produced by the two stator phases offset by a quarter tooth,
and the rotor half tooth offset. The quarter tooth stator offset
in conjunction with drive current timing also defines
direction of rotation.
1/4 tooth offset
\
alignment stator North
PM South
—_
Hybrid stepper motor schematic diagram.
Features of hybrid stepper schematic (Figure above)
The top of the permanent magnet rotor is the south pole,
the bottom north.
The rotor north-south teeth are offset by half a tooth.
If the @-1 stator is temporarily energized north top,
south bottom.
The top g-1 stator teeth align north to rotor top south
teeth.
The bottom g-1' stator teeth align south to rotor bottom
north teeth.
Enough torque applied to the shaft to overcome the
hold-in torque would move the rotor by one tooth.
If the polarity of -1 were reversed, the rotor would move
by one-half tooth, direction unknown. The alignment
would be south stator top to north rotor bottom, north
stator bottom to south rotor.
The g-2 stator teeth are not aligned with the rotor teeth
when q-1 is energized. In fact, the @-2 stator teeth are
offset by one-quarter tooth. This will allow for rotation by
that amount if @-1 is de-energized and @-2 energized.
Polarity of g-1 and g-2 drive determines direction of
rotation.
eae CEPR
off
off
ae Men
(c)
align top align right align bottom
Hybrid stepper motor rotation sequence.
Hybrid stepper motor rotation (Figure above)
e Rotor top is permanent magnet south, bottom north.
Fields @-1, @-2 are switchable: on, off, reverse.
e (a) g-1=on=north-top, o-2=off. Align (top to
bottom): @-1 stator-N:rotor-top-S, @-1' stator-S: rotor-
bottom-N. Start position, rotation=0.
e (b) g-1=off, o-2=on. Align (right to left): o-2 stator-N-
right:rotor-top-S, @-2' stator-S: rotor-bottom-N. Rotate
1/4 tooth, total rotation=1/4 tooth.
e (c) o-1=reverse(on), o-2=off. Align (bottom to top):
-1 stator-S:rotor-bottom-N, o-1' stator-N:rotor-top-S.
Rotate 1/4 tooth from last position. Total rotation from
start: 1/2 tooth.
e Not shown: g-1=off, @-2=reverse(on). Align (left to
right): Total rotation: 3/4 tooth.
e Not shown: @-1=on, 9-2=off (same as (a)). Align (top
to bottom): Total rotation 1-tooth.
An un-powered stepper motor with detent torque is either a
permanent magnet stepper or a hybrid stepper. The hybrid
stepper will have a small step angle, much less than the 7.5°
of permanent magnet steppers. The step angle could bea
fraction of a degree, corresponding to a few hundred steps
per revolution.
Summary: hybrid stepper motor
e The step angle is smaller than variable reluctance or
permanent magnet steppers.
e The rotor is a permanent magnet with fine teeth. North
and south teeth are offset by half a tooth for a smaller
step angle.
The stator poles have matching fine teeth of the same
pitch as the rotor.
e The stator windings are divided into no less than two
phases.
e The poles of one stator windings are offset by a quarter
tooth for an even smaller step angle.
Brushless DC motor
Brushless DC motors were developed from conventional
brushed DC motors with the availability of solid state power
semiconductors. So, why do we discuss brushless DC motors
in a chapter on AC motors? Brushless DC motors are similar
to AC synchronous motors. The major difference is that
synchronous motors develop a sinusoidal back EMF, as
compared to a rectangular, or trapezoidal, back EMF for
brushless DC motors. Both have stator created rotating
magnetic fields producing torque in a magnetic rotor.
Synchronous motors are usually large multi-kilowatt size,
often with electromagnet rotors. True synchronous motors
are considered to be single speed, a submultiple of the
powerline frequency. Brushless DC motors tend to be small-
a few watts to tens of watts, with permanent magnet rotors.
The speed of a brushless DC motor is not fixed unless driven
by a phased locked loop slaved to a reference frequency. The
style of construction is either cylindrical or pancake. (Figures
and below)
Stator
Cylindrical construction: (a) outside rotor, (b) inside rotor.
The most usual construction, cylindrical, can take on two
forms (Figure above). The most common cylindrical style is
with the rotor on the inside, above right. This style motor is
used in hard disk drives. It is also possible to put the rotor on
the outside surrounding the stator. Such is the case with
brushless DC fan motors, sans the shaft. This style of
construction may be short and fat. However, the direction of
the magnetic flux is radial with respect to the rotational axis.
Pancake motor construction: (a) single stator, (b) double
stator.
High torque pancake motors may have stator coils on both
sides of the rotor (Figure above-b). Lower torque applications
like floppy disk drive motors suffice with a stator coil on one
side of the rotor, (Figure above-a). The direction of the
magnetic flux is axial, that is, parallel to the axis of rotation.
The commutation function may be performed by various
shaft position sensors: optical encoder, magnetic encoder
(resolver, synchro, etc), or Hall effect magnetic sensors.
Small inexpensive motors use Hall effect sensors. (Figure
below) A Hall effect sensor is a semiconductor device where
the electron flow is affected by a magnetic field
perpendicular to the direction of current flow.. It looks like a
four terminal variable resistor network. The voltages at the
two outputs are complementary. Application of a magnetic
field to the sensor causes a small voltage change at the
output. The Hall output may drive a comparator to provide
for more stable drive to the power device. Or, it may drive a
compound transistor stage if properly biased. More modern
Hall effect sensors may contain an integrated amplifier, and
digital circuitry. This 3-lead device may directly drive the
power transistor feeding a phase winding. The sensor must
be mounted close to the permanent magnet rotor to sense
its position.
Hall effect sensors commutate 3-9 brushless DC motor.
The simple cylindrical 3-@ motor Figure above is
commutated by a Hall effect device for each of the three
stator phases. The changing position of the permanent
magnet rotor is sensed by the Hall device as the polarity of
the passing rotor pole changes. This Hall signal is amplified
so that the stator coils are driven with the proper current.
Not shown here, the Hall signals may be processed by
combinatorial logic for more efficient drive waveforms.
The above cylindrical motor could drive a harddrive if it were
equipped with a phased locked loop (PLL) to maintain
constant speed. Similar circuitry could drive the pancake
floppy disk drive motor (Figure below). Again, it would need
a PLL to maintain constant speed.
Brushless pancake motor
The 3-@ pancake motor (Figure above) has 6-stator poles and
8-rotor poles. The rotor is a flat ferrite ring magnetized with
eight axially magnetized alternating poles. We do not show
that the rotor is capped by a mild steel plate for mounting to
the bearing in the middle of the stator. The steel plate also
helps complete the magnetic circuit. The stator poles are
also mounted atop a steel plate, helping to close the
magnetic circuit. The flat stator coils are trapezoidal to more
closely fit the coils, and approximate the rotor poles. The 6-
stator coils comprise three winding phases.
If the three stator phases were successively energized, a
rotating magnetic field would be generated. The permanent
magnet rotor would follow as in the case of a synchronous
motor. A two pole rotor would follow this field at the same
rotation rate as the rotating field. However, our 8-pole rotor
will rotate at a submultiple of this rate due the the extra
poles in the rotor.
The brushless DC fan motor (Figure below) has these
feature:
: 2- brushless fan motor
Brushless fan motor, 2-@.
The stator has 2-phases distributed between 4-poles
e There are 4-salient poles with no windings to eliminate
zero torque points.
e The rotor has four main drive poles.
e The rotor has 8-poles superimposed to help eliminate
zero torque points.
e The Hall effect sensors are spaced at 45° physical.
e The fan housing is placed atop the rotor, which is placed
over the stator.
The goal of a brushless fan motor is to minimize the cost of
manufacture. This is an incentive to move lower
performance products from a 3-@ to a 2-@ configuration.
Depending on how it is driven, it may be called a 4-@ motor.
You may recall that conventional DC motors cannot have an
even number of armature poles (2,4, etc) if they are to be
self-starting, 3,5,7 being common. Thus, it is possible for a
hypothetical 4-pole motor to come to rest at a torque
minima, where it cannot be started from rest. The addition of
the four small salient poles with no windings superimposes a
ripple torque upon the torque vs position curve. When this
ripple torque is added to normal energized-torque curve, the
result is that torque minima are partially removed. This
makes it possible to start the motor for all possible stopping
positions. The addition of eight permanant magnet poles to
the normal 4-pole permanent magnet rotor superimposes a
small second harmonic ripple torque upon the normal 4-pole
ripple torque. This further removes the torque minima. As
long as the torque minima does not drop to zero, we should
be able to start the motor. The more successful we are in
removing the torque minima, the easier the motor starting.
The 2-@ stator requires that the Hall sensors be spaced apart
by 90° electrical. If the rotor was a 2-pole rotor, the Hall
sensors would be placed 90° physical. Since we have a 4-
pole permanent magnet rotor, the sensors must be placed
45° physical to achieve the 90° electrical spacing. Note Hall
Spacing above. The majority of the torque is due to the
interaction of the inside stator 2-@ coils with the 4-pole
section of the rotor. Moreover, the 4-pole section of the rotor
must be on the bottom so that the Hall sensors will sense
the proper commutation signals. The 8-poles rotor section is
only for improving motor starting.
Brushless DC motor 2-9 push-pull drive.
In Figure above, the 2-@ push-pull drive (also known as 4-@
drive) uses two Hall effect sensors to drive four windings.
The sensors are spaced 90° electrical apart, which is 90°
physical for a single pole rotor. Since the Hall sensor has two
complementary outputs, one sensor provides commutation
for two opposing windings.
Tesla polyphase induction motors
Most AC motors are induction motors. Induction motors are
favored due to their ruggedness and simplicity. In fact, 90%
of industrial motors are induction motors.
Nikola Tesla conceived the basic principals of the polyphase
induction motor in 1883, and had a half horsepower (400
watt) model by 1888. Tesla sold the manufacturing rights to
George Westinghouse for $65,000.
Most large ( > 1 hp or 1 kW) industrial motors are poly-
phase induction motors. By poly-phase, we mean that the
stator contains multiple distinct windings per motor pole,
driven by corresponding time shifted sine waves. In practice,
this is two or three phases. Large industrial motors are 3-
phase. While we include numerous illustrations of two-phase
motors for simplicity, we must emphasize that nearly all
poly-phase motors are three-phase. By induction motor, we
mean that the stator windings induce a current flow in the
rotor conductors, like a transformer, unlike a brushed DC
commutator motor.
Construction
An induction motor is composed of a rotor, Known as an
armature, and a stator containing windings connected to a
poly-phase energy source as shown in Figure below. The
simple 2-phase induction motor below is similar to the 1/2
horsepower motor which Nikola Tesla introduced in 1888.
Rotor
Stator
Tesla polyphase induction motor.
The stator in Figure above is wound with pairs of coils
corresponding to the phases of electrical energy available.
The 2-phase induction motor stator above has 2-pairs of
coils, one pair for each of the two phases of AC. The
individual coils of a pair are connected in series and
correspond to the opposite poles of an electromagnet. That
is, one coil corresponds to a N-pole, the other to a S-pole
until the phase of AC changes polarity. The other pair of coils
is oriented 90° in space to the first pair. This pair of coils is
connected to AC shifted in time by 90° in the case of a 2-
phase motor. In Tesla's time, the source of the two phases of
AC was a 2-phase alternator.
The stator in Figure above has sa/ent, obvious protruding
poles, as used on Tesla's early induction motor. This design
is used to this day for sub-fractional horsepower motors
(<50 watts). However, for larger motors less torque
pulsation and higher efficiency results if the coils are
embedded into slots cut into the stator laminations. (Figure
below)
Stator frame showing slots for windings.
The stator laminations are thin insulated rings with slots
punched from sheets of electrical grade steel. A stack of
these is secured by end screws, which may also hold the end
housings.
Stator with (a) 2-9 and (b) 3-9 windings.
In Figure above, the windings for both a two-phase motor
and a three-phase motor have been installed in the stator
slots. The coils are wound on an external fixture, then
worked into the slots. Insulation wedged between the coil
periphery and the slot protects against abrasion.
Actual stator windings are more complex than the single
windings per pole in Figure above. Comparing the 2-@ motor
to Tesla's 2-@ motor with salient poles, the number of coils is
the same. In actual large motors, a pole winding, is divided
into identical coils inserted into many smaller slots than
above. This group is called a phase belt. See Figure below.
The distributed coils of the phase belt cancel some of the
odd harmonics, producing a more sinusoidal magnetic field
distribution across the pole. This is shown in the
synchronous motor section. The slots at the edge of the pole
may have fewer turns than the other slots. Edge slots may
contain windings from two phases. That is, the phase belts
overlap.
n0 op
, phi’ phl
3- distributed winding Single phase belt
The key to the popularity of the AC induction motor is
simplicity as evidenced by the simple rotor (Figure below).
The rotor consists of a shaft, a steel laminated rotor, and an
embedded copper or aluminum squirrel cage, shown at (b)
removed from the rotor. As compared to a DC motor
armature, there is no commutator. This eliminates the
brushes, arcing, sparking, graphite dust, brush adjustment
and replacement, and re-machining of the commutator.
(a)
Laminated rotor with (a) embedded squirrel cage, (b)
conductive cage removed from rotor.
The squirrel cage conductors may be skewed, twisted, with
respsect to the shaft. The misalignment with the stator slots
reduces torque pulsations.
Both rotor and stator cores are composed of a stack of
insulated laminations. The laminations are coated with
insulating oxide or varnish to minimize eddy current losses.
The alloy used in the laminations is selected for low
hysteresis losses.
Theory of operation
A short explanation of operation is that the stator creates a
rotating magnetic field which drags the rotor around.
The theory of operation of induction motors is based ona
rotating magnetic field. One means of creating a rotating
magnetic field is to rotate a permanent magnet as shown in
Figure below. If the moving magnetic lines of flux cuta
conductive disk, it will follow the motion of the magnet. The
lines of flux cutting the conductor will induce a voltage, and
consequent current flow, in the conductive disk. This current
flow creates an electromagnet whose polarity opposes the
motion of the permanent magnet- Lenz's Law. The polarity
of the electromagnet is such that it pulls against the
permanent magnet. The disk follows with a little less speed
than the permanent magnet.
Rotating magnetic field produces torque in conductive disk.
The torque developed by the disk is proportional to the
number of flux lines cutting the disk and the rate at which it
cuts the disk. If the disk were to spin at the same rate as the
permanent magnet, there would be no flux cutting the disk,
no induced current flow, no electromagnet field, no torque.
Thus, the disk speed will always fall behind that of the
rotating permanent magnet, so that lines of flux cut the disk
induce a current, create an electromagnetic field in the disk,
which follows the permanent magnet. If a load is applied to
the disk, slowing it, more torque will be developed as more
lines of flux cut the disk. Torque is proportional to s/ip, the
degree to which the disk falls behind the rotating magnet.
More slip corresponds to more flux cutting the conductive
disk, developing more torque.
An analog automotive eddy current soeedometer is based
on the principle illustrated above. With the disk restrained
by aspring, disk and needle deflection is proportional to
magnet rotation rate.
A rotating magnetic field is created by two coils placed at
right angles to each other, driven by currents which are 90°
out of phase. This should not be surprising if you are familiar
with oscilloscope Lissajous patterns.
OSCILLOSCOPE
vertical
Out of phase (90°) sine waves produce circular Lissajous
pattern.
In Figure above, a circular Lissajous is produced by driving
the horizontal and vertical oscilloscope inputs with 90° out
of phase sine waves. Starting at (a) with maximum “X” and
minimum “Y” deflection, the trace moves up and left toward
(b). Between (a) and (b) the two waveforms are equal to
0.707 V_, at 45°. This point (0.707, 0.707) falls on the radius
of the circle between (a) and (b) The trace moves to (b) with
minimum “X” and maximum “Y” deflection. With maximum
negative “X” and minimum “Y” deflection, the trace moves
to (c). Then with minimum “X” and maximum negative “Y”, it
moves to (d), and on back to (a), completing one cycle.
, horizontal
a deflection
b Y verticall
eflecti
X-axis sine and Y-axis cosine trace circle.
Figure above shows the two 90° phase shifted sine waves
applied to oscilloscope deflection plates which are at right
angles in space. If this were not the case, a one dimensional
line would display. The combination of 90° phased sine
waves and right angle deflection, results in a two
dimensional pattern- a circle. This circle is traced out by a
counterclockwise rotating electron beam.
For reference, Figure belowshows why in-phase sine waves
will not produce a circular pattern. Equal “X” and “Y”
deflection moves the illuminated spot from the origin at (a)
up to right (1,1) at (b), back down left to origin at (c),down
left to (-1.-1) at (d), and back up right to origin. The line is
produced by equal deflections along both axes; y=x is a
Straight line.
OSCILLOSCOPE
No circular motion from in-phase waveforms.
If a pair of 90° out of phase sine waves produces a circular
Lissajous, a similar pair of currents should be able to
produce a circular rotating magnetic field. Such is the case
for a 2-phase motor. By analogy three windings placed 120°
apart in space, and fed with corresponding 120° phased
currents will also produce a rotating magnetic field.
Rotating magnetic field from 90° phased sinewaves.
As the 90° phased sinewaves, Figure above, progress from
points (a) through (d), the magnetic field rotates
counterclockwise (figures a-d) as follows:
(a) o-1 maximum, @-2 zero
(a') o-1 70%, 9-2 70%
(b) m-1 zero, g-2 maximum
(c) g-1 maximum negative, @-2 zero
(d) g-1 zero, g-2 maximum negative
Motor speed
The rotation rate of a stator rotating magnetic field is related
to the number of pole pairs per stator phase. The “full
speed” Figure below has a total of six poles or three pole-
pairs and three phases. However,there is but one pole pair
per phase- the number we need. The magnetic field will
rotate once per sine wave cycle. In the case of 60 Hz power,
the field rotates at 60 times per second or 3600 revolutions
per minute (rpm). For 50 Hz power, it rotates at 50 rotations
per second, or 3000 rpm. The 3600 and 3000 rpm, are the
synchronous speed of the motor. Though the rotor of an
induction motor never achieves this speed, it certainly is an
upper limit. If we double the number of motor poles, the
synchronous speed is cut in half because the magnetic field
rotates 180° in space for 360° of electrical sine wave.
full speed hail ape
Doubling the stator poles halves the synchronous speed.
The synchronous speed is given by:
N, = 120-f/P
N, = synchronous speed in rom
f = frequency of applied power, Hz
P = total number of poles per phase, a multiple of 2
Example:
The “half soeed” Figure above has four poles per phase (3-
phase). The synchronous speed for 50 Hz power is:
S = 120:50/4 = 1500 rpm
The short explanation of the induction motor is that the
rotating magnetic field produced by the stator drags the
rotor around with it.
The longer more correct explanation is that the stator's
magnetic field induces an alternating current into the rotor
squirrel cage conductors which constitutes a transformer
secondary. This induced rotor current in turn creates a
magnetic field. The rotating stator magnetic field interacts
with this rotor field. The rotor field attempts to align with the
rotating stator field. The result is rotation of the squirrel
cage rotor. If there were no mechanical motor torque load,
no bearing, windage, or other losses, the rotor would rotate
at the synchronous speed. However, the s/ip between the
rotor and the synchronous speed stator field develops
torque. It is the magnetic flux cutting the rotor conductors
as it slips which develops torque. Thus, a loaded motor will
slip in proportion to the mechanical load. If the rotor were to
run at synchronous speed, there would be no stator flux
cutting the rotor, no current induced in the rotor, no torque.
Torque
When power is first applied to the motor, the rotor is at rest,
while the stator magnetic field rotates at the synchronous
speed N.. The stator field is cutting the rotor at the
synchronous speed N,.. The current induced in the rotor
shorted turns is maximum, as is the frequency of the
current, the line frequency. As the rotor speeds up, the rate
at which stator flux cuts the rotor is the difference between
synchronous speed N, and actual rotor speed N, or (N, - N).
The ratio of actual flux cutting the rotor to synchronous
speed is defined as s/ip:
S= (N, e N)/N,
where: N, = synchronous speed, N = rotor speed
The frequency of the current induced into the rotor
conductors is only as high as the line frequency at motor
start, decreasing as the rotor approaches synchronous
speed. Rotor frequency is given by:
f= sf
where: s = slip, f = stator power line frequency
Slip at 100% torque is typically 5% or less in induction
motors. Thus for f = 50 Hz line frequency, the frequency of
the induced current in the rotor f, = 0.05-50 = 2.5 Hz. Why
is it so low? The stator magnetic field rotates at 50 Hz. The
rotor speed is 5% less. The rotating magnetic field is only
cutting the rotor at 2.5 Hz. The 2.5 Hz is the difference
between the synchronous speed and the actual rotor speed.
If the rotor spins a little faster, at the synchronous speed, no
flux will cut the rotor at all, f, = 0.
breakdown torque
pullup torque
%full load torque & current
full load torque/ current——~_
locked rotor torque
100 80 60 40 20 0 %Sslip
0 20 40 60 80 100 % Ns
Torque and speed vs %Slip. %N;=%Synchronous Speed.
The Figure above graph shows that starting torque known as
locked rotor torque (LRT) is higher than 100% of the ful/ load
torque (FLT), the safe continuous torque rating. The locked
rotor torque is about 175% of FLT for the example motor
graphed above. Starting current known as /ocked rotor
current (LRC) is 500% of ful/ load current (FLC), the safe
running current. The current is high because this is
analogous to a shorted secondary on a transformer. As the
rotor starts to rotate the torque may decrease a bit for
certain classes of motors to a value known as the pul/ up
torque. This is the lowest value of torque ever encountered
by the starting motor. As the rotor gains 80% of synchronous
speed, torque increases from 175% up to 300% of the full
load torque. This breakdown torque is due to the larger than
normal 20% slip. The current has decreased only slightly at
this point, but will decrease rapidly beyond this point. As the
rotor accelerates to within a few percent of synchronous
speed, both torque and current will decrease substantially.
Slip will be only a few percent during normal operation. For a
running motor, any portion of the torque curve below 100%
rated torque is normal. The motor load determines the
operating point on the torque curve. While the motor torque
and current may exceed 100% for a few seconds during
starting, continuous operation above 100% can damage the
motor. Any motor torque load above the breakdown torque
will stall the motor. The torque, slip, and current will
approach zero for a “no mechanical torque” load condition.
This condition is analogous to an open secondary
transformer.
There are several basic induction motor designs (Figure
below) showing consideable variation from the torque curve
above. The different designs are optimized for starting and
running different types of loads. The locked rotor torque
(LRT) for various motor designs and sizes ranges from 60%
to 350% of full load torque (FLT). Starting current or locked
rotor current (LRC) can range from 500% to 1400% of full
load current (FLC). This current draw can present a starting
problem for large induction motors.
NEMA design classes
Various standard classes (or designs) for motors,
corresponding to the torque curves (Figure below) have
been developed to better drive various type loads. The
National Electrical Manufacturers Association (NEMA) has
specified motor classes A, B, C, and D to meet these drive
requirements. Similar International Electrotechnical
Commission (IEC) classes N and H correspond to NEMA B
and C designs respectively.
400%
wo
3
r=
32
%full load torque
100 80 60 40 20 0 %Sslip
0 20 40 60 80 100 % Ns
Characteristics for NEMA designs.
All motors, except class D, operate at %5 slip or less at full
load.
e Class B (IEC Class N) motors are the default motor to
use in most applications. With a starting torque of LRT =
150% to 170% of FLT, it can start most loads, without
excessive starting current (LRT). Efficiency and power
factor are high. It typically drives pumps, fans, and
machine tools.
Class A starting torque is the same as class B. Drop out
torque and starting current (LRT)are higher. This motor
handles transient overloads as encountered in injection
molding machines.
Class C (IEC Class H) has higher starting torque than
class A and B at LRT = 200% of FLT. This motor is applied
to hard-starting loads which need to be driven at
constant speed like conveyors, crushers, and
reciprocating pumps and compressors.
e Class D motors have the highest starting torque (LRT)
coupled with low starting current due to high slip (5%
to 13% at FLT). The high slip results in lower speed.
Speed regulation is poor. However, the motor excels at
driving highly variable speed loads like those requiring
an energy storage flywheel. Applications include punch
presses, shears, and elevators.
Class E motors are a higher efficiency version of class B.
Class F motors have much lower LRC, LRT, and break
down torque than class B. They drive constant easily
started loads.
Power factor
Induction motors present a lagging (inductive) power factor
to the power line.The power factor in large fully loaded high
speed motors can be as favorable as 90% for large high
speed motors. At 3/4 full load the largest high speed motor
power factor can be 92%. The power factor for small low
speed motors can be as low as 50%. At starting, the power
factor can be in the range of 10% to 25%, rising as the rotor
achieves speed.
Power factor (PF) varies considerably with the motor
mechanical load (Figure below). An unloaded motor is
analogous to a transformer with no resistive load on the
secondary. Little resistance is reflected from the secondary
(rotor) to the primary (stator). Thus the power line sees a
reactive load, as low as 10% PF. As the rotor is loaded an
increasing resistive component is reflected from rotor to
stator, increasing the power factor.
efficience,
0 20 40 60 80 100 % load
Induction motor power factor and efficiency.
Efficiency
Large three phase motors are more efficient than smaller 3-
phase motors, and most all single phase motors. Large
induction motor efficiency can be as high as 95% at full
load, though 90% is more common. Efficiency for a lightly
loaded or no-load induction motor is poor because most of
the current is involved with maintaining magnetizing flux.
As the torque load is increased, more current is consumed in
generating torque, while current associated with
magnetizing remains fixed. Efficiency at 75% FLT can be
slightly higher than that at 100% FLT. Efficiency is decreased
a few percent at 50% FLT, and decreased a few more percent
at 25% FLT. Efficiency only becomes poor below 25% FLT.
The variation of efficiency with loading is shown in Figure
above
Induction motors are typically oversized to guarantee that
their mechanical load can be started and driven under all
operating conditions. If a polyphase motor is loaded at less
than 75% of rated torque where efficiency peaks, efficiency
suffers only slightly down to 25% FLT.
Nola power factor corrector
Frank Nola of NASA proposed a power factor corrector (PFC)
as an energy saving device for single phase induction
motors in the late 1970's. It is based on the premise that a
less than fully loaded induction motor is less efficient and
has a lower power factor than a fully loaded motor. Thus,
there is energy to be saved in partially loaded motors, 1-9
motors in particular. The energy consumed in maintaining
the stator magnetic field is relatively fixed with respect to
load changes. While there is nothing to be saved in a fully
loaded motor, the voltage to a partially loaded motor may
be reduced to decrease the energy required to maintain the
magnetic field. This will increase power factor and efficiency.
This was a good concept for the notoriously inefficient single
phase motors for which it was intended.
This concept is not very applicable to large 3-phase motors.
Because of their high efficiency (90%+), there is not much
energy to be saved. Moreover, a 95% efficient motor is still
94% efficient at 50% full load torque (FLT) and 90% efficient
at 25% FLT. The potential energy savings in going from
100% FLT to 25% FLT is the difference in efficiency 95% -
90% = 5%. This is not 5% of the full load wattage but 5% of
the wattage at the reduced load. The Nola power factor
corrector might be applicable to a 3-phase motor which idles
most of the time (below 25% FLT), like a punch press. The
pay-back period for the expensive electronic controller has
been estimated to be unattractive for most applications.
Though, it might be economical as part of an electronic
motor starter or soeed Control. [7]
Induction motor alternator
An induction motor may function as an alternator if it is
driven by a torque at greater than 100% of the synchronous
speed. (Figure below) This corresponds to a few % of
“negative” slip, say -1% slip. This means that as we are
rotating the motor faster than the synchronous speed, the
rotor is advancing 1% faster than the stator rotating
magnetic field. It normally lags by 1% in a motor. Since the
rotor is cutting the stator magnetic field in the opposite
direction (leading), the rotor induces a voltage into the
stator feeding electrical energy back into the power line.
8
%full load torque & current
i] w +
8 8 8
re) 2 2.
TT
ray
3
Generator mde
-20 +0 40 -80 -100 %sip
200 % Ns
- 100%
-200%
-300%
%full bad torque & current
-400%
-500%
Negative torque makes induction motor into generator.
Such an induction generator must be excited by a “live”
source of 50 or 60 Hz power. No power can be generated in
the event of a power company power failure. This type of
alternator appears to be unsuited as a standby power
source. As an auxiliary power wind turbine generator, it has
the advantage of not requiring an automatic power failure
disconnect switch to protect repair crews. It is fail-safe.
Small remote (from the power grid) installations may be
made self-exciting by placing capacitors in parallel with the
stator phases. If the load is removed residual magnetism
may generate a small amount of current flow. This current is
allowed to flow by the capacitors without dissipating power.
As the generator is brought up to full speed, the current flow
increases to supply a magnetizing current to the stator. The
load may be applied at this point. Voltage regulation is poor.
An induction motor may be converted to a self-excited
generator by the addition of capacitors.[6]
Start up procedure is to bring the wind turbine up to speed
in motor mode by application of normal power line voltage
to the stator. Any wind induced turbine speed in excess of
synchronous speed will develop negative torque, feeding
power back into the power line, reversing the normal
direction of the electric kilowatt-hour meter. Whereas an
induction motor presents a lagging power factor to the
power line, an induction alternator presents a leading power
factor. Induction generators are not widely used in
conventional power plants. The speed of the steam turbine
drive is steady and controllable as required by synchronous
alternators. Synchronous alternators are also more efficient.
The speed of a wind turbine is difficult to control, and
subject to wind speed variation by gusts. An induction
alternator is better able to cope with these variations due to
the inherent slip. This stresses the gear train and
mechanical components less than a synchronous genertor.
However, this allowable speed variation only amounts to
about 1%. Thus, a direct line connected induction generator
is considered to be fixed-speed in a wind turbine. See
Doubly-fed induction generator for a true variable speed
alternator. Multiple generators or multiple windings on a
common shaft may be switched to provide a high and low
speed to accomodate variable wind conditions.
Motor starting and speed control
Some induction motors can draw over 1000% of full load
current during starting; though, a few hundred percent is
more common. Small motors of a few kilowatts or smaller
can be started by direct connection to the power line.
Starting larger motors can cause line voltage sag, affecting
other loads. Motor-start rated circuit breakers (analogous to
slow blow fuses) should replace standard circuit breakers for
starting motors of a few kilowatts. This breaker accepts high
over-current for the duration of starting.
2 S = start, R=run
Autotransformer induction motor starter.
Motors over 50 kW use motor starters to reduce line current
from several hundred to a few hundred percent of full load
current. An intermittent duty autotransformer may reduce
the stator voltage for a fraction of a minute during the start
interval, followed by application of full line voltage as in
Figure above. Closure of the S contacts applies reduced
voltage during the start interval. The S contacts open and
the R contacts close after starting. This reduces starting
current to, say, 200% of full load current. Since the
autotransformer is only used for the short start interval, it
may be sized considerably smaller than a continuous duty
unit.
Running 3-phase motors on 1-phase
Three-phase motors will run on single phase as readily as
single phase motors. The only problem for either motor is
starting. Sometimes 3-phase motors are purchased for use
on single phase if three-phase provisioning is anticipated.
The power rating needs to be 50% larger than for a
comparable single phase motor to make up for one unused
winding. Single phase is applied to a pair of windings
simultanous with a start capacitor in series with the third
winding. The start switch is opened in Figure below upon
motor start. Sometimes a smaller capacitor than the start
Capacitor is retained while running.
R
2. ol 1
O.ynthetic
aaeal ’ 3
hate Pe optional run capacitor ol
S = start, R=ron start capacitor synthetic 3-p standard 3-)p
Starting a three-phase motor on single phase.
The circuit in Figure above for running a three-phase motor
on single phase is known as a Static phase converter if the
motor shaft is not loaded. Moreover, the motor acts as a 3-
phase generator. Three phase power may be tapped off from
the three stator windings for powering other 3-phase
equipment. The capacitor supplies a synthetic phase
approximately midway Z90° between the Z2180° single
phase power source terminals for starting. While running,
the motor generates approximately standard 3-@, as shown
in Figure above. Matt Isserstedt shows a complete design for
powering a home machine shop. [8]
220V
single phase in
L20———"o
Run capacitor = 25-30 UF per HP
Self-starting static phase converter. Run capacitor = 25-
30uF per HP. Adapted from Figure 7, Hanrahan [9]
Since a static phase converter has no torque load, it may be
started with a capacitor considerably smaller than a normal
start capacitor. If it is small enough, it may be left in circuit
aS a run-capacitor. See Figure above. However, smaller run-
Capacitors result in better 3-phase power output as in Figure
below. Moreover, adjustment of these capacitors to equalize
the currents as measured in the three phases results in the
most efficient machine.[9] However, a large start capacitor is
required for about a second to quickly start the converter.
Hanrahan provides construction details.[9]
220V
single phase
in
Start capacitor = 50-100 uF/HP. Run capacitors = 12-16 uF/HP.
More efficient static phase converter. Start capacitor = 50-
100uF/HP. Run capacitors = 12-16uF/HP. Adapted from
Figure 1, Hanrahan [9]
Multiple fields
Induction motors may contain multiple field windings, for
example a 4-pole and an 8-pole winding corresponding to
1800 and 900 rpm synchronous speeds. Energizing one field
or the other is less complex than rewiring the stator coils in
Figure below.
Multiple fields allow speed change.
If the field is segmented with leads brought out, it may be
rewired (or switched) from 4-pole to 2-pole as shown above
for a 2-phase motor. The 22.5° segments are switchable to
45° segments. Only the wiring for one phase is shown above
for clarity. Thus, our induction motor may run at multiple
speeds. When switching the above 60 Hz motor from 4 poles
to 2 poles the synchronous speed increases from 1800 rpm
to 3600 rpm. If the motor is driven by 50 Hz, what would be
the corresponding 4-pole and 2-pole synchronous speeds?
N, = 120f/P = 120*50/4 = 1500 rpm (4-pole)
N, = 3000 rpm (2-pole)
Variable voltage
The speed of small squirrel cage induction motors for
applications such as driving fans, may be changed by
reducing the line voltage. This reduces the torque available
to the load which reduces the speed. (Figure below)
reduced
50% V =—s Spee vai
|
_——————— |
100 80 60 40 20 0 %slip
0 20 40 60 80 100 % Ns
Variable voltage controls induction motor speed.
Electronic speed control
Modern solid state electronics increase the options for speed
control. By changing the 50 or 60 Hz line frequency to
higher or lower values, the synchronous speed of the motor
may be changed. However, decreasing the frequency of the
current fed to the motor also decreases reactance X, which
increases the stator current. This may cause the stator
magnetic circuit to saturate with disastrous results. In
practice, the voltage to the motor needs to be decreased
when frequency is decreased.
Inverter,
variable
frequency
& voltage
AC line
Electronic variable speed drive.
Conversely, the drive frequency may be increased to
increase the synchronous speed of the motor. However, the
voltage needs to be increased to overcome increasing
reactance to keep current up to a normal value and maintain
torque. The inverter (Figure above) approximates sinewaves
to the motor with pulse width modulation outputs. This is a
chopped waveform which is either on or off, high or low, the
percentage of “on” time corresponds to the instantaneous
sine wave voltage.
Once electronics is applied to induction motor control, many
control methods are available, varying from the simple to
complex:
Summary: Speed control
e Scaler Contro/ Low cost method described above to
control only voltage and frequency, without feedback.
e Vector Control! Also known as vector phase control. The
flux and torque producing components of stator current
are measured or estimated on a real-time basis to
enhance the motor torque-speed curve. This is
computation intensive.
Direct Torque Contro/ An elaborate adaptive motor
model allows more direct control of flux and torque
without feedback. This method quickly responds to load
changes.
Summary: Tesla polyphase induction motors
e A polyphase induction motor consists of a polyphase
winding embedded in a laminated stator and a
conductive squirrel cage embedded in a laminated rotor.
e Three phase currents flowing within the stator create a
rotating magnetic field which induces a current, and
consequent magnetic field in the rotor. Rotor torque is
developed as the rotor slips a little behind the rotating
stator field.
e Unlike single phase motors, polyphase induction motors
are se/f-starting.
e Motor starters minimize loading of the power line while
providing a larger starting torque than required during
running. Line current reducing starters are only required
for large motors.
Three phase motors will run on single phase, if started.
A static phase converter is a three phase motor running
on single phase having no shaft load, generating a 3-
phase output.
Multiple field windings can be rewired for multiple
discrete motor speeds by changing the number of poles.
Linear induction motor
The wound stator and the squirrel cage rotor of an induction
motor may be cut at the circumference and unrolled into a
linear induction motor. The direction of linear travel is
controlled by the sequence of the drive to the stator phases.
The linear induction motor has been proposed as a drive for
high speed passenger trains. Up to this time, the linear
induction motor with the accompanying magnetic repulsion
levitation system required for a smooth ride has been too
costly for all but experimental installations. However, the
linear induction motor is scheduled to replace steam driven
catapult aircraft launch systems on the next generation of
naval aircraft carrier, CVNX-1, in 2013. This will increase
efficiency and reduce maintenance.[4] [5]
Wound rotor induction motors
A wound rotor induction motor has a stator like the squirrel
cage induction motor, but a rotor with insulated windings
brought out via slip rings and brushes. However, no power is
applied to the slip rings. Their sole purpose is to allow
resistance to be placed in series with the rotor windings
while starting. (Figure below) This resistance is shorted out
once the motor is started to make the rotor look electrically
like the squirrel cage counterpart.
Stator Rotor Start resistance
ol
Wound rotor induction motor.
Why put resistance in series with the rotor? Squirrel cage
induction motors draw 500% to over 1000% of full load
current (FLC) during starting. While this is not a severe
problem for small motors, it is for large (10's of kW) motors.
Placing resistance in series with the rotor windings not only
decreases start current, locked rotor current (LRC), but also
increases the starting torque, locked rotor torque (LRT).
Figure below shows that by increasing the rotor resistance
from Rg to R; to R>, the breakdown torque peak is shifted
left to zero speed.Note that this torque peak is much higher
than the starting torque available with no rotor resistance
(Ro). Slip is proportional to rotor resistance, and pullout
torque is proportional to slip. Thus, high torque is produced
while starting.
breakdown torque
SEO
%full load torque
100 80 60 40 20 0 %Sslip
0 20 40 60 80 100 % Ns
Breakdown torque peak Is shifted to zero speed by
increasing rotor resistance.
The resistance decreases the torque available at full running
speed. But that resistance is shorted out by the time the
rotor is started. A shorted rotor operates like a squirrel cage
rotor. Heat generated during starting is mostly dissipated
external to the motor in the starting resistance. The
complication and maintenance associated with brushes and
Slip rings is a disadvantage of the wound rotor as compared
to the simple squirrel cage rotor.
This motor is suited for starting high inertial loads. A high
starting resistance makes the high pull out torque available
at zero speed. For comparison, a squirrel cage rotor only
exhibits pull out (peak) torque at 80% of its synchronous
speed.
Speed control
Motor speed may be varied by putting variable resistance
back into the rotor circuit. This reduces rotor current and
speed. The high starting torque available at zero speed, the
down shifted break down torque, is not available at high
speed. See R> curve at 90% Ns, Figure below. Resistors
RoR R2R3 increase in value from zero. A higher resistance at
R3 reduces the speed further. Speed regulation is poor with
respect to changing torque loads. This speed control
technique is only useful over a range of 50% to 100% of full
speed. Speed control works well with variable speed loads
like elevators and printing presses.
Ro, 1, 2. 3 = Motor torque
Ro
speed
reduction
aN
eS eee (a ls
100 80 60 40 20 0 %Sslip
0 20 40 60 80 100 % Ns
%full load torque
Rotor resistance controls speed of wound rotor induction
motor.
Doubly-fed induction generator
We previously described a squirrel cage induction motor
acting like a generator if driven faster than the synchronous
speed. (See Induction motor alternator) This is a singly-fed
induction generator, having electrical connections only to
the stator windings. A wound rotor induction motor may also
act as a generator when driven above the synchronous
speed. Since there are connections to both the stator and
rotor, such a machine is known as a doubly-fed induction
generator (DFIG).
Over-speed 1
torqure Stator 2
C) 70% :
Electric energy
100% : 3
Torque energy
30% Waste heat
Rotor resistance allows over-speed of doubly-fed induction
generator.
The singly-fed induction generator only had a usable slip
range of 1% when driven by troublesome wind torque. Since
the speed of a wound rotor induction motor may be
controlled over a range of 50-100% by inserting resistance
in the rotor, we may expect the same of the doubly-fed
induction generator. Not only can we slow the rotor by 50%,
we can also overspeed it by 50%. That is, we can vary the
speed of a doubly fed induction generator by +50% from the
synchronous speed. In actual practice, +30% is more
practical.
If the generator over-speeds, resistance placed in the rotor
circuit will absorb excess energy while the stator feeds
constant 60 Hz to the power line. (Figure above) In the case
of under-speed, negative resistance inserted into the rotor
circuit can make up the energy deficit, still allowing the
stator to feed the power line with 60 Hz power.
Over-speed ol
torqure Stator 2
; C) 70% |
100% fa aa lal
Torque energy 263 ——>
30% Electric energy
Converter recovers energy from rotor of doubly-fed
induction generator.
In actual practice, the rotor resistance may be replaced by a
converter (Figure above) absorbing power from the rotor,
and feeding power into the power line instead of dissipating
it. This improves the efficiency of the generator.
Under-speed ol
torqure | Stator 2
; C) : 130%
100% ia ——
Torque energy
3 <—
30% Electric energy
Converter borrows energy from power line for rotor of
doubly fed induction generator, allowing it to function well
under synchronous speed.
The converter may “borrow” power from the line for the
under-speed rotor, which passes it on to the stator. (Figure
above) The borrowed power, along with the larger shaft
energy, passes to the stator which is connected to the power
line. The stator appears to be supplying 130% of power to
the line. Keep in mind that the rotor “borrows” 30%, leaving,
leaving the line with 100% for the theoretical lossless DFIG.
Wound rotor induction motor qualities.
Excellent starting torque for high inertia loads.
Low starting current compared to squirrel cage induction
motor.
e Speed is resistance variable over 50% to 100% full
speed.
e Higher maintenance of brushes and slip rings compared
to squirrel cage motor.
e The generator version of the wound rotor machine is
known as a doubly-fed induction generator, a variable
speed machine.
Single-phase induction motors
A three phase motor may be run from a single phase power
source. (Figure below) However, it will not self-start. It may
be hand started in either direction, coming up to speed ina
few seconds. It will only develop 2/3 of the 3-@ power rating
because one winding is not used.
no-start 1-) motor
3- motor. 1-) powered
3-gmotor runs from 1-g power, but does not start.
The single coil of a single phase induction motor does not
produce a rotating magnetic field, but a pulsating field
reaching maximum intensity at 0° and 180° electrical.
(Figure below)
Single phase stator produces a nonrotating, pulsating
magnetic field.
Another view is that the single coil excited by a single phase
Current produces two counter rotating magnetic field
phasors, coinciding twice per revolution at 0° (Figure above-
a) and 180° (figure e). When the phasors rotate to 90° and
-90° they cancel in figure b. At 45° and -45° (figure c) they
are partially additive along the +x axis and cancel along the
y axis. An analogous situation exists in figure d. The sum of
these two phasors is a phasor stationary in space, but
alternating polarity in time. Thus, no starting torque is
developed.
However, if the rotor is rotated forward at a bit less than the
synchronous speed, it will develop maximum torque at 10%
Slip with respect to the forward rotating phasor. Less torque
will be developed above or below 10% slip. The rotor will see
200% - 10% slip with respect to the counter rotating
magnetic field phasor. Little torque (See torque vs slip curve)
other than a double frequency ripple is developed from the
counter rotating phasor. Thus, the single phase coil will
develop torque, once the rotor is started. If the rotor is
started in the reverse direction, it will develop a similar large
torque as it nears the speed of the backward rotating phasor.
Single phase induction motors have a copper or aluminum
squirrel cage embedded in a cylinder of steel laminations,
typical of poly-phase induction motors.
Permanent-split capacitor motor
One way to solve the single phase problem is to build a 2-
phase motor, deriving 2-phase power from single phase. This
requires a motor with two windings spaced apart 90°
electrical, fed with two phases of current displaced 90° in
time. This is called a permanent-split capacitor motor in
Figure below.
~ BO
Permanent-split capacitor induction motor.
This type of motor suffers increased current magnitude and
backward time shift as the motor comes up to speed, with
torque pulsations at full soeed. The solution is to keep the
Capacitor (impedance) small to minimize losses. The losses
are less than for a shaded pole motor. This motor
configuration works well up to 1/4 horsepower (200watt),
though, usually applied to smaller motors. The direction of
the motor is easily reversed by switching the capacitor in
series with the other winding. This type of motor can be
adapted for use as a Servo motor, described elsewhere is this
chapter.
Single phase induction motor with embedded stator coils.
Single phase induction motors may have coils embedded
into the stator as shown in Figure above for larger size
motors. Though, the smaller sizes use less complex to build
concentrated windings with salient poles.
Capacitor-start induction motor
In Figure below a larger capacitor may be used to start a
single phase induction motor via the auxiliary winding if it is
switched out by a centrifugal switch once the motor is up to
speed. Moreover, the auxiliary winding may be many more
turns of heavier wire than used in a resistance split-phase
motor to mitigate excessive temperature rise. The result is
that more starting torque Is available for heavy loads like air
conditioning compressors. This motor configuration works so
well that it is available in multi-horsepower (multi-kilowatt)
sizes.
aS
Capacitor-start induction motor.
Capacitor-run motor induction motor
A variation of the capacitor-start motor (Figure below) is to
start the motor with a relatively large capacitor for high
starting torque, but leave a smaller value capacitor in place
after starting to improve running characteristics while not
drawing excessive current. The additional complexity of the
Capacitor-run motor is justified for larger size motors.
ASE Y
Capacitor-run motor induction motor.
A motor starting capacitor may be a double-anode non-polar
electrolytic capacitor which could be two + to + (or - to -)
series connected polarized electrolytic capacitors. Such AC
rated electrolytic capacitors have such high losses that they
can only be used for intermittent duty (1 second on, 60
seconds off) like motor starting. A capacitor for motor
running must not be of electrolytic construction, but a lower
loss polymer type.
Resistance split-phase induction motor
If an auxiliary winding of much fewer turns of smaller wire is
placed at 90° electrical to the main winding, it can start a
single phase induction motor. (Figure below) With lower
inductance and higher resistance, the current will
experience less phase shift than the main winding. About
30° of phase difference may be obtained. This coil produces
a moderate starting torque, which is disconnected by a
centrifugal switch at 3/4 of synchronous speed. This simple
(no capacitor) arrangement serves well for motors up to 1/3
horsepower (250 watts) driving easily started loads.
Resistance split-phase induction motor.
This motor has more starting torque than a shaded pole
motor (next section), but not as much as a two phase motor
built from the same parts. The current density in the
auxiliary winding is so high during starting that the
consequent rapid temperature rise precludes frequent
restarting or slow starting loads.
Nola power factor corrector
Frank Nola of NASA proposed a power factor corrector for
improving the efficiency of AC induction motors in the mid
1970's. It is based on the premise that induction motors are
inefficient at less than full load. This inefficiency correlates
with a low power factor. The less than unity power factor is
due to magnetizing current required by the stator. This fixed
current is a larger proportion of total motor current as motor
load is decreased. At light load, the full magnetizing current
is not required. It could be reduced by decreasing the
applied voltage, improving the power factor and efficiency.
The power factor corrector senses power factor, and
decreases motor voltage, thus restoring a higher power
factor and decreasing losses.
Since single-phase motors are about 2 to 4 times as
inefficient as three-phase motors, there is potential energy
savings for 1-@ motors. There is no savings for a fully loaded
motor since all the stator magnetizing current is required.
The voltage cannot be reduced. But there is potential
Savings from a less than fully loaded motor. A nominal 117
VAC motor is designed to work at as high as 127 VAC, as low
as 104 VAC. That means that it is not fully loaded when
operated at greater than 104 VAC, for example, a 117 VAC
refrigerator. It is safe for the power factor controller to lower
the line voltage to 104-110 VAC. The higher the initial line
voltage, the greater the potential savings. Of course, if the
power company delivers closer to 110 VAC, the motor will
operate more efficiently without any add-on device.
Any substantially idle, 25% FLC or less, single phase
induction motor is a candidate for a PFC. Though, it needs to
operate a large number of hours per year. And the more time
it idles, as in a lumber saw, punch press, or conveyor, the
greater the possibility of paying for the controller in a few
years operation. It should be easier to pay for it by a factor
of three as compared to the more efficient 3-@-motor. The
cost of a PFC cannot be recovered for a motor operating only
a few hours per day. [7]
Summary: Single-phase induction motors
e Single-phase induction motors are not self-starting
without an auxiliary stator winding driven by an out of
phase current of near 90°. Once started the auxiliary
winding Is optional.
e The auxiliary winding of a permanent-split capacitor
motor has a capacitor in series with it during starting
and running.
e A capacitor-start induction motor only has a capacitor in
series with the auxiliary winding during starting.
e A capacitor-run motor typically has a large non-polarized
electrolytic capacitor in series with the auxiliary winding
for starting, then a smaller non-electrolytic capacitor
during running.
e The auxiliary winding of a resistance split-ophase motor
develops a phase difference versus the main winding
during starting by virtue of the difference in resistance.
Other specialized motors
Shaded pole induction motor
An easy way to provide starting torque to a single phase
motor is to embed a shorted turn in each pole at 30° to 60°
to the main winding. (Figure below) Typically 1/3 of the pole
is enclosed by a bare copper strap. These shading coils
produce a time lagging damped flux spaced 30° to 60° from
the main field. This lagging flux with the undamped main
component, produces a rotating field with a small torque to
start the rotor.
shorting bars
stator coils
Shaded pole induction motor, (a) dual coil design, (b)
smaller single coil version.
Starting torque is so low that shaded pole motors are only
manufactured in smaller sizes, below 50 watts. Low cost and
simplicity suit this motor to small fans, air circulators, and
other low torque applications. Motor speed can be lowered
by switching reactance in series to limit current and torque,
or by switching motor coil taps as in Figure below.
Qs Os u
ph ep
Speed control of shaded pole motor.
_2-phase servo motor
A servo motor is typically part of a feedback loop containing
electronic, mechanical, and electrical components. The
servo loop is a means of controlling the motion of an object
via the motor. A requirement of many such systems is fast
response. To reduce acceleration robbing inertia, the iron
core is removed from the rotor leaving only a shaft mounted
aluminum cup to rotate. (Figure below) The iron core is
reinserted within the cup as a static (non-rotating)
component to complete the magnetic circuit. Otherwise, the
construction is typical of a two phase motor. The low mass
rotor can accelerate more rapidly than a squirrel cage rotor.
WX 9 oI
EO)
—2
| rotor cup
High acceleration 2-g AC servo motor.
iron core
One phase is connected to the single phase line; the other is
driven by an amplifier. One of the windings is driven by a
90° phase shifted waveform. In the above figure, this is
accomplished by a series capacitor in the power line
winding. The other winding Is driven by a variable amplitude
sine wave to control motor speed. The phase of the
waveform may invert (180° phase shift) to reverse the
direction of the motor. This variable sine wave is the output
of an error amplifier. See synchro CT section for example.
Aircraft control surfaces may be positioned by 400 Hz 2-0
servo motors.
_Hysteresis motor
If the low hysteresis Si-steel laminated rotor of an induction
motor is replaced by a slotless windingless cylinder of
hardened magnet steel, hysteresis, or lagging behind of
rotor magnetization, is greatly accentuated. The resulting
low torque synchronous motor develops constant torque
from stall to synchronous speed. Because of the low torque,
the hysteresis motor is only available in very small sizes,
and is only used for constant speed applications like clock
drives, and formerly, phonograph turntables.
Eddy current clutch
If the stator of an induction motor or a synchronous motor is
mounted to rotate independently of the rotor, an eddy
current clutch results. The coils are excited with DC and
attached to the mechanical load. The squirrel cage rotor is
attached to the driving motor. The drive motor is started
with no DC excitation to the clutch. The DC excitation is
adjusted from zero to the desired final value providing a
continuously and smoothly variable torque. The operation of
the eddy current clutch is similar to an analog eddy current
automotive speedometer.
Summary: Other specialized motors
e The shaded pole induction motor, used in under 50 watt
low torque applications, develops a second phase from
shorted turns in the stator.
e Hysteresis motors are a small low torque synchronous
motor once used in clocks and phonographs.
e The eddy current clutch provides an adjustable torque.
Normally, the rotor windings of a wound rotor induction
motor are shorted out after starting. During starting,
resistance may be placed in series with the rotor windings to
limit starting current. If these windings are connected toa
common starting resistance, the two rotors will remain
synchronized during starting. (Figure below) This is useful
for printing presses and draw bridges, where two motors
need to be synchronized during starting. Once started, and
the rotors are shorted, the synchronizing torque is absent.
The higher the resistance during starting, the higher the
synchronizing torque for a pair of motors. If the starting
resistors are removed, but the rotors still paralleled, there is
no starting torque. However there is a substantial
synchronizing torque. This is a se/syn, which is an
abbreviation for “self synchronous”.
ol
3
Ml C) C) M2
2
Stator Rotor Start resistance Rotor Stator
Starting wound rotor induction motors from common
resistors.
The rotors may be stationary. If one rotor is moved through
an angle 9, the other selsyn shaft will move through an
angle 9. If drag is applied to one selsyn, this will be felt
when attempting to rotate the other shaft. While multi-
horsepower (multi-kilowatt) selsyns exist, the main
appplication is small units of a few watts for instrumentation
applications- remote position indication.
M Ww
2 Ye |e
Stator Rotor Rotor Stator
Selsyns without starting resistance.
Instrumentation selsyns have no use for starting resistors.
(Figure above) They are not intended to be self rotating.
Since the rotors are not shorted out nor resistor loaded, no
starting torque is developed. However, manual rotation of
one shaft will produce an unbalance in the rotor currents
until the parallel unit's shaft follows. Note that a common
source of three phase power is applied to both stators.
Though we show three phase rotors above, a single phase
powered rotor is sufficient as shown in Figure below.
Transmitter - receiver
Small instrumentation selsyns, also known as synchros, use
single phase paralleled, AC energized rotors, retaining the 3-
phase paralleled stators, which are not externally energized.
(Figure below) Synchros function as rotary transformers. If
the rotors of both the torque transmitter (TX) and torque
receiver (RX) are at the same angle, the phases of the
induced stator voltages will be identical for both, and no
current will flow. Should one rotor be displaced from the
other, the stator phase voltages will differ between
transmitter and receiver. Stator current will flow developing
torque. The receiver shaft is electrically slaved to the
transmitter shaft. Either the transmitter or receiver shaft
may be rotated to turn the opposite unit.
Stator Rotor Rotor Stator Alternate abreviated symbols
Torque transmitter - TX Torque receiver - RX
Synchros have single phase powered rotors.
Synchro stators are wound with 3-phase windings brought
out to external terminals. The single rotor winding of a
torque transmitter or receiver is brought out by brushed slip
rings. Synchro transmitters and receivers are electrically
identical. However, a synchro receiver has inertial damping
built in. A synchro torque transmitter may be substituted for
a torque receiver.
Remote position sensing is the main synchro application.
(Figure below) For example, a synchro transmitter coupled to
a radar antenna indicates antenna position on an indicator
in a control room. A synchro transmitter coupled toa
weather vane indicates wind direction at a remote console.
Synchros are available for use with 240 Vac 50 Hz, 115 Vac
60 Hz, 115 Vac 400 Hz, and 26 Vac 400 Hz power.
transmitter
receiver
Synchro application: remote position indication.
Differential transmitter - receiver
A synchro differential transmitter (TDX) has both a three
phase rotor and stator. (Figure below) A synchro differential
transmitter adds a shaft angle input to an electrical angle
input on the rotor inputs, outputting the sum on the stator
outputs. This stator electrical angle may be displayed by
sending it to an RX. For example, a synchro receiver displays
the position of a radar antenna relative to a ship's bow. The
addition of a ship's compass heading by a synchro
differential transmitter, displays antenna postion on an RX
relative to true north, regardless of ship's heading.
Reversing the S1-S3 pair of stator leads between a TX and
TDX subtracts angular positions.
7 pL j
Tx *®
( Torque
Differential M2
Ml Transmitter M3
R2
TX
Torque TR
Transmitter Torque
Receiver
Torque differential transmitter (TDX).
A shipboard radar antenna coupled to a synchro transmitter
encodes the antenna angle with respect to ship's bow.
(Figure below) It is desired to display the antenna position
with respect to true north. We need to add the ships heading
from a gyrocompass to the bow-relative antenna position to
display antenna angle with respect to true north. Zantenna
+ Zgyro
differential
~ transmitter
receiver
- Ix
Zrx = Z1x + Zgy
transmitter - tx
gyrocompass - gy
Torque differential transmitter application: angular addition.
Zantenna-N = Zantenna + Zgyro
Zrx = Ztx + Lgy
For example, ship's heading is 230°, antenna position
relative to ship's bow is Z0°, Zantenna-N is:
Zrx = Ztx + Lgy
£30° = £30° + Z0°
Example, ship's heading is 230°, antenna position relative
to ship's bow is 215°, Zantenna-N is:
245° = 230° + 215°
Addition vs subtraction
For reference we show the wiring diagrams for subtraction
and addition of shaft angles using both TDX's (Torque
Differential Transmitter) and TDR's (Torque Differential
Receiver). The TDX has a torque angle input on the shaft, an
electrical angle input on the three stator connections, and
an electrical angle output on the three rotor connections.
The TDR has electrical angle inputs on both the stator and
rotor. The angle output is a torque on the TDR shaft. The
difference between a TDX and a TDR is that the TDX is a
torque transmitter and the TDR a torque receiver.
TDX subtraction: ZTX - ZTDX = ZTR
TDX subtraction.
The torque inputs in Figure above are TX and TDX. The
torque output angular difference is TR.
O) (~) BX OEY CY)
TDX addition: ZTX + ZTDX = ZTR
TDX Addition.
The torque inputs in Figure above are TX and TDX. The
torque output angular sum is TR.
Og (ror | Jie ) | BY stot B87
(ron ca aeene ? eeme a
TDR subtraction: ZTDR = =a - ZTX,
TDR subtraction.
The torque inputs in Figure above are TX, and TX>. The
torque output angular difference is TDR.
TDR addition: ZTDR = pa Ne + ZTX,
TDR addition.
The torque inputs in Figure above are TX, and TX>. The
torque output angular sum is TDR.
Control transformer
A variation of the synchro transmitter is the control
transformer. It has three equally spaced stator windings like
a TX. Its rotor is wound with more turns than a transmitter or
receiver to make it more sensitive at detecting a null as it is
rotated, typically, by a servo system. The CT (Control
Transformer) rotor output is zero when it is oriented at a
angle right angle to the stator magnetic field vector. Unlike
a TX or RX, the CT neither transmits nor receives torque. It is
simply a sensitive angular position detector.
input Ge
Control transformer (CT) detects servo null.
In Figure above, the shaft of the TX is set to the desired
position of the radar antenna. The servo system will cause
the servo motor to drive the antenna to the commanded
position. The CT compares the commanded to actual
position and signals the servo amplifier to drive the motor
until that commanded angle is achieved.
Servo uses CT to sense antenna position null
When the control transformer rotor detects a null at 90° to
the axis of the stator field, there is no rotor output. Any rotor
displacement produces an AC error voltage proportional to
displacement. A servo (Figure above) seeks to minimize the
error between a commanded and measured variable due to
negative feedback. The control transformer compares the
shaft angle to the stator magnetic field angle, sent by the TX
stator. When it measures a minimum, or null, the servo has
driven the antenna and control transformer rotor to the
commanded position. There is no error between measured
and commanded position, no CT, control transformer, output
to be amplified. The servo motor, a 2-phase motor, stops
rotating. However, any CT detected error drives the amplifier
which drives the motor until the error is minimized. This
corresponds to the servo system having driven the antenna
coupled CT to match the angle commanded by the TX.
The servo motor may drive a reduction gear train and be
large compared to the TX and CT synchros. However, the
poor efficiency of AC servo motors limits them to smaller
loads. They are also difficult to control since they are
constant speed devices. However, they can be controlled to
some extent by varying the voltage to one phase with line
voltage on the other phase. Heavy loads are more efficiently
driven by large DC servo motors.
Airborne applications use 400Hz components- TX, CT, and
servo motor. Size and weight of the AC magnetic
components is inversely proportional to frequency.
Therefore, use of 400 Hz components for aircraft
applications, like moving control surfaces, saves size and
weight.
Resolver
A resolver (Figure below) has two stator windings placed at
90° to each other, and a single rotor winding driven by
alternating current. A resolver is used for polar to
rectangular conversion. An angle input at the rotor shaft
produces rectangular co-ordinates sin®8 and cos@
proportional voltages on the stator windings.
C) <——
: ONG
INS
A 1
1 = cosé
Resolver converts shaft angle to sine and cosine of angle.
For example, a black-box within a radar encodes the
distance to a target as a sine wave proportional voltage V,
with the bearing angle as a shaft angle. Convert to X and Y
co-ordinates. The sine wave is fed to the rotor of a resolver.
The bearing angle shaft is coupled to the resolver shaft. The
coordinates (X, Y) are available on the resolver stator coils:
X=V(cos(Zbearing))
Y=V(sin(Zbearing))
The Cartesian coordinates (X, Y) may be plotted on a map
display.
A TX (torque transmitter) may be adapted for service asa
resolver. (Figure below)
vant (6+90°)
=Vcos( (8)
Sl
5 TTX
: sui)
Scott-T | Lol =Vsin(@)
synchro i Y
transformer
Scott-T converts 3-g to 2-9 enabling TX to perform resolver
function.
It is possible to derive resolver-like quadrature angular
components from a synchro transmitter by using a Scott-T
transformer. The three TX outputs, 3-phases, are processed
by a Scott-T transformer into a pair of quadrature
components. See Scott-T chapter 9 for details.
There is also a linear version of the resolver Known as an
inductosyn. The rotary version of the /nductosyn has a finer
resolution than a resolver.
Summary: Selsyn (synchro) motors
A synchro, also known as a se/syn, is a rotary
transformer used to transmit shaft torque.
A TX, torque transmitter, accepts a torque input at its
shaft for transmission on three-phase electrical outputs.
An RX, torque receiver, accepts a three-phase electrical
representation of an angular input for conversion to a
torque output at its shaft. Thus, TX transmits a torque
from an input shaft to a remote RX output shaft.
A TDX, torque differential transmitter, sums an electrical
angle input with a shaft angle input producing an
electrical angle output
A TDR, torque differential receiver, sums two electrical
angle inputs producing a shaft angle output
A CT, control transformer, detects a null when the rotor
is positioned at a right angle to the stator angle input. A
CT is typically a component of a servo- feedback
system.
A Reso/ver outputs a quadrature sin(@) and cos(6)
representation of the shaft angle input instead of a
three-phase output.
The three-phase output of a TX is converted to a resolver
style output by a Scott-T transformer.
AC commutator motors
Charles Proteus Steinmetz's first job after arriving in America
was to investigate problems encountered in the design of
the alternating current version of the brushed commutator
motor. The situation was so bad that motors could not be
designed ahead of the actual construction. The success or
failure of a motor design was not known until after it was
actually built at great expense and tested. He formulated
the laws of magnetic hysteresis in finding a solution.
Hysteresis is a lagging behind of the magnetic field strength
as compared to the magnetizing force. This produces a loss
not present in DC magnetics. Low hysteresis alloys and
breaking the alloy into thin insulated /aminations made it
possible to accurately design AC commutator motors before
building.
AC commutator motors, like comparable DC motors, have
higher starting torque and higher speed than AC induction
motors. The series motor operates well above the
synchronous speed of a conventional AC motor. AC
commutator motors may be either single-phase or poly-
phase. The single-phase AC version suffers a double line
frequency torque pulsation, not present in poly-phase motor.
Since a commutator motor can operate at much higher
speed than an induction motor, it can output more power
than a similar size induction motor. However commutator
motors are not as maintenance free as induction motors, due
to brush and commutator wear.
Single phase series motor
If a DC series motor equipped with a laminated field is
connected to AC, the lagging reactance of the field coil will
considerably reduce the field current. While such a motor
will rotate, operation is marginal. While starting, armature
windings connected to commutator segments shorted by the
brushes look like shorted transformer turns to the field. This
results in considerable arcing and sparking at the brushes as
the armature begins to turn. This is less of a problem as
speed increases, which shares the arcing and sparking
between commutator segments. The lagging reactance and
arcing brushes are only tolerable in very small
uncompensated series AC motors operated at high speed.
Series AC motors smaller than hand drills and kitchen mixers
may be uncompensated. (Figure below)
field
= field
Uncompensated series AC motor.
Compensated series motor
The arcing and sparking is mitigated by placing a
compensating winding in the stator in series with the
armature positioned so that its magnetomotive force (mmf)
cancels out the armature AC mmf. (Figure below) A smaller
motor air gap and fewer field turns reduces lagging
reactance in series with the armature improving the power
factor. All but very small AC commutator motors employ
compensating windings. Motors as large as those employed
in a kitchen mixer, or larger, use compensated stator
windings.
compensating field
winding
Compensated series AC motor.
Universal motor
It is possible to design small (under 300 watts) universal
motors which run from either DC or AC. Very small universal
motors may be uncompensated. Larger higher speed
universal motors use a compensating winding. A motor will
run slower on AC than DC due to the reactance encountered
with AC. However, the peaks of the sine waves saturate the
magnetic path reducing total flux below the DC value,
increasing the speed of the “series” motor. Thus, the
offsetting effects result in a nearly constant speed from DC
to 60 Hz. Small line operated appliances, such as drills,
vacuum cleaners, and mixers, requiring 3000 to 10,000 rom
use universal motors. Though, the development of solid
state rectifiers and inexpensive permanent magnets is
making the DC permanent magnet motor a viable
alternative.
Repulsion motor
A repulsion motor (Figure below) consists of a field directly
connected to the AC line voltage and a pair of shorted
brushes offset by 15°to 25° from the field axis. The field
induces a current flow into the shorted armature whose
magnetic field opposes that of the field coils. Soeed can be
conrolled by rotating the brushes with respect to the field
axis. This motor has superior commutation below
synchronous speed, inferior commutation above
synchronous speed. Low starting current produces high
starting torque.
shorting
brushes
field
compensating
winding
Repulsion AC motor.
Repulsion start induction motor
When an induction motor drives a hard starting load like a
compressor, the high starting torque of the repulsion motor
may be put to use. The induction motor rotor windings are
brought out to commutator segments for starting by a pair
of shorted brushes. At near running speed, a centrifugal
switch shorts out all commutator segments, giving the effect
of a squirrel cage rotor . The brushes may also be lifted to
prolong bush life. Starting torque is 300% to 600% of the
full soeed value as compared to under 200% for a pure
induction motor.
Summary: AC commutator motors
The single phase series motor is an attempt to build a
motor like a DC commutator motor. The resulting motor
iS only practical in the smallest sizes.
The addition of a compensating winding yields the
compensated series motor, overcoming excessive
commutator sparking. Most AC commutator motors are
this type. At high speed this motor provides more power
than a same-size induction motor, but is not
maintenance free.
It is possible to produce small appliance motors powered
by either AC or DC. This is known as a universal motor.
e The AC line is directly connected to the stator of a
repulsion motor with the commutator shorted by the
brushes.
Retractable shorted brushes may start a wound rotor
induction motor. This is known as a repulsion start
induction motor.
Bibliography
1. [1]“American Superconductor achieves full power of
5MW Ship motor”, at www.spacedaily.com/news/energy-
tech-04zzn.html
2.[2]“Linear motor applications guide”, (Aerotech, Inc.,
Pittsburg, PA)
www.aerotech.com/products/PDF/LMAppGuide.pdfopt_tx
t
3. [3]“Linear motor outperforms steam-piston catapults”,
Design News, www.designnews.com/index.asp?
layout=article&articleid=CA151563&cfd=1
4. [4]“Future Aircraft Carrier - CVF, Navy Matters”,
http://navy-matters.beedall.com/cvf3-2.htm
5. [5]Bill Schweber, “Electronics poised to replace steam-
powered aircraft launch system”, EDN, (4/11/2002).
www.edn.com/article/CA207108.html?
pubdate=04%2F11%2F2002
6. [6]“Operating 60 cycle motors as generators”, Red Rock
Energy www.redrok.com/cimtext.pdf
7.[7]“Energy Saver systems for Induction motors”, M
Photonics Ltd, P.O. Box 13 076, Christchurch, New
Zealand at
http://www.Imphotonics.com/energy.htm
8. [8] Matt Isserstedt“Building an Auto-Start Rotary Three
Phase Converter”, May 2008, at
http://www.metalwebnews.com/howto/phase-
converter/phase-converter.html
9. [9]Jim Hanrahan“Building a Phase Converter”, December
1995, at http://www.metalwebnews.com/howto/ph-
conv/ph-conv.html
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] 4]\—
—||+]l—
Lessons In Electric Circuits
-- Volume Il
Chapter 14
TRANSMISSION LINES
e A 50-ohm cable?
Circuits and the speed of light
Characteristic impedance
Finite-length transmission lines
“Long” and “short” transmission lines
Standing waves and resonance
Impedance transformation
Waveguides
A 50-ohm cable?
Early in my explorations of electricity, | came across a length
of coaxial cable with the label “50 ohms” printed along its
outer sheath. (Figure below) Now, coaxial cable is a two-
conductor cable made of a single conductor surrounded by a
braided wire jacket, with a plastic insulating material
separating the two. As such, the outer (braided) conductor
completely surrounds the inner (single wire) conductor, the
two conductors insulated from each other for the entire
length of the cable. This type of cabling is often used to
conduct weak (low-amplitude) voltage signals, due to its
excellent ability to shield such signals from external
interference.
Inner
conductor
| Outer
conductor
Jf (wire braid)
=
Protective jacket
Insulation (polyvinyl chloride)
(polyethylene)
Coaxial cable contruction.
| was mystified by the “50 ohms” label on this coaxial cable.
How could two conductors, insulated from each other by a
relatively thick layer of plastic, have 50 ohms of resistance
between them? Measuring resistance between the outer and
inner conductors with my ohmmeter, | found it to be infinite
(open-circuit), just as | would have expected from two
insulated conductors. Measuring each of the two conductors'
resistances from one end of the cable to the other indicated
nearly zero ohms of resistance: again, exactly what | would
have expected from continuous, unbroken lengths of wire.
Nowhere was | able to measure 50 OQ of resistance on this
cable, regardless of which points | connected my ohmmeter
between.
What | didn't understand at the time was the cable's response
to short-duration voltage “pulses” and high-frequency AC
signals. Continuous direct current (DC) -- such as that used by
my ohmmeter to check the cable's resistance -- shows the two
conductors to be completely insulated from each other, with
nearly infinite resistance between the two. However, due to
the effects of capacitance and inductance distributed along
the length of the cable, the cable's response to rapidly-
changing voltages is such that it acts as a finite impedance,
drawing current proportional to an applied voltage. What we
would normally dismiss as being just a pair of wires becomes
an important circuit element in the presence of transient and
high-frequency AC signals, with characteristic properties all
its own. When expressing such properties, we refer to the wire
pair as a transmission line.
This chapter explores transmission line behavior. Many
transmission line effects do not appear in significant measure
in AC circuits of powerline frequency (50 or 60 Hz), or in
continuous DC circuits, and so we haven't had to concern
ourselves with them in our study of electric circuits thus far.
However, in circuits involving high frequencies and/or
extremely long cable lengths, the effects are very significant.
Practical applications of transmission line effects abound in
radio-frequency (“RF”) communication circuitry, including
computer networks, and in low-frequency circuits subject to
voltage transients (“surges”) such as lightning strikes on
power lines.
Circuits and the speed of light
Suppose we had a simple one-battery, one-lamp circuit
controlled by a switch. When the switch is closed, the lamp
immediately lights. When the switch is opened, the lamp
immediately darkens: (Figure below)
Switch \
Lamp
Battery ——
Lamp appears to immediately respond to switch.
Actually, an incandescent lamp takes a short time for its
filament to warm up and emit light after receiving an electric
current of sufficient magnitude to power it, so the effect is not
instant. However, what I'd like to focus on is the immediacy of
the electric current itself, not the response time of the lamp
filament. For all practical purposes, the effect of switch action
is instant at the lamp's location. Although electrons move
through wires very slowly, the overall effect of electrons
pushing against each other happens at the speed of light
(approximately 186,000 miles per second!).
What would happen, though, if the wires carrying power to
the lamp were 186,000 miles long? Since we know the effects
of electricity do have a finite speed (albeit very fast), a set of
very long wires should introduce a time delay into the circuit,
delaying the switch's action on the lamp: (Figure below)
|~—_______ 186,000 miles —_____ +
Switch \
Lamp
Battery ——
At the speed of light, lamp responds after 1 second.
Assuming no warm-up time for the lamp filament, and no
resistance along the 372,000 mile length of both wires, the
lamp would light up approximately one second after the
switch closure. Although the construction and operation of
superconducting wires 372,000 miles in length would pose
enormous practical problems, it is theoretically possible, and
so this “thought experiment” is valid. When the switch is
opened again, the lamp will continue to receive power for one
second of time after the switch opens, then it will de-energize.
One way of envisioning this is to imagine the electrons within
a conductor as rail cars in a train: linked together with a small
amount of “slack” or “play” in the couplings. When one rail
car (electron) begins to move, it pushes on the one ahead of it
and pulls on the one behind it, but not before the slack is
relieved from the couplings. Thus, motion is transferred from
car to car (from electron to electron) at a maximum velocity
limited by the coupling slack, resulting in a much faster
transfer of motion from the left end of the train (circuit) to the
right end than the actual speed of the cars (electrons):
(Figure below)
First car begins to move
... then the second car moves...
... and then the last car moves!
Motion is transmitted sucessively from one car to next.
Another analogy, perhaps more fitting for the subject of
transmission lines, is that of waves in water. Suppose a flat,
wall-shaped object is suddenly moved horizontally along the
surface of water, so as to produce a wave ahead of it. The
wave will travel as water molecules bump into each other,
transferring wave motion along the water's surface far faster
than the water molecules themselves are actually traveling:
(Figure below)
Object
water surface
7, water molecule
>
a
=
| wave
=>
| Wave
Wave motion in water.
Likewise, electron motion “coupling” travels approximately at
the speed of light, although the electrons themselves don't
move that quickly. In a very long circuit, this “coupling” speed
would become noticeable to a human observer in the form of
a short time delay between switch action and lamp action.
e REVIEW:
e In an electric circuit, the effects of electron motion travel
approximately at the speed of light, although electrons
within the conductors do not travel anywhere near that
velocity.
Characteristic impedance
Suppose, though, that we had a set of parallel wires of infinite
length, with no lamp at the end. What would happen when we
close the switch? Being that there is no longer a load at the
end of the wires, this circuit is open. Would there be no
current at all? (Figure below)
|x-_____—— infinite length
|
Switch \
Battery ——
Driving an infinite transmission line.
Despite being able to avoid wire resistance through the use of
Superconductors in this “thought experiment,” we cannot
eliminate capacitance along the wires' lengths. Any pair of
conductors separated by an insulating medium creates
Capacitance between those conductors: (Figure below)
|~—___—__—_— infinite length —_——_—_—_—___+
Switch \ 1
Battery ——
Equivalent circuit showing stray capacitance between
conductors.
Voltage applied between two conductors creates an electric
field between those conductors. Energy is stored in this
electric field, and this storage of energy results in an
opposition to change in voltage. The reaction of a capacitance
against changes in voltage is described by the equation i =
C(de/dt), which tells us that current will be drawn
proportional to the voltage's rate of change over time. Thus,
when the switch is closed, the capacitance between
conductors will react against the sudden voltage increase by
charging up and drawing current from the source. According
to the equation, an instant rise in applied voltage (as
produced by perfect switch closure) gives rise to an infinite
charging current.
However, the current drawn by a pair of parallel wires will not
be infinite, because there exists series impedance along the
wires due to inductance. (Figure below) Remember that
current through any conductor develops a magnetic field of
proportional magnitude. Energy is stored in this magnetic
field, (Figure below) and this storage of energy results in an
opposition to change in current. Each wire develops a
magnetic field as it carries charging current for the
Capacitance between the wires, and in so doing drops voltage
according to the inductance equation e = L(di/dt). This
voltage drop limits the voltage rate-of-change across the
distributed capacitance, preventing the current from ever
reaching an infinite magnitude:
|——_—_—_———_ infinite length ————————_>
Battery ——
Equivalent circuit showing stray capacitance and inductance.
|-—___———_ infinite length —________- |
Switch =
électric field
Battery —— 3
magnetic field
Voltage charges capacitance, current charges inductance.
Because the electrons in the two wires transfer motion to and
from each other at nearly the speed of light, the “wave front”
of voltage and current change will propagate down the length
of the wires at that same velocity, resulting in the distributed
Capacitance and inductance progressively charging to full
voltage and current, respectively, like this: (Figures below,
below, below, below)
|
Uncharged transmission line.
Switch closes!
COOP OCOR COS COE SH. COC OST
> -
Begin wave propagation.
cececc
SUI SIU SI SI SU SU
COFCO TONING
eS
Continue wave propagation.
JU SUS SUS UUs
Propagate at speed of light.
The end result of these interactions is a constant current of
limited magnitude through the battery source. Since the wires
are infinitely long, their distributed capacitance will never
fully charge to the source voltage, and their distributed
inductance will never allow unlimited charging current. In
other words, this pair of wires will draw current from the
source so long as the switch is closed, behaving as a constant
load. No longer are the wires merely conductors of electrical
current and carriers of voltage, but now constitute a circuit
component in themselves, with unique characteristics. No
longer are the two wires merely a pair of conductors, but
rather a transmission line.
As a constant load, the transmission line's response to applied
voltage is resistive rather than reactive, despite being
comprised purely of inductance and capacitance (assuming
superconducting wires with zero resistance). We can say this
because there is no difference from the battery's perspective
between a resistor eternally dissipating energy and an infinite
transmission line eternally absorbing energy. The impedance
(resistance) of this line in ohms is called the characteristic
impedance, and it is fixed by the geometry of the two
conductors. For a parallel-wire line with air insulation, the
characteristic impedance may be calculated as such:
1 SEIS —
d
= ees -
ie ee
—
—
Where,
Z, = Characteristic impedance of line
d = Distance between conductor centers
r = Conductor radius
k = Relative permittivity of insulation
between conductors
If the transmission line is coaxial in construction, the
characteristic impedance follows a different equation:
Where,
Z, = Characteristic impedance of line
d, = Inside diameter of outer conductor
d, = Outside diameter of inner conductor
k = Relative permittivity of insulation
between conductors
In both equations, identical units of measurement must be
used in both terms of the fraction. If the insulating material is
other than air (or a vacuum), both the characteristic
impedance and the propagation velocity will be affected. The
ratio of a transmission line's true propagation velocity and the
speed of light in a vacuum is called the velocity factor of that
line.
Velocity factor is purely a factor of the insulating material's
relative permittivity (otherwise known as its dielectric
constant), defined as the ratio of a material's electric field
permittivity to that of a pure vacuum. The velocity factor of
any cable type -- coaxial or otherwise -- may be calculated
quite simply by the following formula:
GFE Et eee tee es
— Velocity of wave propagation =
Vv l
Velocity factor = —- = ——
Vk
Where,
v = Velocity of wave propagation
c = Velocity of light in a vacuum
k = Relative permittivity of insulation
between conductors
Characteristic impedance is also Known as natural
impedance, and it refers to the equivalent resistance of a
transmission line if it were infinitely long, owing to distributed
Capacitance and inductance as the voltage and current
“waves” propagate along its length at a propagation velocity
equal to some large fraction of light speed.
It can be seen in either of the first two equations that a
transmission line's characteristic impedance (Zo) increases as
the conductor spacing increases. If the conductors are moved
away from each other, the distributed capacitance will
decrease (greater spacing between capacitor “plates”), and
the distributed inductance will increase (less cancellation of
the two opposing magnetic fields). Less parallel capacitance
and more series inductance results in a smaller current drawn
by the line for any given amount of applied voltage, which by
definition is a greater impedance. Conversely, bringing the
two conductors closer together increases the parallel
Capacitance and decreases the series inductance. Both
changes result in a larger current drawn for a given applied
voltage, equating to a lesser impedance.
Barring any dissipative effects such as dielectric “leakage”
and conductor resistance, the characteristic impedance of a
transmission line is equal to the square root of the ratio of the
line's inductance per unit length divided by the line's
Capacitance per unit length:
Where,
Z, = Characteristic impedance of line
L = Inductance per unit length of line
C = Capacitance per unit length of line
e REVIEW:
e A transmission line is a pair of parallel conductors
exhibiting certain characteristics due to distributed
Capacitance and inductance along its length.
e When a voltage is suddenly applied to one end of a
transmission line, both a voltage “wave” and a current
“wave” propagate along the line at nearly light speed.
e If a DC voltage is applied to one end of an infinitely long
transmission line, the line will draw current from the DC
source as though it were a constant resistance.
¢ The characteristic impedance (Zy) of a transmission line is
the resistance it would exhibit if it were infinite in length.
This is entirely different from leakage resistance of the
dielectric separating the two conductors, and the metallic
resistance of the wires themselves. Characteristic
impedance is purely a function of the capacitance and
inductance distributed along the line's length, and would
exist even if the dielectric were perfect (infinite parallel
resistance) and the wires superconducting (zero series
resistance).
e Velocity factor is a fractional value relating a transmission
line's propagation speed to the speed of light ina
vacuum. Values range between 0.66 and 0.80 for typical
two-wire lines and coaxial cables. For any cable type, it is
equal to the reciprocal (1/x) of the square root of the
relative permittivity of the cable's insulation.
Finite-length transmission lines
A transmission line of infinite length is an interesting
abstraction, but physically impossible. All transmission lines
have some finite length, and as such do not behave precisely
the same as an infinite line. If that piece of 50 O “RG-58/U”
cable | measured with an ohmmeter years ago had been
infinitely long, | actually would have been able to measure 50
Q worth of resistance between the inner and outer
conductors. But it was not infinite in length, and so it
measured as “open” (infinite resistance).
Nonetheless, the characteristic impedance rating of a
transmission line is important even when dealing with limited
lengths. An older term for characteristic impedance, which |
like for its descriptive value, is surge impedance. If a transient
voltage (a “surge”) is applied to the end of a transmission
line, the line will draw a current proportional to the surge
voltage magnitude divided by the line's surge impedance
(I=E/Z). This simple, Ohm's Law relationship between current
and voltage will hold true for a limited period of time, but not
indefinitely.
If the end of a transmission line is open-circuited -- that is, left
unconnected -- the current “wave” propagating down the
line's length will have to stop at the end, since electrons
cannot flow where there is no continuing path. This abrupt
cessation of current at the line's end causes a “pile-up” to
occur along the length of the transmission line, as the
electrons successively find no place to go. Imagine a train
traveling down the track with slack between the rail car
couplings: if the lead car suddenly crashes into an immovable
barricade, it will come to a stop, causing the one behind it to
come to a stop as soon as the first coupling slack is taken up,
which causes the next rail car to stop as soon as the next
coupling's slack is taken up, and so on until the last rail car
stops. The train does not come to a halt together, but rather
in sequence from first car to last: (Figure below)
First car stops
... then the second car stops .. .
. ..and then the last car stops!
Reflected wave.
A signal propagating from the source-end of a transmission
line to the load-end is called an incident wave. The
propagation of a signal from load-end to source-end (such as
what happened in this example with current encountering the
end of an open-circuited transmission line) is called a
reflected wave.
When this electron “pile-up” propagates back to the battery,
current at the battery ceases, and the line acts as a simple
open circuit. All this happens very quickly for transmission
lines of reasonable length, and so an ohmmeter measurement
of the line never reveals the brief time period where the line
actually behaves as a resistor. For a mile-long cable with a
velocity factor of 0.66 (signal propagation velocity is 66% of
light speed, or 122,760 miles per second), it takes only
1/122,760 of a second (8.146 microseconds) for a signal to
travel from one end to the other. For the current signal to
reach the line's end and “reflect” back to the source, the
round-trip time is twice this figure, or 16.292 us.
High-speed measurement instruments are able to detect this
transit time from source to line-end and back to source again,
and may be used for the purpose of determining a cable's
length. This technique may also be used for determining the
presence and location of a break in one or both of the cable's
conductors, since a current will “reflect” off the wire break
just as it will off the end of an open-circuited cable.
Instruments designed for such purposes are called time-
domain reflectometers (TDRs). The basic principle is identical
to that of sonar range-finding: generating a sound pulse and
measuring the time it takes for the echo to return.
A similar phenomenon takes place if the end of a transmission
line is short-circuited: when the voltage wave-front reaches
the end of the line, it is reflected back to the source, because
voltage cannot exist between two electrically common points.
When this reflected wave reaches the source, the source sees
the entire transmission line as a short-circuit. Again, this
happens as quickly as the signal can propagate round-trip
down and up the transmission line at whatever velocity
allowed by the dielectric material between the line's
conductors.
A simple experiment illustrates the phenomenon of wave
reflection in transmission lines. Take a length of rope by one
end and “whip” it with a rapid up-and-down motion of the
wrist. A wave may be seen traveling down the rope's length
until it dissipates entirely due to friction: (Figure below)
= Wave
—
— map Wave
==> Wave
Lossy transmission line.
This is analogous to a long transmission line with internal
loss: the signal steadily grows weaker as it propagates down
the line's length, never reflecting back to the source.
However, if the far end of the rope is secured to a solid object
at a point prior to the incident wave's total dissipation, a
second wave will be reflected back to your hand: (Figure
below)
= Wave
wave ==>
‘ems Wave
Reflected wave.
Usually, the purpose of a transmission line is to convey
electrical energy from one point to another. Even if the
signals are intended for information only, and not to power
some significant load device, the ideal situation would be for
all of the original signal energy to travel from the source to
the load, and then be completely absorbed or dissipated by
the load for maximum signal-to-noise ratio. Thus, “loss” along
the length of a transmission line is undesirable, as are
reflected waves, since reflected energy is energy not
delivered to the end device.
Reflections may be eliminated from the transmission line if
the load's impedance exactly equals the characteristic
(“surge”) impedance of the line. For example, a 50 Q coaxial
cable that is either open-circuited or short-circuited will
reflect all of the incident energy back to the source. However,
ifa50 QO resistor is connected at the end of the cable, there
will be no reflected energy, all signal energy being dissipated
by the resistor.
This makes perfect sense if we return to our hypothetical,
infinite-length transmission line example. A transmission line
of 50 Q characteristic impedance and infinite length behaves
exactly like a 50 Q resistance as measured from one end.
(Figure below) If we cut this line to some finite length, it will
behave as a 50 OQ resistor to a constant source of DC voltage
for a brief time, but then behave like an open- or a short-
circuit, depending on what condition we leave the cut end of
the line: open (Figure below) or shorted. (Figure below)
However, if we terminate the line with a 50 Q resistor, the line
will once again behave as a 50 O resistor, indefinitely: the
same as if it were of infinite length again: (Figure below)
50 Q coaxial cable
Cable’s behavior from perspective of battery:
Exactly like a 50 Q resistor
Infinite transmission line looks like resistor.
SF ET
50 Q coaxial cable
Velocity factor = 0.66
Spach |< 1 mile —_____ +
Battery ——
Cable’s behavior from perspective of battery:
Like a 50 Q resistor for 16.292 ts,
then like an open (infinite resistance)
One mile transmission.
Switch
ae eae
50 2 coaxial cable
Battery —— Velocity factor = 0.66
Cable’s behavior from perspective of battery:
Like a 50 Q resistor for 16.292 us,
then like a short (Zero resistance)
Shorted transmission line.
Switch
Battery ——
50 Q coaxial cable
Velocity factor = 0.66
Cable’s behavior from perspective of battery:
Exactly like a 50 Q resistor
Line terminated in characteristic impedance.
In essence, a terminating resistor matching the natural
impedance of the transmission line makes the line “appear”
infinitely long from the perspective of the source, because a
resistor has the ability to eternally dissipate energy in the
same way a transmission line of infinite length is able to
eternally absorb energy.
Reflected waves will also manifest if the terminating
resistance isn't precisely equal to the characteristic
impedance of the transmission line, not just if the line is left
unconnected (open) or jumpered (shorted). Though the
energy reflection will not be total with a terminating
impedance of slight mismatch, it will be partial. This happens
whether or not the terminating resistance is greater or /ess
than the line's characteristic impedance.
Re-reflections of a reflected wave may also occur at the
source end of a transmission line, if the source's internal
impedance (Thevenin equivalent impedance) is not exactly
equal to the line's characteristic impedance. A reflected wave
returning back to the source will be dissipated entirely if the
source impedance matches the line's, but will be reflected
back toward the line end like another incident wave, at least
partially, if the source impedance does not match the line.
This type of reflection may be particularly troublesome, as it
makes it appear that the source has transmitted another
pulse.
e REVIEW:
e Characteristic impedance is also known as surge
impedance, due to the temporarily resistive behavior of
any length transmission line.
e A finite-length transmission line will appear to a DC
voltage source as a constant resistance for some short
time, then as whatever impedance the line is terminated
with. Therefore, an open-ended cable simply reads “open”
when measured with an ohmmeter, and “shorted” when
its end is short-circuited.
e A transient (“surge”) signal applied to one end of an
open-ended or short-circuited transmission line will
“reflect” off the far end of the line as a secondary wave. A
signal traveling on a transmission line from source to load
is called an incident wave; a signal “bounced” off the end
of a transmission line, traveling from load to source, is
called a reflected wave.
e Reflected waves will also appear in transmission lines
terminated by resistors not precisely matching the
characteristic impedance.
e A finite-length transmission line may be made to appear
infinite in length if terminated by a resistor of equal value
to the line's characteristic impedance. This eliminates all
signal reflections.
e A reflected wave may become re-reflected off the source-
end of a transmission line if the source's internal
impedance does not match the line's characteristic
impedance. This re-reflected wave will appear, of course,
like another pulse signal transmitted from the source.
“Long” and “short” transmission lines
In DC and low-frequency AC circuits, the characteristic
impedance of parallel wires is usually ignored. This includes
the use of coaxial cables in instrument circuits, often
employed to protect weak voltage signals from being
corrupted by induced “noise” caused by stray electric and
magnetic fields. This is due to the relatively short timespans
in which reflections take place in the line, as compared to the
period of the waveforms or pulses of the significant signals in
the circuit. AS we saw in the last section, if a transmission line
is connected to a DC voltage source, it will behave asa
resistor equal in value to the line's characteristic impedance
only for as long as it takes the incident pulse to reach the end
of the line and return as a reflected pulse, back to the source.
After that time (a brief 16.292 us for the mile-long coaxial
cable of the last example), the source “sees” only the
terminating impedance, whatever that may be.
If the circuit in question handles low-frequency AC power,
such short time delays introduced by a transmission line
between when the AC source outputs a voltage peak and
when the source “sees” that peak loaded by the terminating
impedance (round-trip time for the incident wave to reach the
line's end and reflect back to the source) are of little
consequence. Even though we know that signal magnitudes
along the line's length are not equal at any given time due to
signal propagation at (nearly) the speed of light, the actual
phase difference between start-of-line and end-of-line signals
is negligible, because line-length propagations occur within a
very small fraction of the AC waveform's period. For all
practical purposes, we can say that voltage along all
respective points on a low-frequency, two-conductor line are
equal and in-phase with each other at any given point in
time.
In these cases, we can say that the transmission lines in
question are electrically short, because their propagation
effects are much quicker than the periods of the conducted
signals. By contrast, an electrically long line is one where the
propagation time is a large fraction or even a multiple of the
signal period. A “long” line is generally considered to be one
where the source's signal waveform completes at least a
quarter-cycle (90° of “rotation”) before the incident signal
reaches line's end. Up until this chapter in the Lessons In
Electric Circuits book series, all connecting lines were
assumed to be electrically short.
To put this into perspective, we need to express the distance
traveled by a voltage or current signal along a transmission
line in relation to its source frequency. An AC waveform with a
frequency of 60 Hz completes one cycle in 16.66 ms. At light
speed (186,000 mile/s), this equates to a distance of 3100
miles that a voltage or current signal will propagate in that
time. If the velocity factor of the transmission line is less than
1, the propagation velocity will be less than 186,000 miles
per second, and the distance less by the same factor. But
even if we used the coaxial cable's velocity factor from the
last example (0.66), the distance is still a very long 2046
miles! Whatever distance we calculate for a given frequency
is called the wavelength of the signal.
A simple formula for calculating wavelength is as follows:
;
f
Where,
i = Wavelength
v = Velocity of propagation
f = Frequency of signal
The lower-case Greek letter “lambda” (A) represents
wavelength, in whatever unit of length used in the velocity
figure (if miles per second, then wavelength in miles; if
meters per second, then wavelength in meters). Velocity of
propagation is usually the speed of light when calculating
signal wavelength in open air or in a vacuum, but will be less
if the transmission line has a velocity factor less than 1.
If a “long” line is considered to be one at least 1/4 wavelength
in length, you can see why all connecting lines in the circuits
discussed thusfar have been assumed “short.” For a 60 Hz AC
power system, power lines would have to exceed 775 miles in
length before the effects of propagation time became
significant. Cables connecting an audio amplifier to speakers
would have to be over 4.65 miles in length before line
reflections would significantly impact a 10 kHz audio signal!
When dealing with radio-frequency systems, though,
transmission line length is far from trivial. Consider a 100 MHz
radio signal: its wavelength is a mere 9.8202 feet, even at the
full propagation velocity of light (186,000 mile/s). A
transmission line carrying this signal would not have to be
more than about 2-1/2 feet in length to be considered “long!”
With a cable velocity factor of 0.66, this critical length shrinks
to 1.62 feet.
When an electrical source is connected to a load via a “short”
transmission line, the load's impedance dominates the circuit.
This is to say, when the line is short, its own characteristic
impedance is of little consequence to the circuit's behavior.
We see this when testing a coaxial cable with an ohmmeter:
the cable reads “open” from center conductor to outer
conductor if the cable end is left unterminated. Though the
line acts as a resistor for a very brief period of time after the
meter is connected (about 50 QO for an RG-58/U cable), it
immediately thereafter behaves as a simple “open circuit:”
the impedance of the line's open end. Since the combined
response time of an ohmmeter and the human being using it
greatly exceeds the round-trip propagation time up and down
the cable, it is “electrically short” for this application, and we
only register the terminating (load) impedance. It is the
extreme speed of the propagated signal that makes us unable
to detect the cable's 50 Q transient impedance with an
ohmmeter.
If we use a Coaxial cable to conduct a DC voltage or current to
a load, and no component in the circuit is capable of
measuring or responding quickly enough to “notice” a
reflected wave, the cable is considered “electrically short”
and its impedance is irrelevant to circuit function. Note how
the electrical “shortness” of a cable is relative to the
application: in a DC circuit where voltage and current values
change slowly, nearly any physical length of cable would be
considered “short” from the standpoint of characteristic
impedance and reflected waves. Taking the same length of
cable, though, and using it to conduct a high-frequency AC
signal could result in a vastly different assessment of that
cable's “shortness!”
When a source is connected to a load via a “long”
transmission line, the line's own characteristic impedance
dominates over load impedance in determining circuit
behavior. In other words, an electrically “long” line acts as the
principal component in the circuit, its own characteristics
overshadowing the load's. With a source connected to one
end of the cable and a load to the other, current drawn from
the source is a function primarily of the line and not the load.
This is increasingly true the longer the transmission line is.
Consider our hypothetical 50 Q cable of infinite length, surely
the ultimate example of a “long” transmission line: no matter
what kind of load we connect to one end of this line, the
source (connected to the other end) will only see 50 O of
impedance, because the line's infinite length prevents the
signal from ever reaching the end where the load is
connected. In this scenario, line impedance exclusively
defines circuit behavior, rendering the load completely
irrelevant.
The most effective way to minimize the impact of
transmission line length on circuit behavior is to match the
line's characteristic impedance to the load impedance. If the
load impedance is equal to the line impedance, then any
signal source connected to the other end of the line will “
the exact same impedance, and will have the exact same
amount of current drawn from it, regardless of line length. In
this condition of perfect impedance matching, line length
only affects the amount of time delay from signal departure at
the source to signal arrival at the load. However, perfect
matching of line and load impedances is not always practical
or possible.
see”
The next section discusses the effects of “long” transmission
lines, especially when line length happens to match specific
fractions or multiples of signal wavelength.
REVIEW:
Coaxial cabling is sometimes used in DC and low-
frequency AC circuits as well as in high-frequency circuits,
for the excellent immunity to induced “noise” that it
provides for signals.
When the period of a transmitted voltage or current
signal greatly exceeds the propagation time for a
transmission line, the line is considered electrically short.
Conversely, when the propagation time is a large fraction
or multiple of the signal's period, the line is considered
electrically long.
A signal's wavelength is the physical distance it will
propagate in the timespan of one period. Wavelength is
calculated by the formula A=v/f, where “A” is the
wavelength, “v” is the propagation velocity, and “f” is the
signal frequency.
A rule-of-thumb for transmission line “shortness” is that
the line must be at least 1/4 wavelength before it is
considered “long.”
In a circuit with a “short” line, the terminating (load)
impedance dominates circuit behavior. The source
effectively sees nothing but the load's impedance,
barring any resistive losses in the transmission line.
In a circuit with a “long” line, the line's own characteristic
impedance dominates circuit behavior. The ultimate
example of this is a transmission line of infinite length:
since the signal will never reach the load impedance, the
source only “sees” the cable's characteristic impedance.
When a transmission line is terminated by a load
precisely matching its impedance, there are no reflected
waves and thus no problems with line length.
Standing waves and resonance
Whenever there is a mismatch of impedance between
transmission line and load, reflections will occur. If the
incident signal is a continuous AC waveform, these reflections
will mix with more of the oncoming incident waveform to
produce stationary waveforms called standing waves.
The following illustration shows how a triangle-shaped
incident waveform turns into a mirror-image reflection upon
reaching the line's unterminated end. The transmission line in
this illustrative sequence is shown as a single, thick line
rather than a pair of wires, for simplicity's sake. The incident
wave is shown traveling from left to right, while the reflected
wave travels from right to left: (Figure below)
Direction of propagation —>
[ Source |__ urce a a
line
Incident wave”
Reflected wave
Time | Incident wave”
Reflected wave
Incident wave reflects off end of unterminated transmission
line.
If we add the two waveforms together, we find that a third,
stationary waveform is created along the line's length: (Figure
below)
Direction of propagation —>
wa Unterminated
\ y, ~*~ line
Incident wave
Reflected wave
_
Time | Incident wave
Reflected wave
fi nil
Wo
FS
The sum of the incident and reflected waves Is a stationary
wave.
This third, “standing” wave, in fact, represents the only
voltage along the line, being the representative sum of
incident and reflected voltage waves. It oscillates in
instantaneous magnitude, but does not propagate down the
cable's length like the incident or reflected waveforms
causing it. Note the dots along the line length marking the
“Zero” points of the standing wave (where the incident and
reflected waves cancel each other), and how those points
never change position: (Figure below)
Source oars
a nated
Time |
Plt
The standing wave does not propgate along the transmission
line.
Standing waves are quite abundant in the physical world.
Consider a string or rope, shaken at one end, and tied down
at the other (only one half-cycle of hand motion shown,
moving downward): (Figure below)
Standing waves on a rope.
Both the nodes (points of little or no vibration) and the
antinodes (points of maximum vibration) remain fixed along
the length of the string or rope. The effect is most pronounced
when the free end is shaken at just the right frequency.
Plucked strings exhibit the same “standing wave” behavior,
with “nodes” of maximum and minimum vibration along their
length. The major difference between a plucked string and a
shaken string is that the plucked string supplies its own
“correct” frequency of vibration to maximize the standing-
wave effect: (Figure below)
Plucked string
sr
~
Standing waves on a plucked string.
Wind blowing across an open-ended tube also produces
standing waves; this time, the waves are vibrations of air
molecules (sound) within the tube rather than vibrations of a
solid object. Whether the standing wave terminates in a node
(minimum amplitude) or an antinode (maximum amplitude)
depends on whether the other end of the tube is open or
closed: (Figure below)
Standing sound waves in open-ended tubes
1/4 wave 1/2 wave
3/4 wave 1 wave
Standing sound waves in open ended tubes.
A closed tube end must be a wave node, while an open tube
end must be an antinode. By analogy, the anchored end of a
vibrating string must be a node, while the free end (if there is
any) must be an antinode.
Note how there is more than one wavelength suitable for
producing standing waves of vibrating air within a tube that
precisely match the tube's end points. This is true for all
standing-wave systems: standing waves will resonate with the
system for any frequency (wavelength) correlating to the
node/antinode points of the system. Another way of saying
this is that there are multiple resonant frequencies for any
system supporting standing waves.
All higher frequencies are integer-multiples of the lowest
(fundamental) frequency for the system. The sequential
progression of harmonics from one resonant frequency to the
next defines the overtone frequencies for the system: (Figure
below)
1/4 wave 1/2 wave
i i
harmonic harmonic
3/4 wave st
a" a=
harmonic overtone harmonic
th gm rd
harmonic overtone harmonic
th slag ath
harmonic overtone harmonic
gh 4h 5th
harmonic overtone harmonic
Harmonics (overtones) in open ended pipes
The actual frequencies (measured in Hertz) for any of these
harmonics or overtones depends on the physical length of the
tube and the waves' propagation velocity, which is the speed
of sound in air.
Because transmission lines support standing waves, and force
these waves to possess nodes and antinodes according to the
type of termination impedance at the load end, they also
exhibit resonance at frequencies determined by physical
length and propagation velocity. Transmission line resonance,
though, is a bit more complex than resonance of strings or of
air in tubes, because we must consider both voltage waves
and current waves.
This complexity is made easier to understand by way of
computer simulation. To begin, let's examine a perfectly
matched source, transmission line, and load. All components
have an impedance of 75 Q: (Figure below)
Transmission line
(75 Q)
Perfectly matched transmission line.
Using SPICE to simulate the circuit, we'll specify the
transmission line (t1) with a 75 QO characteristic impedance
(z0=75) and a propagation delay of 1 microsecond (td=1u). This
iS a convenient method for expressing the physical length of
a transmission line: the amount of time it takes a wave to
propagate down its entire length. If this were a real 75 O
cable -- perhaps a type “RG-59B/U” coaxial cable, the type
commonly used for cable television distribution -- with a
velocity factor of 0.66, it would be about 648 feet long. Since
1 us is the period of a 1 MHz signal, I'll choose to sweep the
frequency of the AC source from (nearly) zero to that figure,
to see how the system reacts when exposed to signals
ranging from DC to 1 wavelength.
Here is the SPICE netlist for the circuit shown above:
Transmission line
vl 10 ac 1 sin
rsource 1 2 75
tl 2 0 3 0 z0=75 td=1u
rload 3 0 75
.ac Lin 101 1m 1meg
* Using “Nutmeg” program to plot analysis
.end
Running this simulation and plotting the source impedance
drop (as an indication of current), the source voltage, the
line's source-end voltage, and the load voltage, we see that
the source voltage -- shown as vm(1) (voltage magnitude
between node 1 and the implied ground point of node 0) on
the graphic plot -- registers a steady 1 volt, while every other
voltage registers a steady 0.5 volts: (Figure below)
Units — vm¢3> — vmei>
— vm2> — vm¢1,2)
vm¢2>) ss vm(1,2)
No resonances on a matched transmission line.
In a system where all impedances are perfectly matched,
there can be no standing waves, and therefore no resonant
“peaks” or “valleys” in the Bode plot.
Now, let's change the load impedance to 999 MQ, to simulate
an open-ended transmission line. (Figure below) We should
definitely see some reflections on the line now as the
frequency is swept from 1 MHz to 1 MHz: (Figure below)
Transmission line R 999 MQ
(75 Q) cad (open)
Open ended transmission line.
Transmission line
vl 10 ac 1 sin
rsource 1 2 75
tl 2 0 3 0 z0=75 td=1lu
rload 3 0 999meg
.ac Lin 101 1m 1meg
* Using “Nutmeg” program to plot analysis
.end
Units — vm¢3> — vmed>
— vm¢2> — vm¢1,2)
vm¢1)>
Resonances on open transmission line.
Here, both the supply voltage vm(1) and the line's load-end
voltage vm(3) remain steady at 1 volt. The other voltages dip
and peak at different frequencies along the sweep range of 1
mHz to 1 MHz. There are five points of interest along the
horizontal axis of the analysis: O Hz, 250 kHz, 500 kHz, 750
kHz, and 1 MHz. We will investigate each one with regard to
voltage and current at different points of the circuit.
At 0 Hz (actually 1 mHz), the signal is practically DC, and the
circuit behaves much as it would given a 1-volt DC battery
source. There is no circuit current, as indicated by zero
voltage drop across the source impedance (Zeouyrce! vm(1,2)),
and full source voltage present at the source-end of the
transmission line (voltage measured between node 2 and
node 0: vm(2)). (Figure below)
Z source
[ 75. Q Transmission line
1 V1
— (75 Q)
Bou toe
At f=0: input: V=1, |=0; end: V=1, /=0.
At 250 kHz, we see zero voltage and maximum current at the
source-end of the transmission line, yet still full voltage at the
load-end: (Figure below)
1 V = 13.33 mA
Zz
75 Q Transmission line
(75 Q)
At f=250 KHz: input: V=0, 1=13.33 mA; end: V=1 /=0.
source
You might be wondering, how can this be? How can we get
full source voltage at the line's open end while there is zero
voltage at its entrance? The answer is found in the paradox of
the standing wave. With a source frequency of 250 kHz, the
line's length is precisely right for 1/4 wavelength to fit from
end to end. With the line's load end open-circuited, there can
be no current, but there will be voltage. Therefore, the load-
end of an open-circuited transmission line is a current node
(zero point) and a voltage antinode (maximum amplitude):
(Figure below)
Se aaa
z - Maximum E
SoUm@€ PPO
E source
250 kHz
etawsi i. S
-
-
in-
Open end of transmission line shows current node, voltage
antinode at open end.
At 500 kHz, exactly one-half of a standing wave rests on the
transmission line, and here we see another point in the
analysis where the source current drops off to nothing and the
source-end voltage of the transmission line rises again to full
voltage: (Figure below)
Maximum E Maximum E
Full standing wave on half wave open transmission line.
At 750 kHz, the plot looks a lot like it was at 250 kHz: zero
source-end voltage (vm(2)) and maximum current (vm(1,2)).
This is due to 3/4 of a wave poised along the transmission
line, resulting in the source “seeing” a short-circuit where it
connects to the transmission line, even though the other end
of the line is open-circuited: (Figure below)
Maximum E Maximum E
Zero E ZeroE ~*--
“se. =
E source
750 kHz
—_-
Maximum1l __-
---"" Zerol Zero 1
Maximum 1
1 1/2 standing waves on 3/4 wave open transmission line.
When the supply frequency sweeps up to 1 MHz, a full
standing wave exists on the transmission line. At this point,
the source-end of the line experiences the same voltage and
current amplitudes as the load-end: full voltage and zero
current. In essence, the source “Sees” an open circuit at the
point where it connects to the transmission line. (Figure
below)
Maximum E
Zero E
a
E source
| MHz
“"-""" Zerol
Maximum1 Maximum 1
Double standing waves on full wave open transmission line.
In a similar fashion, a short-circuited transmission line
generates standing waves, although the node and antinode
assignments for voltage and current are reversed: at the
shorted end of the line, there will be zero voltage (node) and
maximum current (antinode). What follows is the SPICE
simulation (circuit Figure below and illustrations of what
happens (Figure 2nd-below at resonances) at all the
interesting frequencies: 0 Hz (Figure below) , 250 kHz (Figure
below), 500 kHz (Figure below), 750 kHz (Figure below), and
1 MHz (Figure below). The short-circuit jumper is simulated by
a 1 yO load impedance: (Figure below)
Shorted transmission line.
Transmission line
vl 10 ac 1 sin
rsource 1 2 75
tl 2 0 3 0 z0=75 td=1u
rload 3 0 lu
.ac Lin 101 1m 1meg
* Using “Nutmeg” program to plot analysis
.end
Units — vm¢3> — vmei>
— vm2> — vm(1,2)
Resonances on shorted transmission line
LV = 13.33 mA
Z
source
Transmission line
(75 9)
At f=0 Hz: input: V=0, /=13.33 MA; end: V=0, 1=13.33 MA.
Maximum E
-
=
a
--
enunaseaqee=]=”
FE ource
250 kHz (V)
~
~
~
~
=e
<CMaximumi
Half wave standing wave pattern on 1/4 wave shorted
transmission line.
Maximum E
OE eee E
Full wave standing wave pattern on half wave shorted
transmission line.
: Maximum E
Maximum E Jer E
~ -
- -
~—-=<=—
Maximum 1 Zero |
1 1/2 standing wavepattern on 3/4 wave shorted transmission
line.
Maximum E Maximum E
Zero E Zerok .--.. ZeroE
Pica
E 752
source
1 MHz (V)
Maximum 1 Maximum 1
--“"Zerol —_ _Zerol ~~~
Maximum 1
Double standing waves on full wave shorted transmission
line.
In both these circuit examples, an open-circuited line and a
short-circuited line, the energy reflection is total: 100% of the
incident wave reaching the line's end gets reflected back
toward the source. If, however, the transmission line is
terminated in some impedance other than an open or a short,
the reflections will be less intense, as will be the difference
between minimum and maximum values of voltage and
current along the line.
Suppose we were to terminate our example line with a 100 O
resistor instead of a 75 Q resistor. (Figure below) Examine the
results of the corresponding SPICE analysis to see the effects
of impedance mismatch at different source frequencies:
(Figure below)
Lars tce 2 3
Transmission line
(75 Q)
Foon 1ce (“)
0
Transmission line terminated in a mismatch
Transmission line
vl 10 ac 1 sin
rsource 1 2 75
tl 2 0 3 0 z0=75 td=1lu
rload 3 0 100
.ac Lin 101 1m 1meg
* Using “Nutmeg” program to plot analysis
.end
Units — vm¢3> — vme1>
— vm¢2> — vm¢1,2>
Weak resonances on a mismatched transmission line
If we run another SPICE analysis, this time printing numerical
results rather than plotting them, we can discover exactly
what is happening at all the interesting frequencies: (DC,
Figure below; 250 kHz, Figure below; 500 kHz, Figure below;
750 kHz, Figure below; and 1 MHz, Figure below).
Transmission line
vl 10 ac 1 sin
rsource 1 2 75
tl 2 0 3 0 z0=75 td=1lu
rload 3 0 100
.ac Lin 5 1m 1meg
.print ac v(1,2) v(1) v(2) v(3)
.end
freq v(1,2) v(1) v(2) v(3)
1.000E-03 4.286E-01 1.000E+00 5.714E-01 5.714E-01
2.500E+05 5.714E-01 1.000E+00 4.286E-01 5.714E-01
5.000E+05 4.286E-01 1.000E+00 5.714E-01 5.714E-01
7.500E+05 5.714E-01 1.000E+00 4.286E-01 5.714E-01
1.000E+06 4.286E-01 1.000E+00 5.714E-01 5.714E-01
At all frequencies, the source voltage, v(1), remains steady at
1 volt, as it should. The load voltage, v(3), also remains
steady, but at a lesser voltage: 0.5714 volts. However, both
the line input voltage (v(2)) and the voltage dropped across
the source's 75 O impedance (v(1,2), indicating current
drawn from the source) vary with frequency.
0.4286 V =
5.715 mA
Z
source
Transmission line
Esme = [05714 V (75 Q) 0.5714 V 100 Q
O Hz
At f=0 Hz: input: V=0.57.14, 1=5.715 mA; end: V=0.5714,
/=5.715 MA.
0.5714 V =
7.619 mA
Z source
75 ot eae line
fo4zs6v ] (752 05714 V |
At f=250 KHz: input: V=0.4286, |1=7.619 mA; end: V=0.5714,
/=7.619 MA.
100 Q
a toe
250 kHz
0.4286 V =
5.715 mA
Z source
75 at Transmission line
fos7l4v ] (92 [0.5714 V |
At f=500 KHz: input: V=0.5714, 1=5.715 MA; end: V=5.714,
/=5.715 MA.
100 Q
source
500 kHz
0.5714 V=
7.619 mA
Z source
75 at Transmission line
lo4286v |] (792 [0.5714 V |
At f=750 KHz: input: V=0.4286, 1=7.619 mA; end: V=0.5714,
/=7.619 MA.
100 Q
source
750 kHz
0.4286 V =
5.715 mA
7502 Transmission line
0.5714 V (75 2) 0.5714V
At f=1 MHz: input: V=0.5714, 1=5.715 mA; end: V=0.5714,
/=0.5715 MA.
At odd harmonics of the fundamental frequency (250 kHz,
Figure 3rd-above and 750 KHz, Figure above) we see differing
levels of voltage at each end of the transmission line, because
at those frequencies the standing waves terminate at one end
in a node and at the other end in an antinode. Unlike the
open-circuited and short-circuited transmission line
examples, the maximum and minimum voltage levels along
this transmission line do not reach the same extreme values
of 0% and 100% source voltage, but we still have points of
“minimum” and “maximum” voltage. (Figure 6th-above) The
same holds true for current: if the line's terminating
impedance is mismatched to the line's characteristic
impedance, we will have points of minimum and maximum
Current at certain fixed locations on the line, corresponding to
the standing current wave's nodes and antinodes,
respectively.
One way of expressing the severity of standing waves is asa
ratio of maximum amplitude (antinode) to minimum
amplitude (node), for voltage or for current. When a line is
terminated by an open ora short, this standing wave ratio, or
SWR is valued at infinity, since the minimum amplitude will
be zero, and any finite value divided by zero results in an
infinite (actually, “undefined”) quotient. In this example, with
a75 Q line terminated by a 100 Q impedance, the SWR will
be finite: 1.333, calculated by taking the maximum line
voltage at either 250 kHz or 750 kHz (0.5714 volts) and
dividing by the minimum line voltage (0.4286 volts).
Standing wave ratio may also be calculated by taking the
line's terminating impedance and the line's characteristic
impedance, and dividing the larger of the two values by the
smaller. In this example, the terminating impedance of 100 Q
divided by the characteristic impedance of 75 QO yields a
quotient of exactly 1.333, matching the previous calculation
very closely.
Basi L axi
, maximum maximum
SWR= ———— =
minimum 1 minimum
Z
S WR = load - Lo
Lo Lioad
which ever is greater
A perfectly terminated transmission line will have an SWR of
1, since voltage at any location along the line's length will be
the same, and likewise for current. Again, this is usually
considered ideal, not only because reflected waves constitute
energy not delivered to the load, but because the high values
of voltage and current created by the antinodes of standing
waves may over-stress the transmission line's insulation (high
voltage) and conductors (high current), respectively.
Also, a transmission line with a high SWR tends to act as an
antenna, radiating electromagnetic energy away from the
line, rather than channeling all of it to the load. This is usually
undesirable, as the radiated energy may “couple” with nearby
conductors, producing signal interference. An interesting
footnote to this point is that antenna structures -- which
typically resemble open- or short-circuited transmission lines -
- are often designed to operate at high standing wave ratios,
for the very reason of maximizing signal radiation and
reception.
The following photograph (Figure below) shows a set of
transmission lines at a junction point in a radio transmitter
system. The large, copper tubes with ceramic insulator caps
at the ends are rigid coaxial transmission lines of 50 O
characteristic impedance. These lines carry RF power from the
radio transmitter circuit to a small, wooden shelter at the base
of an antenna structure, and from that shelter on to other
shelters with other antenna structures:
Flexible coaxial cables connected to rigid lines.
Flexible coaxial cable connected to the rigid lines (also of 50
Q characteristic impedance) conduct the RF power to
Capacitive and inductive “phasing” networks inside the
shelter. The white, plastic tube joining two of the rigid lines
together carries “filling” gas from one sealed line to the other.
The lines are gas-filled to avoid collecting moisture inside
them, which would be a definite problem for a coaxial line.
Note the flat, copper “straps” used as jumper wires to connect
the conductors of the flexible coaxial cables to the conductors
of the rigid lines. Why flat straps of copper and not round
wires? Because of the skin effect, which renders most of the
cross-sectional area of a round conductor useless at radio
frequencies.
Like many transmission lines, these are operated at low SWR
conditions. As we will see in the next section, though, the
phenomenon of standing waves in transmission lines is not
always undesirable, as it may be exploited to perform a useful
function: impedance transformation.
e REVIEW:
e Standing waves are waves of voltage and current which
do not propagate (i.e. they are stationary), but are the
result of interference between incident and reflected
waves along a transmission line.
e A node is a point on a standing wave of minimum
amplitude.
e An antinode is a point on a standing wave of maximum
amplitude.
e Standing waves can only exist in a transmission line when
the terminating impedance does not match the line's
characteristic impedance. In a perfectly terminated line,
there are no reflected waves, and therefore no standing
waves at all.
e At certain frequencies, the nodes and antinodes of
standing waves will correlate with the ends of a
transmission line, resulting in resonance.
e The lowest-frequency resonant point on a transmission
line is where the line is one quarter-wavelength long.
Resonant points exist at every harmonic (integer-
multiple) frequency of the fundamental (quarter-
wavelength).
e Standing wave ratio, or SWR, is the ratio of maximum
standing wave amplitude to minimum standing wave
amplitude. It may also be calculated by dividing
termination impedance by characteristic impedance, or
vice versa, which ever yields the greatest quotient. A line
with no standing waves (perfectly matched: Zj53q to Zo)
has an SWR equal to 1.
e Transmission lines may be damaged by the high
maximum amplitudes of standing waves. Voltage
antinodes may break down insulation between
conductors, and current antinodes may overheat
conductors.
Impedance transformation
Standing waves at the resonant frequency points of an open-
or short-circuited transmission line produce unusual effects.
When the signal frequency is such that exactly 1/2 wave or
some multiple thereof matches the line's length, the source
“sees” the load impedance as it is. The following pair of
illustrations shows an open-circuited line operating at 1/2
(Figure below) and 1 wavelength (Figure below) frequencies:
Maximum E Maximum E
— Maximum?
-
~ o
ZO "9 scnnn coun? --" Zerol
Source sees open, same as end of half wavelength line.
Maximum E
“"-""" Zerol
Maximum1 Maximum 1
Source sees open, same as end of full wavelength (2x half
wavelength line).
In either case, the line has voltage antinodes at both ends,
and current nodes at both ends. That is to say, there is
maximum voltage and minimum current at either end of the
line, which corresponds to the condition of an open circuit.
The fact that this condition exists at both ends of the line tells
us that the line faithfully reproduces its terminating
impedance at the source end, so that the source “sees” an
open circuit where it connects to the transmission line, just as
if it were directly open-circuited.
The same is true if the transmission line is terminated by a
short: at signal frequencies corresponding to 1/2 wavelength
(Figure below) or some multiple (Figure below) thereof, the
source “sees” a short circuit, with minimum voltage and
maximum current present at the connection points between
source and transmission line:
Maximum E
ssi oguete—il E
~
-
7 a aie
~ a
source "=
- --
“see =~
E source
500 kHz
Source sees short, same as end of half wave length line.
Maximum E Maximum E
Zero E ZeroE _--.. Zero
Z
source
~~
E 75Q
Miz ©)
Maximum 1 > Maximum 1
- £601 ~ Zefol."~
Maximum 1
-~a—=e,
Source sees short, same as end of full wavelength line (2x
half wavelength).
However, if the signal frequency is such that the line
resonates at 1/4 wavelength or some multiple thereof, the
source will “see” the exact opposite of the termination
impedance. That is, if the line is open-circuited, the source
will “see” a short-circuit at the point where it connects to the
line; and if the line is short-circuited, the source will “see” an
open circuit: (Figure below)
Line open-circuited; source “sees” a short circuit: at
quarter wavelength line (Figure below), at three-quarter
wavelength line (Figure below)
slr amar
Maximum E
Z source
-
-
~aw
~ oe wo
E source
250 kHz
eee 8 |
-
-
--
Source sees short, reflected from open at end of quarter
wavelength line.
Maximum E
Zero E Zero E Maximum E
source
7 eg ae
75.Q
Boa (ce
750 kHz
a~-=-.~
of ~
Maximum 1 _
---"" Zerol Zero 1
Maximum 1
Source sees short, reflected from open at end of three-
quarter wavelength line.
Line short-circuited; source “sees” an open circuit: at
quarter wavelength line (Figure below), at three-quarter
wavelength line (Figure below)
Maximum E
--
Z source
752
Fource (V)
250 kHz
<OMaximums
~
~
a
Source sees open, reflected from short at end of quarter
wavelength line.
: Maximum E
Maximum E Zero E
-
- 2a
> Zero E
E source
750 kHz
-
=e we =
Maximum 1 Zero |
Source sees open, reflected from short at end of three-
quarter wavelength line.
At these frequencies, the transmission line is actually
functioning as an /mpedance transformer, transforming an
infinite impedance into zero impedance, or vice versa. Of
course, this only occurs at resonant points resulting ina
standing wave of 1/4 cycle (the line's fundamental, resonant
frequency) or some odd multiple (3/4, 5/4, 7/4, 9/4 .. .), but if
the signal frequency is known and unchanging, this
phenomenon may be used to match otherwise unmatched
impedances to each other.
Take for instance the example circuit from the last section
where a 75 Q source connects to a 75 Q transmission line,
terminating in a 100 Q load impedance. From the numerical
figures obtained via SPICE, let's determine what impedance
the source “sees” at its end of the transmission line at the
line's resonant frequencies: quarter wavelength (Figure
below), halfwave length (Figure below), three-quarter
wavelength (Figure below) full wavelength (Figure below)
0.5714 V= Fundamental frequency
7.619 mA (1” harmonic)
Z
source
Q Transmission line
0.4286 V (75 Q) 0.5714 V
a (44
Source "sees" 0:-4286V_ _ 56.250
7.619 mA
fs
E source
250 kHz
Source sees 100 Q reflected from 100 Q load at end of
quarter wavelength line.
0.4286 V = d ;
Z
source
Q Transmission line
aD) 0.5714 V
a
0.5714 V
Source "sees" ————— = 100 Q
5.715 mA
42
a toe
500 kHz
Source sees 100 Q reflected from 100 Q load at end of half
wavelength line.
0.5714 V= d .
aieich 3° harmonic
Z,
source
752 Transmission line
0.4286 V (75 Q) 0.5714 V
a 344.
Source "sees" 0:-4286V_ _ 56.25.09
7.619 mA
E source
750 kHz
Source sees 56.25 Q reflected from 100 Q load at end of
three-quarter wavelength line (same as quarter wavelength).
0.4286 V = th .
5.715 mA 4° harmonic
Z source
752 Transmission line
0.5714 V (75 Q) 05714V
-+¥— 14 ——>
0.5714 V
" » d= 1002
Source "sees 5715 mA
source
| MHz
Source sees 56.25 Q reflected from 100 Q load at end of full-
wavelength line (same as half-wavelength).
A simple equation relates line impedance (Z 9), load
impedance (Zj,aq), and input impedance (Zjnput) for an
unmatched transmission line operating at an odd harmonic of
its fundamental frequency:
Z= V Zinput Ziad
One practical application of this principle would be to match
a 300 Q load to a75 QO signal source at a frequency of 50
MHz. All we need to do is calculate the proper transmission
line impedance (Z,9), and length so that exactly 1/4 of a wave
will “stand” on the line at a frequency of 50 MHz.
First, calculating the line impedance: taking the 75 Q we
desire the source to “see” at the source-end of the
transmission line, and multiplying by the 300 Q load
resistance, we obtain a figure of 22,500. Taking the square
root of 22,500 yields 150 QO for a characteristic line
impedance.
Now, to calculate the necessary line length: assuming that
our cable has a velocity factor of 0.85, and using a speed-of-
light figure of 186,000 miles per second, the velocity of
propagation will be 158,100 miles per second. Taking this
velocity and dividing by the signal frequency gives usa
wavelength of 0.003162 miles, or 16.695 feet. Since we only
need one-quarter of this length for the cable to support a
quarter-wave, the requisite cable length is 4.17 38 feet.
Here is a schematic diagram for the circuit, showing node
numbers for the SPICE analysis we're about to run: (Figure
below)
Transmission line
Z, = 1502 =
E source
50 MHz
150 = -/(75)(300)
Quarter wave section of 150 Q transmission line matches 75
Q source to 300 Q load.
We can specify the cable length in SPICE in terms of time
delay from beginning to end. Since the frequency is 50 MHz,
the signal period will be the reciprocal of that, or 20 nano-
seconds (20 ns). One-quarter of that time (5 ns) will be the
time delay of a transmission line one-quarter wavelength
long:
Transmission line
v1 10 ac 1 sin
rsource 1 2 75
tl 2 0 3 0 z0=150 td=5n
rload 3 0 300
.ac Lin 1 50meg 50meg
.print ac v(1,2) v(1) v(2) v(3)
.end
freq v(1,2) v(1) v(2) v(3)
5.Q00E+07 5.000E-01 1.000E+00 5.000E-01 1.000E+00
At a frequency of 50 MHz, our 1-volt signal source drops half
of its voltage across the series 75 Q impedance (v(1,2)) and
the other half of its voltage across the input terminals of the
transmission line (v(2)). This means the source “thinks” it is
powering a75 Q load. The actual load impedance, however,
receives a full 1 volt, as indicated by the 1.000 figure at v(3).
With 0.5 volt dropped across 75 Q, the source is dissipating
3.333 mW of power: the same as dissipated by 1 volt across
the 300 Q load, indicating a perfect match of impedance,
according to the Maximum Power Transfer Theorem. The 1/4-
wavelength, 150 Q, transmission line segment has
successfully matched the 300 O load to the 75 QO source.
Bear in mind, of course, that this only works for 50 MHz and
its odd-numbered harmonics. For any other signal frequency
to receive the same benefit of matched impedances, the 150
Q line would have to lengthened or shortened accordingly so
that it was exactly 1/4 wavelength long.
Strangely enough, the exact same line can also match a 75 O
load to a 300 Q source, demonstrating how this phenomenon
of impedance transformation is fundamentally different in
principle from that of a conventional, two-winding
transformer:
Transmission line
v1 10 ac 1 sin
rsource 1 2 300
tl 2 0 3 0 z0=150 td=5n
rload 3 0 75
.ac Lin 1 50meg 50meg
.print ac v(1,2) v(1) v(2) v(3)
.end
freq v(1,2) v(1) v(2) v(3)
5.0Q00E+07 5.000E-01 1.000E+00 5.000E-01 2.500E-01
Here, we see the 1-volt source voltage equally split between
the 300 Q source impedance (v(1,2)) and the line's input
(v(2)), indicating that the load “appears” as a 300 QO
impedance from the source's perspective where it connects to
the transmission line. This 0.5 volt drop across the source's
300 Q internal impedance yields a power figure of 833.33 UW,
the same as the 0.25 volts across the 75 Q load, as indicated
by voltage figure v(3). Once again, the impedance values of
source and load have been matched by the transmission line
segment.
This technique of impedance matching is often used to match
the differing impedance values of transmission line and
antenna in radio transmitter systems, because the
transmitter's frequency is generally well-known and
unchanging. The use of an impedance “transformer” 1/4
wavelength in length provides impedance matching using the
shortest conductor length possible. (Figure below)
a ee
Dipole
. antenna
Transmitter pag 300 Q
Impedance
"transformer"
Quarter wave 150 Q transmission line section matches 75 Q
line to 300 Q antenna.
e REVIEW:
e A transmission line with standing waves may be used to
match different impedance values if operated at the
correct frequency(ies).
e When operated at a frequency corresponding toa
standing wave of 1/4-wavelength along the transmission
line, the line's characteristic impedance necessary for
impedance transformation must be equal to the square
root of the product of the source's impedance and the
load's impedance.
Waveguides
A waveguide is a special form of transmission line consisting
of a hollow, metal tube. The tube wall provides distributed
inductance, while the empty space between the tube walls
provide distributed capacitance: Figure below
Wave.
propagation
Wave guides conduct microwave energy at lower loss than
coaxial cables.
Waveguides are practical only for signals of extremely high
frequency, where the wavelength approaches the cross-
sectional dimensions of the waveguide. Below such
frequencies, waveguides are useless as electrical transmission
lines.
When functioning as transmission lines, though, waveguides
are considerably simpler than two-conductor cables --
especially coaxial cables -- in their manufacture and
maintenance. With only a single conductor (the waveguide's
“shell”), there are no concerns with proper conductor-to-
conductor spacing, or of the consistency of the dielectric
material, since the only dielectric in a waveguide is air.
Moisture is not as severe a problem in waveguides as it is
within coaxial cables, either, and so waveguides are often
spared the necessity of gas “filling.”
Waveguides may be thought of as conduits for
electromagnetic energy, the waveguide itself acting as
nothing more than a “director” of the energy rather than asa
signal conductor in the normal sense of the word. In a sense,
all transmission lines function as conduits of electromagnetic
energy when transporting pulses or high-frequency waves,
directing the waves as the banks of a river direct a tidal wave.
However, because waveguides are single-conductor elements,
the propagation of electrical energy down a waveguide is of a
very different nature than the propagation of electrical
energy down a two-conductor transmission line.
All electromagnetic waves consist of electric and magnetic
fields propagating in the same direction of travel, but
perpendicular to each other. Along the length of a normal
transmission line, both electric and magnetic fields are
perpendicular (transverse) to the direction of wave travel.
This is Known as the principal mode, or TEM (Transverse
Electric and Magnetic) mode. This mode of wave propagation
can exist only where there are two conductors, and it is the
dominant mode of wave propagation where the cross-
sectional dimensions of the transmission line are small
compared to the wavelength of the signal. (Figure below)
Wave
TEM mode propagation
Magnetic field
Magnetic field
Both field planes perpendicular (transverse) to
direction of signal propagation.
Twin lead transmission line propagation: TEM mode.
At microwave signal frequencies (between 100 MHz and 300
GHz), two-conductor transmission lines of any substantial
length operating in standard TEM mode become impractical.
Lines small enough in cross-sectional dimension to maintain
TEM mode signal propagation for microwave signals tend to
have low voltage ratings, and suffer from large, parasitic
power losses due to conductor “skin” and dielectric effects.
Fortunately, though, at these short wavelengths there exist
other modes of propagation that are not as “lossy,” if a
conductive tube is used rather than two parallel conductors.
It is at these high frequencies that waveguides become
practical.
When an electromagnetic wave propagates down a hollow
tube, only one of the fields -- either electric or magnetic -- will
actually be transverse to the wave's direction of travel. The
other field will “loop” longitudinally to the direction of travel,
but still be perpendicular to the other field. Whichever field
remains transverse to the direction of travel determines
whether the wave propagates in 7E mode (Transverse
Electric) or TM (Transverse Magnetic) mode. (Figure below)
Magnetic
field Magnetic
‘ field
Wave.
Electric propagation
TE mode field TM mode
Magnetic flux lines appear as continuous loops
Electric flux lines appear with beginning and end points
Waveguide (TE) transverse electric and (TM) transverse
magnetic modes.
Many variations of each mode exist for a given waveguide,
and a full discussion of this is subject well beyond the scope
of this book.
Signals are typically introduced to and extracted from
waveguides by means of small antenna-like coupling devices
inserted into the waveguide. Sometimes these coupling
elements take the form of a dipole, which is nothing more
than two open-ended stub wires of appropriate length. Other
times, the coupler is a single stub (a half-dipole, similar in
principle to a “whip” antenna, 1/4A in physical length), ora
short loop of wire terminated on the inside surface of the
waveguide: (Figure below)
Waveguide Waveguide
Coaxial
cable
Coaxial
cable
Stub and loop coupling to waveguide.
In some cases, such as a class of vacuum tube devices called
inductive output tubes (the so-called k/ystron tube falls into
this category), a “cavity” formed of conductive material may
intercept electromagnetic energy from a modulated beam of
electrons, having no contact with the beam itself: (Figure
below below)
The inductive output tube (IOT)
coaxial
output
cable
RF power
|— output
“— toroidal
cavity
Doo
DC supply
Klystron inductive output tube.
Just as transmission lines are able to function as resonant
elements in a circuit, especially when terminated by a short-
circuit or an open-circuit, a dead-ended waveguide may also
resonate at particular frequencies. When used as such, the
device is called a cavity resonator. Inductive output tubes use
toroid-shaped cavity resonators to maximize the power
transfer efficiency between the electron beam and the output
cable.
A cavity's resonant frequency may be altered by changing its
physical dimensions. To this end, cavities with movable
plates, screws, and other mechanical elements for tuning are
manufactured to provide coarse resonant frequency
adjustment.
If a resonant cavity is made open on one end, it functions as a
unidirectional antenna. The following photograph shows a
home-made waveguide formed from a tin can, used as an
antenna for a 2.4 GHz signal in an “802.11b” computer
communication network. The coupling element is a quarter-
wave stub: nothing more than a piece of solid copper wire
about 1-1/4 inches in length extending from the center of a
coaxial cable connector penetrating the side of the can:
(Figure below)
Can-tenna illustrates stub coupling to waveguide.
A few more tin-can antennae may be seen in the background,
one of them a “Pringles” potato chip can. Although this can is
of cardboard (paper) construction, its metallic inner lining
provides the necessary conductivity to function as a
waveguide. Some of the cans in the background still have
their plastic lids in place. The plastic, being nonconductive,
does not interfere with the RF signal, but functions as a
physical barrier to prevent rain, snow, dust, and other
physical contaminants from entering the waveguide. “Real”
waveguide antennae use similar barriers to physically enclose
the tube, yet allow electromagnetic energy to pass
unimpeded.
e REVIEW:
e Waveguides are metal tubes functioning as “conduits” for
carrying electromagnetic waves. They are practical only
for signals of extremely high frequency, where the signal
wavelength approaches the cross-sectional dimensions of
the waveguide.
e Wave propagation through a waveguide may be classified
into two broad categories: 7E (Transverse Electric), or 7M
(Transverse Magnetic), depending on which field (electric
or magnetic) is perpendicular (transverse) to the direction
of wave travel. Wave travel along a standard, two-
conductor transmission line is of the TEM (Transverse
Electric and Magnetic) mode, where both fields are
oriented perpendicular to the direction of travel. TEM
mode is only possible with two conductors and cannot
exist in a waveguide.
e A dead-ended waveguide serving as a resonant element
in a microwave circuit is called a cavity resonator.
e A cavity resonator with an open end functions as a
unidirectional antenna, sending or receiving RF energy
to/from the direction of the open end.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+]l\—
— 4 —
Appendix 1
ABOUT THIS BOOK
Purpose
They say that necessity is the mother of invention. At least
in the case of this book, that adage is true. As an industrial
electronics instructor, | was forced to use a sub-standard
textbook during my first year of teaching. My students were
daily frustrated with the many typographical errors and
obscure explanations in this book, having spent much time
at home struggling to comprehend the material within.
Worse yet were the many incorrect answers in the back of
the book to selected problems. Adding insult to injury was
the $100+ price.
Contacting the publisher proved to be an exercise in futility.
Even though the particular text | was using had been in
print and in popular use for a couple of years, they claimed
my complaint was the first they'd ever heard. My request to
review the draft for the next edition of their book was met
with disinterest on their part, and | resolved to find an
alternative text.
Finding a Suitable alternative was more difficult than | had
imagined. Sure, there were plenty of texts in print, but the
really good books seemed a bit too heavy on the math and
the less intimidating books omitted a lot of information | felt
was important. Some of the best books were out of print, and
those that were still being printed were quite expensive.
It was out of frustration that | compiled Lessons in Electric
Circuits from notes and ideas | had been collecting for years.
My primary goal was to put readable, high-quality
information into the hands of my students, but a secondary
goal was to make the book as affordable as possible. Over
the years, | had experienced the benefit of receiving free
instruction and encouragement in my pursuit of learning
electronics from many people, including several teachers of
mine in elementary and high school. Their selfless
assistance played a key role in my own studies, paving the
way for a rewarding career and fascinating hobby. If only |
could extend the gift of their help by giving to other people
what they gavetome...
So, | decided to make the book freely available. More than
that, | decided to make it “open” following the same
development model used in the making of free software
(most notably the various UNIX utilities released by the Free
Software Foundation, and the Linux operating system,
whose fame Is growing even as | write). The goal was to
copyright the text -- so as to protect my authorship -- but
expressly allow anyone to distribute and/or modify the text
to suit their own needs with a minimum of legal
encumbrance. This willful and formal revoking of standard
distribution limitations under copyright is whimsically
termed copyleft. Anyone can “copyleft” their creative work
simply by appending a notice to that effect on their work,
but several Licenses already exist, covering the fine legal
points in great detail.
The first such License | applied to my work was the GPL --
General Public License -- of the Free Software Foundation
(GNU). The GPL, however, is intended to copyleft works of
computer software, and although its introductory language
is broad enough to cover works of text, its wording is not as
clear as it could be for that application. When other, less
specific copyleft Licenses began appearing within the free
software community, | chose one of them (the Design
Science License, or DSL) as the official notice for my project.
In “copylefting” this text, | guaranteed that no instructor
would be limited by a text insufficient for their needs, as |
had been with error-ridden textbooks from major publishers.
I'm sure this book in its initial form will not satisfy everyone,
but anyone has the freedom to change it, leveraging my
efforts to suit variant and individual requirements. For the
beginning student of electronics, learn what you can from
this book, editing it as you feel necessary if you come across
a useful piece of information. Then, if you pass it on to
someone else, you will be giving them something better
than what you received. For the instructor or electronics
professional, feel free to use this as a reference manual,
adding or editing to your heart's content. The only “catch” is
this: if you plan to distribute your modified version of this
text, you must give credit where credit is due (to me, the
Original author, and anyone else whose modifications are
contained in your version), and you must ensure that
whoever you give the text to is aware of their freedom to
similarly share and edit the text. The next chapter covers
this process in more detail.
It must be mentioned that although | strive to maintain
technical accuracy in all of this book's content, the subject
matter is broad and harbors many potential dangers.
Electricity maims and kills without provocation, and
deserves the utmost respect. | strongly encourage
experimentation on the part of the reader, but only with
circuits powered by small batteries where there is no risk of
electric shock, fire, explosion, etc. High-power electric
circuits should be left to the care of trained professionals!
The Design Science License clearly states that neither | nor
any contributors to this book bear any liability for what is
done with its contents.
The use of SPICE
One of the best ways to learn how things work is to follow
the inductive approach: to observe specific instances of
things working and derive general conclusions from those
observations. In science education, labwork is the
traditionally accepted venue for this type of learning,
although in many cases labs are designed by educators to
reinforce principles previously learned through lecture or
textbook reading, rather than to allow the student to learn
on their own through a truly exploratory process.
Having taught myself most of the electronics that | know, |
appreciate the sense of frustration students may have in
teaching themselves from books. Although electronic
components are typically inexpensive, not everyone has the
means or opportunity to set up a laboratory in their own
homes, and when things go wrong there's no one to ask for
help. Most textbooks seem to approach the task of education
from a deductive perspective: tell the student how things
are supposed to work, then apply those principles to specific
instances that the student may or may not be able to
explore by themselves. The inductive approach, as useful as
it is, is hard to find in the pages of a book.
However, textbooks don't have to be this way. | discovered
this when | started to learn a computer program called
SPICE. It is a text-based piece of software intended to model
circuits and provide analyses of voltage, current, frequency,
etc. Although nothing is quite as good as building real
circuits to gain knowledge in electronics, computer
simulation is an excellent alternative. In learning how to use
this powerful tool, | made a discovery: SPICE could be used
within a textbook to present circuit simulations to allow
students to “observe”the phenomena for themselves. This
way, the readers could learn the concepts inductively (by
interpreting SPICE's output) as well as deductively (by
interpreting my explanations). Furthermore, in seeing SPICE
used over and over again, they should be able to
understand how to use it themselves, providing a perfectly
safe means of experimentation on their own computers with
circuit simulations of their own design.
Another advantage to including computer analyses in a
textbook is the empirical verification it adds to the concepts
presented. Without demonstrations, the reader is left to take
the author's statements on faith, trusting that what has
been written is indeed accurate. The problem with faith, of
course, is that it is only as good as the authority in which it
is placed and the accuracy of interpretation through which it
is understood. Authors, like all human beings, are liable to
err and/or communicate poorly. With demonstrations,
however, the reader can immediately see for themselves
that what the author describes is indeed true.
Demonstrations also serve to clarify the meaning of the text
with concrete examples.
SPICE is introduced early in volume | (DC) of this book
series, and hopefully in a gentle enough way that it doesn't
create confusion. For those wishing to learn more, a chapter
in the Reference volume (volume V) contains an overview of
SPICE with many example circuits. There may be more flashy
(graphic) circuit simulation programs in existence, but SPICE
is free, a virtue complementing the charitable philosophy of
this book very nicely.
Acknowledgements
First, | wish to thank my wife, whose patience during those
many and long evenings (and weekends!) of typing has
been extraordinary.
| also wish to thank those whose open-source software
development efforts have made this endeavor all the more
affordable and pleasurable. The following is a list of various
free computer software used to make this book, and the
respective programmers:
e GNU/Linux Operating System -- Linus Torvalds, Richard
Stallman, and a host of others too numerous to mention.
Vim text editor -- Bram Moolenaar and others.
Xcircuit drafting program -- Tim Edwards.
SPICE circuit simulation program -- too many
contributors to mention.
e Nutmeg post-processor program for SPICE -- Wayne
Christopher.
e T-X text processing system -- Donald Knuth and others.
e Texinfo document formatting system -- Free Software
Foundation.
¢ LATEX document formatting system -- Leslie Lamport and
others.
e Gimp image manipulation program -- too many
contributors to mention.
e Winscope signal analysis software -- Dr. Constantin
Zeldovich. (Free for personal and academic use.)
Appreciation is also extended to Robert L. Boylestad, whose
first edition of Introductory Circuit Analysis taught me more
about electric circuits than any other book. Other important
texts in my electronics studies include the 1939 edition of
The “Radio” Handbook, Bernard Grob's second edition of
Introduction to Electronics I, and Forrest Mims' original
Engineer's Notebook.
Thanks to the staff of the Bellingham Antique Radio
Museum, who were generous enough to let me terrorize their
establishment with my camera and flash unit. Similar thanks
to Jim Swartos and KARI radio in Blaine, Washington for a
very informative tour of their expanded (50 kW) facilities as
well as their vintage transmitter equipment.
| wish to specifically thank Jeffrey Elkner and all those at
Yorktown High School for being willing to host my book as
part of their Open Book Project, and to make the first effort
in contributing to its form and content. Thanks also to David
Sweet (website: [*]) and Ben Crowell (website: [*]) for
providing encouragement, constructive criticism, and a
wider audience for the online version of this book.
Thanks to Michael Stutz for drafting his Design Science
License, and to Richard Stallman for pioneering the concept
of copyleft.
Last but certainly not least, many thanks to my parents and
those teachers of mine who saw in me a desire to learn
about electricity, and who kindled that flame into a passion
for discovery and intellectual adventure. | honor you by
helping others as you have helped me.
Tony Kuphaldt, April 2002
“A candle loses nothing of its light when lighting
another”
Kahlil Gibran
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—|/]|+4]\l\—
—| | +]
Appendix 2
CONTRIBUTOR LIST
How to contribute to this book
As a copylefted work, this book is open to revision and expansion by
any interested parties. The only “catch” is that credit must be given
where credit is due. This /s a copyrighted work: it is notin the public
domain!
If you wish to cite portions of this book in a work of your own, you
must follow the same guidelines as for any other copyrighted work.
Here is a Sample from the Design Science License:
The Work is copyright the Author. All rights to the Work are reserved
by the Author, except as specifically described below. This License
describes the terms and conditions under which the Author permits you
to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by “fair use”) quote from it, just as you would any
copyrighted material under copyright Law.
Your right to operate, perform, read or otherwise interpret and/or
execute the Work is unrestricted; however, you do so at your own risk,
because the Work comes WITHOUT ANY WARRANTY -- see Section 7 (“NO
WARRANTY” ) below.
If you wish to modify this book in any way, you must document the
nature of those modifications in the “Credits” section along with your
name, and ideally, information concerning how you may be
contacted. Again, the Design Science License:
Permission is granted to modify or sample from a copy of the Work,
producing a derivative work, and to distribute the derivative work
under the terms described in the section for distribution above,
provided that the following terms are met:
(a) The new, derivative work is published under the terms of this
License.
(b) The derivative work is given a new name, so that its name or
title can not be confused with the Work, or with a version of
the Work, in any way.
(c) Appropriate authorship credit is given: for the differences
between the Work and the new derivative work, authorship is
attributed to you, while the material sampled or used from
the Work remains attributed to the original Author; appropriate
notice must be included with the new work indicating the nature
and the dates of any modifications of the Work made by you.
Given the complexities and security issues surrounding the
maintenance of files comprising this book, it is recommended that
you submit any revisions or expansions to the original author (Tony R.
Kuphaldt). You are, of course, welcome to modify this book directly by
editing your own personal copy, but we would all stand to benefit
from your contributions if your ideas were incorporated into the
online “master copy” where all the world can see it.
Credits
All entries arranged in alphabetical order of surname. Major
contributions are listed by individual name with some detail on the
nature of the contribution(s), date, contact info, etc. Minor
contributions (typo corrections, etc.) are listed by name only for
reasons of brevity. Please understand that when | classify a
contribution as “minor,” it is in no way inferior to the effort or value of
a “major” contribution, just smaller in the sense of less text changed.
Any and all contributions are gratefully accepted. | am indebted to all
those who have given freely of their own knowledge, time, and
resources to make this a better book!
Tony R. Kuphaldt
« Date(s) of contribution(s): 1996 to present
¢ Nature of contribution: Original author.
e Contact at: liec0@lycos.com
Jason Starck
« Date(s) of contribution(s): May-June 2000
¢ Nature of contribution: HTML formatting, some error
corrections.
¢ Contact at: jstarck@yhslug.tux.org
Dennis Crunkilton
« Date(s) of contribution(s): April 2005 to present
e Nature of contribution: Spice-Nutmeg plots, gnuplot Fourier
plots chapters 6, 7, 8, 9, 10; 04/2005.
¢ Nature of contribution: Broke “Special transformers and
applications” section into subsections. Scott-T and LVDT
subsections inserted, added to Air core transformers subsections
chapter 9; 09/2005.
e Nature of contribution: Chapter 13: AC motors; 01/2006.
¢ Nature of contribution: Mini table of contents, all chapters
except appedicies; html, latex, ps, pdf; See Devel/tutorial.AtmI;
01/2006.
e Nature of contribution: Chapters: all; Incremented edition
number to 6 for major format change. Added floating captioned
LaTeX figures for more book-like appearance of .pdf; 06/2006.
Added Doubly-Fed Induction Generator subsection, CH 13.
¢ Nature of contribution: Chapter 13: AC motors,“Running 3-
phase motors on 1-phase”, added to. Ch10, Ch12, minor change,
02/2009.
e Contact at: dcrunkilton(at)att(dot)net
Bill Stoddard, www.billsclockworks.com
« Date(s) of contribution(s): June 2005
e Nature of contribution: Granted permission to reprint
synchronous westclox motor jpg's, Reprinted by permission of
Westclox History at www.clockHistory.com, chapter 13
¢ Contact at: bill(at)billsclockworks (dot) com
Kurt Zierhut
« Date(s) of contribution(s): June 2011
¢ Nature of contribution: Construction of 3-phase distributed
motor windings.
¢ Contact at: kzierhut(at)haascnc.com
Your name here
« Date(s) of contribution(s): Month and year of contribution
e Nature of contribution: Insert text here, describing how you
contributed to the book.
e Contact at: my email@provider.net
Typo corrections and other “minor” contributions
¢ line-allaboutcircuits.com (June 2005) Typographical error
correction in Volumes 1,2,3,5, various chapters, (S/visa-versa/vice
versa/).
e The students of Bellingham Technical College's Instrumentation
program.
Bart Anderson (January 2004) Corrected conceptual and safety
errors regarding Tesla coils.
Ed Beroset (May 2002) Suggested better ways to illustrate the
meaning of the prefix “poly-” in chapter 10.
anonymous (September 2007) Typo correction in Basic AC
chapter, s/Alterantor/Alternator.
Michiel van Bolhuis (April 2007), Corrections numerous
chapters, images: 12008.eps, 02053.eps, 02056.eps, 02062.eps,
02515.eps, 02257.eps, 02258.eps, 02068.eps, 0207 4.eps,
02516.eps, 02516.eps, 02263.eps, text: s/(Figure 8.18/(Figure
8.18), s/dividing it my the/dividing it by the/, s/will be drive it/will
drive it/, S/oecause we can to use/because we can use/, S/phase
shift makes complicates/phase shift complicates/, s/750
kiloWatt/7 50 Watt, s/over 50 Kw use/over 50 kW use/, s/in an
open ended/in open ended/.
Kieran Clancy (August 2006) Ch 4, s/capcitive/capacitive,
S/positive negative/positive or negative.
Richard Cooper (December 2005) Clarification of 02206.eps,
02209.eps 3-phase transformer images. Correction of 02210.eps
open-delta image.
Colin Creitz (May 2007) Chapters: several, s/it's/its.
Duane Damiano (February 2003) Pointed out magnetic polarity
error in DC generator illustration.
Jeff DeFreitas (March 2006)Improve appearance: replace "/" and
”/" various chapters.
Sean Donner (January 2005) Typographical error correction in
“Series resistor-inductor circuits” section, Chapter 3: REACTANCE
AND IMPEDANCE -- INDUCTIVE “Voltage and current” section, (If
we were restrict ourselves /If we were to restrict ourselves),
(Across voltage across the resistor/ Voltage across the resistor);
More on the “skin effect” section, (corrected for the skin
effect/corrected for the skin effect).
(January 2005),Typographical error correction in “AC capacitor
circuits” section, Chapter 4: REACTANCE AND IMPEDANCE --
CAPACITIVE (calculate the phase angle of the inductor's reactive
opposition / calculate the phase angle of the capacitor's reactive
opposition).
(January 2005),Typographical error correction in “ Parallel R, L,
and C” section, Chapter 5: REACTANCE AND IMPEDANCE -- R, L,
AND C, (02083.eps, change Vic to Vir above resistor in image)
(January 2005),Typographical error correction in “Other
waveshapes” section, Chapter 7: MIXED-FREQUENCY AC SIGNALS,
(which only allow passage current in one direction./ which only
allow the passage of current in one direction.)
(January 2005),Typographical error correction in “What is a filter?”
section, Chapter 8: FILTERS, (from others in within mixed-
frequency signals. / from others within mixed-frequency signals.),
(dropping most of the voltage gets across series resistor /
dropping most of the voltage across series resistor)
Brendan Finley (March 2007) Suggested content change in
Transformers chapter, clarified text, changed image 02305.eps
“Mutual inductance and basic operation” section.
Steven Jones (November 2006) Suggested content addition in
Power factor chapter, added graph to “Calculating factor
correction” section.
Harvey Lew (February 2003) Typo correction in Basic AC
chapter: word “circuit” should have been “circle”.
Elmo Mantynen (August 2006) Numerous corrections in
chapters: Resonance, Polyphase AC Circuits, Power Factor, AC
Motors.
Jim Palmer (May 2002) Typo correction on complex number
math.
Bob Schmid (April 20027) Suggested we add Inductosyn, added
to Ch12“AC metering”.
Don Stalkowski (June 2002) Technical help with PostScript-to-
PDF file format conversion.
John Symonds (March 2002) Suggested an improved
explanation of the unit “Hertz.”
Puddy Tat@allaboutcircuits.com (May 2007) Pointed out error
in Form Factor definition and calculation, 3plcs Ch 1.3.
Joseph Teichman (June 2002) Suggestion and technical help
regarding use of PNG images instead of JPEG.
Mark D. Zarella (April 2002) Suggested an improved
explanation for the “average” value of a waveform.
machan@allaboutcircuits.com (April 2007) Transformer
voltage regulation example error, image: 12105.eps.
reccaO2@allaboutcircuits.com (April 2007) Resonance,
Parallel; missing formula, image: 12081.eps.
earsintraining@allaboutcircuits.com (July 2007) Ch 1, “AC
Phase” image 02022.png not displayed in html.
Dave@allaboutcircuits.com (Aug 2007) Ch, s/Vary/Very/ .
jut@allaboutcircuits.com (Sept 2007) Ch 1, s/as a the/as the/,
s/eight white/seven white/ .
rrgibbs@allaboutcircuits.com (Oct 2007) Ch 1, s/100/180
trigonometric sin function table.
Devin Bayer (September 2007) Correction to sml2html.sed, \} to
} in <tabular>.
mike@allaboutcircuits.com (Nov 2007) Ch 13 , Corrected error
concerning Tesla's sale of AC induction motor, Change one million
to to $65,000.
stacymckenna@allaboutcircuits.com (Feb 2008) Ch9,
Clarification of light load as refering to less current.
Unregistered@allaboutcircuits.com (Feb 2008) Ch 2, s/by/be
in "More on AC polaity" section.
Timothy Kingman (Feb 2008) Changed default roman font to
newcent.
Imranullah Syed (Feb 2008) Suggested centering of
uncaptioned schematics.
ShaunManners@allaboutcircuits.com (Feb 2008) Ch 1, Error
in the sign of value in sine table.
Dennis Crunkilton (Feb 2009) Ch 13 , s/Over-speed/Under-
speed 02514.png
peter o@allaboutcircuits.com (Feb 2009) Ch 2 , image
02046.png
Unregistered Guest@allaboutcircuits.com (April 2009) Ch 2
, S/that its/that it is/.
The Electrician@allaboutcircuits.com (November 2009) Ch
1, Clarification: average responding metermovement is a
d'Arsonval movement.
whanes@allaboutcircuits.com (January 2010) Ch 13, image
02419.png 02420.png moved single phase motors for polyphase
to singnle phase tree.
skfir@allaboutcircuits.com (august 2010) Ch 6,
s/prodces/produces/ s/the series resonant circuit looks
inductive/the parallel resonant circuit looks inuctive/ .
zyne@allaboutcircuits.com (August 2010) Ch 5, image
12067 .png 02420.png 2nd row 1st column s/480/480m/.
« Unregistered Guest@allaboutcircuits.com (August 2010) Ch
4 , s/Series capacitor inductor/Series capacitor/.
« Unregistered Guest@allaboutcircuits.com (August 2010) Ch
4 , s/voltage lags currrent in an inductor/voltage lags current in a
Capacitor/ caption for 02073.png.
katterjohn@allaboutcircuits.com (August 2010) Ch 14,
numerous missing links.
D Crunkilton (September 2010) Ch 13 , s/useable/usable/.
Unregistered-F034@allaboutcircuits.com (Feb 2011) Ch4,
sign of angle: s/36.87 /-36.87/
¢ Skfir@allaboutcircuits.com (Feb 2011) Ch13,
s/proviced/provided/ ; s/the put the/to put the/ ;
s/formlated/formulated/
Dave@allaboutcircuits.com (Feb 2011) Ch 6, s/ciruits/circuits/
Dcrunkilton (May 2011) Ch 13 , s/corrrector/corrector/
Dcrunkilton (June 2011) hi.latex ,latex header file -updated link
to openbookproject.net/electricCircuits
« SgtWookie (May 2012) resonant.sml s
/correspondes/corresponds , s/ration/ratio/ .
« theamber@allaboutcircuits.com (January 2014) trans.sml s
Scott-T transmormer voltage subscripts corrected .
Wilibald@allaboutcircuits.com (January 2014) lines.sml
S+m/s+miles/sec+, 2-instances near 186,000 .
¢ granzscientific @allaboutcircuits.com (January 2014)
lines.sml s/100/56.25/ after 02402.png .
chipwitch@allaboutcircuits.com (August 2015) xzl.sml
S/associate/associated/ .
¢ Skfir@allaboutcircuits.com (August 2015) filter.sml s/a input
impedance/an input impedance’ .
e David Winter (Feb 2017) xzc.sml Missing ")" near 88.42 .
Missing"(" above near 1/2mfC .
¢ Stewart Todd Morgan (Feb 2020) See [*] for numerous
corrections .
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt,
under the terms and conditions of the Design Science License.
—/ | 4]
Appendix 3
DESIGN SCIENCE LICENSE
Copyright © 1999-2000 Michael Stutz stutz@dsl.org
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END OF TERMS AND CONDITIONS
[ $Id: dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]
— 4 —
Copyright (C) 2002-2020, Tony R.
Kuphaldt
See the Design Science License (Appendix 3)
for details regarding copying and distribution
Revised January 18, 2010
Chapter 5: DISCRETE SEMICONDUCTOR CIRCUITS
Chapter 6: ANALOG INTEGRATED CIRCUITS
Chapter 7: DIGITAL INTEGRATED CIRCUITS
Chapter 8: 555 TIMER CIRCUITS
Appendix 1: ABOUT THIS BOOK
Appendix 2: CONTRIBUTOR LIST
Appendix 3: DESIGN SCIENCE LICENSE
Download printable versions of this
volume
Adobe PDF format:
EXP. pdf
Adobe PDF
1
Approximately 3.7 megabytes
Adobe PostScript (compressed) format:
EXP.ps.gz
PostScript
1
Approximately 18 megabytes
"How do! view and/or print PostScript documents," you ask?
Easy! Just download some free software at:
www.cs.wisc.edu/~ ghost.
There you'll find GSview and Ghostscript, two progams
necessary to display and print Postscript files (they'll even
display and print compressed PostScript files!). These
programs also display and format Adobe PDF files as a bonus.
Versions for Windows, OS/2, and Linux available.
Download source files for this volume
0 O
EXPsrc.tar.gz
<SubML>
Approximately 24 megabytes
o o
EXPtiny. tar.gz
<SubML>
Approximately 1 megabyte
To "compile" these source files into a viewable format, you
will need the following pieces of software (all available freely
over the internet):
e Make, a project management utility originally intended
as a programming tool, but useful for managing just
about any kind of computer project composed of many
files. /f you cannot obtain a copy of Make for your
computer system, you can get by with a little skill and a
few batch files (also known as shell scripts). The master
"Makefile" in this directory is readable with a text editor
or word processor, and contains all the instructions
carried out by the other utilities.
Sed (stands for Stream EDitor), a common UNIX utility
for performing search-and-replace commands on text
files. Required to convert SUbML source code into HTML,
TeX, LaTeX, and other formats. This is all you need for
generating HTML output!
LaTeX2e, a document formatting system designed as an
extension to TeX, Donald Knuth's outstanding text
processing system. You can also get by with just plain
TeX, but your printed output won't look quite as nice and
it will lack table-of-contents and index entries.
If you opt for the smaller of the two files (EXPtiny.tar.gz),
you'll also need a set of graphic manipulation utilities
released as a package called ImageMagick. Specifically, the
utility you'll need is named Mogrify. The larger of the two
source archive files contains all graphic images in two
formats, Encapsulated PostScript (*.eps) and JPEG (*.jpg).
This makes for a large file. The smaller source archive file
only contains Encapsulated PostScript for schematic
diagrams and JPEG images for photographs. This makes for a
much smaller file, but it requires that you do some image
conversion on your end. If you have access to other image
manipulation software capable of converting hundreds of
files with a batch command, you won't have to use
ImageMagick.
Back to Master Index
—/ | 4]
Lessons In Electric Circuits
-- Volume VI
Chapter 1
INTRODUCTION
e Electronics as science
e Setting up a home lab
o Work area
o Jools
o Supplies
e Contributors
Electronics as science
Electronics is a science, and a very accessible science at
that. With other areas of scientific study, expensive
equipment is generally required to perform any non-trivial
experiments. Not so with electronics. Many advanced
concepts may be explored using parts and equipment
totaling under a few hundred US dollars. This is good,
because hands-on experimentation is vital to gaining
scientific knowledge about any subject.
When | started writing Lessons In Electric Circuits, my intent
was to create a textbook suitable for introductory college
use. However, being mostly self-taught in electronics myself,
| knew the value of a good textbook to hobbyists and
experimenters not enrolled in any formal electronics course.
Many people selflessly volunteered their time and expertise
in helping me learn electronics when | was younger, and my
intent is to honor their service and love by giving back to
the world what they gave to me.
In order for someone to teach themselves a science such as
electronics, they must engage in hands-on experimentation.
Knowledge gleaned from books alone has limited use,
especially in scientific endeavors. If my contribution to
society is to be complete, | must include a guide to
experimentation along with the text(s) on theory, so that the
individual learning on their own has a resource to guide
their experimental adventures.
A formal laboratory course for college electronics study
requires an enormous amount of work to prepare, and
usually must be based around specific parts and equipment
so that the experiments will be sufficient detailed, with
results sufficiently precise to allow for rigorous comparison
between experimental and theoretical data. A process of
assessment, articulated through a qualified instructor, is
also vital to guarantee that a certain level of learning has
taken place. Peer review (comparison of experimental results
with the work of others) is another important component of
college-level laboratory study, and helps to improve the
quality of learning. Since | cannot meet these criteria
through the medium of a book, it is impractical for me to
present a complete laboratory course here. In the interest of
keeping this experiment guide reasonably low-cost for
people to follow, and practical for deployment over the
internet, | am forced to design the experiments at a lower
level than what would be expected for a college lab course.
The experiments in this volume begin at a level appropriate
for someone with no electronics knowledge, and progress to
higher levels. They stress qualitative knowledge over
quantitative knowledge, although they could serve as
templates for more rigorous coursework. If there is any
portion of Lessons /n Electric Circuits that will remain
"incomplete," it is this one: | fully intend to continue adding
experiments ad infinitum so as to provide the experimenter
or hobbyist with a wealth of ideas to explore the science of
electronics. This volume of the book series is also the easiest
to contribute to, for those who would like to help me in
providing free information to people learning electronics. It
doesn't take a tremendous effort to describe an experiment
or two, and | will gladly include it if you email it to me,
giving you full credit for the work. Refer to Appendix 2 for
details on contributing to this book.
When performing these experiments, feel free to explore by
trying different circuit construction and measurement
techniques. If something isn't working as the text describes
it should, don't give up! It's probably due to a simple
problem in construction (loose wire, wrong component
value) or test equipment setup. It can be frustrating working
through these problems on your own, but the knowledge
gained by "troubleshooting" a circuit yourself is at least as
important as the knowledge gained by a properly
functioning experiment. This is one of the most important
reasons why experimentation is so vital to your scientific
education: the real problems you will invariably encounter in
experimentation challenge you to develop practical
problem-solving skills.
In many of these experiments, | offer part numbers for Radio
Shack brand components. This is not an endorsement of
Radio Shack, but simply a convenient reference to an
electronic supply company well-known in North America.
Often times, components of better quality and lower price
may be obtained through mail-order companies and other,
lesser-known supply houses. | strongly recommend that
experimenters obtain some of the more expensive
components such as transformers (see the AC chapter) by
Salvaging them from discarded electrical appliances, both
for economic and ecological reasons.
All experiments shown in this book are designed with safety
in mind. It is nearly impossible to shock or otherwise hurt
yourself by battery-powered experiments or other circuits of
low voltage. However, hazards do exist building anything
with your own two hands. Where there is a greater-than-
normal level of danger in an experiment, | take efforts to
direct the reader's attention toward it. However, it is
unfortunately necessary in this litigious society to disclaim
any and all liability for the outcome of any experiment
presented here. Neither myself nor any contributors bear
responsibility for injuries resulting from the construction or
use of any of these projects, from the mis-handling of
electricity by the experimenter, or from any other unsafe
practices leading to injury. Perform these experiments
at your own risk!
Setting up a home lab
In order to build the circuits described in this volume, you
will need a small work area, as well as a few tools and critical
supplies. This section describes the setup of a home
electronics laboratory.
Work area
A work area should consist of a large workbench, desk, or
table (preferably wooden) for performing circuit assembly,
with household electrical power (120 volts AC) readily
accessible to power soldering equipment, power supplies,
and any test equipment. Inexpensive desks intended for
computer use function very well for this purpose. Avoid a
metal-surface desk, as the electrical conductivity of a metal
surface creates both a shock hazard and the very distinct
possibility of unintentional "short circuits" developing from
circuit components touching the metal tabletop. Vinyl and
plastic bench surfaces are to be avoided for their ability to
generate and store large static-electric charges, which may
damage sensitive electronic components. Also, these
materials melt easily when exposed to hot soldering irons
and molten solder droplets.
If you cannot obtain a wooden-surface workbench, you may
turn any form of table or desk into one by laying a piece of
plywood on top. If you are reasonably skilled with
woodworking tools, you may construct your own desk using
plywood and 2x4 boards.
The work area should be well-lit and comfortable. | have a
small radio set up on my own workbench for listening to
music or news as | experiment. My own workbench has a
"power strip" receptacle and switch assembly mounted to
the underside, into which | plug all 120 volt devices. It is
convenient to have a single switch for shutting off a// power
in case of an accidental short-circuit!
Tools
A few tools are required for basic electronics work. Most of
these tools are inexpensive and easy to obtain. If you desire
to keep the cost as low as possible, you might want to
search for them at thrift stores and pawn shops before
buying them new. As you can tell from the photographs,
some of my own tools are rather old but function well
nonetheless.
First and foremost in your tool collection is a multimeter.
This is an electrical instrument designed to measure voltage,
current, resistance, and often other variables as well.
Multimeters are manufactured in both digital and analog
form. A digital multimeter is preferred for precision work, but
analog meters are also useful for gaining an intuitive
understanding of instrument sensitivity and range.
My own digital multimeter is a Fluke model 27, purchased in
1987:
Digital multimeter
—_
Most analog multimeters sold today are quite inexpensive,
and not necessarily precision test instruments. | recommend
having both digital and analog meter types in your tool
collection, spending as little money as possible on the
analog multimeter and investing in a good-quality digital
multimeter (I highly recommend the Fluke brand).
A test instrument | have found indispensable in my home
work is a sensitive voltage detector, or sensitive audio
detector, described in nearly identical experiments in two
chapters of this book volume. It is nothing more than a
sensitized set of audio headphones, equipped with an
attenuator (volume control) and limiting diodes to limit
sound intensity from strong signals. Its purpose is to audibly
indicate the presence of low-intensity voltage signals, DC or
AC. In the absence of an oscilloscope, this is a most valuable
tool, because it allows you to /isten to an electronic signal,
and thereby determine something of its nature. Few tools
engender an intuitive comprehension of frequency and
amplitude as this! | cite its use in many of the experiments
shown in this volume, so | strongly encourage that you build
your own. Second only to a multimeter, it is the most useful
piece of test equipment in the collection of the budget
electronics experimenter.
Sensitive voltage/audio detector
As you can see, | built my detector using scrap parts
(household electrical switch/receptacle box for the
enclosure, section of brown lamp cord for the test leads).
Even some of the internal components were salvaged from
scrap (the step-down transformer and headphone jack were
taken from an old radio, purchased in non-working condition
from a thrift store). The entire thing, including the
headphones purchased second-hand, cost no more than $15
to build. Of course, one could take much greater care in
choosing construction materials (metal box, shielded test
probe cable), but it probably wouldn't improve its
performance significantly.
The single most influential component with regard to
detector sensitivity is the headphone assembly: generally
speaking, the greater the "dB" rating of the headphones, the
better they will function for this purpose. Since the
headphones need not be modified for use in the detector
circuit, and they can be unplugged from it, you might justify
the purchase of more expensive, high-quality headphones
by using them as part of a home entertainment
(audio/video) system.
Also essential is a so/derless breadboard, sometimes called a
prototyping board, or proto-board. This device allows you to
quickly join electronic components to one another without
having to solder component terminals and wires together.
Solderless breadboard
When working with wire, you need a tool to "strip" the
plastic insulation off the ends so that bare copper metal is
exposed. This tool is called a wire stripper, and it is a special
form of plier with several knife-edged holes in the jaw area
sized just right for cutting through the plastic insulation and
not the copper, for a multitude of wire sizes, or gauges.
Shown here are two different sizes of wire stripping pliers:
Wire stripping pliers
In order to make quick, temporary connections between
some electronic components, you need jumper wires with
small "alligator-jaw" clips at each end. These may be
purchased complete, or assembled from clips and wires.
Jumper wires (as sold by Radio Shack)
Jumper wires (home-made)
The home-made jumper wires with large, uninsulated (bare
metal) alligator clips are okay to use so long as care is taken
to avoid any unintentional contact between the bare clips
and any other wires or components. For use in crowded
breadboard circuits, jumper wires with insulated (rubber-
covered) clips like the jumper shown from Radio Shack are
much preferred.
Needle-nose pliers are designed to grasp small objects, and
are especially useful for pushing wires into stubborn
breadboard holes.
Needle-nose pliers
No tool set would be complete without screwdrivers, and |
recommend a complementary pair (3/16 inch slotted and #2
Phillips) as the starting point for your collection. You may
later find it useful to invest in a set of /ewe/er's screwdrivers
for work with very small screws and screw-head
adjustments.
Screwdrivers
For projects involving printed-circuit board assembly or
repair, a small soldering iron and a spool of "rosin-core"
solder are essential tools. | recommend a 25 watt soldering
iron, no larger for printed circuit board work, and the
thinnest solder you can find. Do not use “acid-core" solder!
Acid-core solder is intended for the soldering of copper
tubes (plumbing), where a small amount of acid helps to
clean the copper of surface impurities and provide a
stronger bond. If used for electrical work, the residual acid
will cause wires to corrode. Also, you should avoid solder
containing the metal /ead, opting instead for silver-alloy
solder. If you do not already wear glasses, a pair of safety
glasses is highly recommended while soldering, to prevent
bits of molten solder from accidently landing in your eye
should a wire release from the joint during the soldering
process and fling bits of solder toward you.
Soldering iron and solder ("rosin core")
Projects requiring the joining of large wires by soldering will
necessitate a more powerful heat source than a 25 watt
soldering iron. A soldering gunis a practical option.
Soldering gun
Knives, like screwdrivers, are essential tools for all kinds of
work. For safety's sake, | recommend a "utility" knife with
retracting blade. These knives are also advantageous to
have for their ability to accept replacement blades.
Utility knife
Pliers other than the needle-nose type are useful for the
assembly and disassembly of electronic device chassis. Two
types | recommend are s/ip-joint and adjustable-joint
("“Channel-lock").
Slip-joint pliers
Adjustable-joint pliers
Drilling may be required for the assembly of large projects.
Although power drills work well, | have found that a simple
hand-crank drill does a remarkable job drilling through
plastic, wood, and most metals. It is certainly safer and
quieter than a power drill, and costs quite a bit less.
Hand drill
As the wear on my drill indicates, it is an often-used tool
around my home!
Some experiments will require a source of audio-frequency
voltage signals. Normally, this type of signal is generated in
an electronics laboratory by a device called a signal
generator or function generator. While building such a
device is not impossible (nor difficult!), it often requires the
use of an oscilloscope to fine-tune, and oscilloscopes are
usually outside the budgetary range of the home
experimenter. A relatively inexpensive alternative toa
commercial signal generator is an e/ectronic keyboard of the
musical type. You need not be a musician to operate one for
the purposes of generating an audio signal (just press any
key on the board!), and they may be obtained quite readily
at second-hand stores for substantially less than new price.
The electronic signal generated by the keyboard is
conducted to your circuit via a headphone cable plugged
into the "headphones" jack. More details regarding the use
of a "Musical Keyboard as a Signal Generator" may be found
in the experiment of that name in chapter 4 (AC).
Supplies
Wire used in solderless breadboards must be 22-gauge, solid
copper. Spools of this wire are available from electronic
supply stores and some hardware stores, in different
insulation colors. Insulation color has no bearing on the
wire's performance, but different colors are sometimes
useful for "color-coding" wire functions in a complex circuit.
Spool of 22-gauge, solid copper wire
al
Note how the last 1/4 inch or so of the copper wire
protruding from the spool has been "stripped" of its plastic
insulation.
An alternative to solderless breadboard circuit construction
iS Wire-wrap, where 30-gauge (very thin!) solid copper wire
is tightly wrapped around the terminals of components
inserted through the holes of a fiberglass board. No
soldering is required, and the connections made are at least
as durable as soldered connections, perhaps more. Wire-
wrapping requires a spool of this very thin wire, and a
special wrapping tool, the simplest kind resembling a small
screwdriver.
Wire-wrap wire and wrapping tool
Large wire (14 gauge and bigger) may be needed for
building circuits that carry significant levels of current.
Though electrical wire of practically any gauge may be
purchased on spools, | have found a very inexpensive source
of stranded (flexible), copper wire, available at any hardware
store: cheap extension cords. Typically comprised of three
wires colored white, black, and green, extension cords are
often sold at prices less than the retail cost of the
constituent wire alone. This is especially true if the cord is
purchased on sale! Also, an extension cord provides you
with a pair of 120 volt connectors: male (plug) and female
(receptacle) that may be used for projects powered by 120
volts.
Extension cord, in package
16 awe
—
To extract the wires, carefully cut the outer layer of plastic
insulation away using a utility knife. With practice, you may
find you can peel away the outer insulation by making a
Short cut in it at one end of the cable, then grasping the
wires with one hand and the insulation with the other and
pulling them apart. This is, of course, much preferable to
Slicing the entire length of the insulation with a knife, both
for safety's sake and for the sake of avoiding cuts in the
individual wires' insulation.
During the course of building many circuits, you will
accumulate a large number of small components. One
technique for keeping these components organized is to
keep them in a plastic "organizer" box like the type used for
fishing tackle.
Component box
In this view of one of my component boxes, you can see
plenty of 1/8 watt resistors, transistors, diodes, and even a
few 8-pin integrated circuits ("chips"). Labels for each
compartment were made with a permanent ink marker.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Michael Warner (April 9, 2002): Suggestions for a section
describing home laboratory setup.
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—|/|+4]l\—
—/ | 4]
Lessons In Electric Circuits
-- Volume VI
Chapter 2
BASIC CONCEPTS AND
TEST EQUIPMENT
Voltmeter usage
Ohmmeter usage
A very simple circuit
Ammeter usage
Ohm's Law
Nonlinear resistance
Power dissipation
Circuit with a switch
Electromagnetism
Electromagnetic induction
Voltmeter usage
PARTS AND MATERIALS
Multimeter, digital or analog
Assorted batteries
One light-emitting diode (Radio Shack catalog # 276-
026 or equivalent)
Small "hobby" motor, permanent-magnet type (Radio
Shack catalog # 273-223 or equivalent)
Two jumper wires with "alligator clip" ends (Radio Shack
catalog # 278-1156, 278-1157, or equivalent)
A multimeter is an electrical instrument capable of
measuring voltage, current, and resistance. Digital
multimeters have numerical displays, like digital clocks, for
indicating the quantity of voltage, current, or resistance.
Analog multimeters indicate these quantities by means of a
moving pointer over a printed scale.
Analog multimeters tend to be less expensive than digital
multimeters, and more beneficial as learning tools for the
first-time student of electricity. | strongly recommend
purchasing an analog multimeter before purchasing a digital
multimeter, but to eventually have both in your tool kit for
these experiments.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 1: "Basic
Concepts of Electricity"
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
LEARNING OBJECTIVES
e How to measure voltage
e Characteristics of voltage: existing between two points
e Selection of proper meter range
ILLUSTRATION
Digital
multimeter
Analog
multimeter
Test leads Test leads
Test probes Test probes
A,*/
A) ,
Light-emitting
diode ("LED")
6-volt "lantern"
battery
Permanent-
magnet motor
1.5-volt "D-cell"
battery
INSTRUCTIONS
In all the experiments in this book, you will be using some
sort of test equipment to measure aspects of electricity you
cannot directly see, feel, hear, taste, or smell. Electricity -- at
least in small, safe quantities -- is insensible by our human
bodies. Your most fundamental "eyes" in the world of
electricity and electronics will be a device called a
multimeter. Multimeters indicate the presence of, and
measure the quantity of, electrical properties such as
voltage, current, and resistance. In this experiment, you will
familiarize yourself with the measurement of voltage.
Voltage is the measure of electrical "push" ready to motivate
electrons to move through a conductor. In scientific terms, it
is the specific energy per unit charge, mathematically
defined as joules per coulomb. It is analogous to pressure in
a fluid system: the force that moves fluid through a pipe,
and is measured in the unit of the Volt (V).
Your multimeter should come with some basic instructions.
Read them well! If your multimeter is digital, it will require a
small battery to operate. If it is analog, it does not need a
battery to measure voltage.
Some digital multimeters are autoranging. An autoranging
meter has only a few selector switch (dial) positions. Manual-
ranging meters have several different selector positions for
each basic quantity: several for voltage, several for current,
and several for resistance. Autoranging is usually found on
only the more expensive digital meters, and is to manual
ranging as an automatic transmission is to a manual
transmission in a car. An autoranging meter "shifts gears"
automatically to find the best measurement range to display
the particular quantity being measured.
Set your multimeter's selector switch to the highest-value
"DC volt" position available. Autoranging multimeters may
only have a single position for DC voltage, in which case you
need to set the switch to that one position. Touch the red
test probe to the positive (+) side of a battery, and the black
test probe to the negative (-) side of the same battery. The
meter should now provide you with some sort of indication.
Reverse the test probe connections to the battery if the
meter's indication is negative (on an analog meter, a
negative value is indicated by the pointer deflecting left
instead of right).
If your meter is a manual-range type, and the selector switch
has been set to a high-range position, the indication will be
small. Move the selector switch to the next lower DC voltage
range setting and reconnect to the battery. The indication
should be stronger now, as indicated by a greater deflection
of the analog meter pointer (need/e), or more active digits
on the digital meter display. For the best results, move the
selector switch to the lowest-range setting that does not
"over-range" the meter. An over-ranged analog meter is said
to be "pegged," as the needle will be forced all the way to
the right-hand side of the scale, past the full-range scale
value. An over-ranged digital meter sometimes displays the
letters "OL", or a series of dashed lines. This indication is
manufacturer-specific.
What happens if you only touch one meter test probe to one
end of a battery? How does the meter have to connect to the
battery in order to provide an indication? What does this tell
us about voltmeter use and the nature of voltage? Is there
such a thing as voltage "at" a single point?
Be sure to measure more than one size of battery, and learn
how to select the best voltage range on the multimeter to
give you maximum indication without over-ranging.
Now switch your multimeter to the lowest DC voltage range
available, and touch the meter's test probes to the terminals
(wire leads) of the light-emitting diode (LED). An LED is
designed to produce light when powered by a small amount
of electricity, but LEDs also happen to generate DC voltage
when exposed to light, somewhat like a solar cell. Point the
LED toward a bright source of light with your multimeter
connected to it, and note the meter's indication:
Light source
Batteries develop electrical voltage through chemical
reactions. When a battery "dies," it has exhausted its
original store of chemical "fuel." The LED, however, does not
rely on an internal "fuel" to generate voltage; rather, it
converts optical energy into electrical energy. So long as
there is light to illuminate the LED, it will produce voltage.
Another source of voltage through energy conversion a
generator. The small electric motor specified in the "Parts
and Materials" list functions as an electrical generator if its
Shaft is turned by a mechanical force. Connect your
voltmeter (your multimeter, set to the "volt" function) to the
motor's terminals just as you connected it to the LED's
terminals, and spin the shaft with your fingers. The meter
should indicate voltage by means of needle deflection
(analog) or numerical readout (digital).
If you find it difficult to maintain both meter test probes in
connection with the motor's terminals while simultaneously
spinning the shaft with your fingers, you may use alligator
clio "jumper" wires like this:
Alligator
clip
Determine the relationship between voltage and generator
shaft soeed? Reverse the generator's direction of rotation
and note the change in meter indication. When you reverse
shaft rotation, you change the po/arity of the voltage
created by the generator. The voltmeter indicates polarity
by direction of needle direction (analog) or sign of numerical
indication (digital). When the red test lead is positive (+)
and the black test lead negative (-), the meter will register
voltage in the normal direction. If the applied voltage is of
the reverse polarity (negative on red and positive on black),
the meter will indicate "backwards."
Ohmmeter usage
PARTS AND MATERIALS
e Multimeter, digital or analog
e Assorted resistors (Radio Shack catalog # 271-312 isa
500-piece assortment)
Rectifying diode (LN4001 or equivalent; Radio Shack
catalog # 276-1101)
e Cadmium Sulphide photocell (Radio Shack catalog #
276-1657)
e Breadboard (Radio Shack catalog # 276-174 or
equivalent)
Jumper wires
Paper
Pencil
Glass of water
Table salt
This experiment describes how to measure the electrical
resistance of several objects. You need not possess a// items
listed above in order to effectively learn about resistance.
Conversely, you need not limit your experiments to these
items. However, be sure to never measure the resistance of
any electrically "live" object or circuit. In other words, do not
attempt to measure the resistance of a battery or any other
source of substantial voltage using a multimeter set to the
resistance ("ohms") function. Failing to heed this warning
will likely result in meter damage and even personal injury.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 1: "Basic
Concepts of Electricity"
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
LEARNING OBJECTIVES
e Determination and comprehension of "electrical
continuity"
e Determination and comprehension of "electrically
common points"
e How to measure resistance
e Characteristics of resistance: existing between two
points
Selection of proper meter range
e Relative conductivity of various components and
materials
ILLUSTRATION
(a) —_ +
= Diode
Incandescent Photocell
lamp
—_—_- -—-—
Resistor Resistor
INSTRUCTIONS
Resistance is the measure of electrical "friction" as electrons
move through a conductor. It is measured in the unit of the
"Ohm," that unit symbolized by the capital Greek letter
omega (Q).
Set your multimeter to the highest resistance range
available. The resistance function is usually denoted by the
unit symbol for resistance: the Greek letter omega (Q), or
sometimes by the word "ohms." Touch the two test probes of
your meter together. When you do, the meter should register
O ohms of resistance. If you are using an analog meter, you
will notice the needle deflect full-scale when the probes are
touched together, and return to its resting position when the
probes are pulled apart. The resistance scale on an analog
multimeter is reverse-printed from the other scales: zero
resistance in indicated at the far right-hand side of the scale,
and infinite resistance is indicated at the far left-hand side.
There should also be a small adjustment knob or "wheel" on
the analog multimeter to calibrate it for "Zero" ohms of
resistance. Touch the test probes together and move this
adjustment until the needle exactly points to zero at the
right-hand end of the scale.
Although your multimeter is capable of providing
quantitative values of measured resistance, it is also useful
for qualitative tests of continuity: whether or not there is a
continuous electrical connection from one point to another.
You can, for instance, test the continuity of a piece of wire by
connecting the meter probes to opposite ends of the wire
and checking to see the the needle moves full-scale. What
would we say about a piece of wire if the ohmmeter needle
didn't move at all when the probes were connected to
opposite ends?
Digital multimeters set to the "resistance" mode indicate
non-continuity by displaying some non-numerical indication
on the display. Some models say "OL" (Open-Loop), while
others display dashed lines.
Use your meter to determine continuity between the holes
on a breadboard: a device used for temporary construction
of circuits, where component terminals are inserted into
holes on a plastic grid, metal spring clips underneath each
hole connecting certain holes to others. Use small pieces of
22-gauge solid copper wire, inserted into the holes of the
breadboard, to connect the meter to these spring clips so
that you can test for continuity:
Continuity!
1 Analog
yy /X meter
4 \
22-gauge wire 22-gauge wire
Breadboard
No continuity
1 Analog
y, / meter
7 ‘\
I
- +
22-gauge wire 22-gauge wire
Breadboard
An important concept in electricity, closely related to
electrical continuity, is that of points being e/ectrically
common to each other. Electrically common points are
points of contact on a device or in a circuit that have
negligible (extremely small) resistance between them. We
could say, then, that points within a breadboard column
(vertical in the illustrations) are e/ectrically common to each
other, because there is electrical continuity between them.
Conversely, breadboard points within a row (horizontal in
the illustrations) are not electrically common, because there
is no continuity between them. Continuity describes what is
between points of contact, while commonality describes how
the points themselves relate to each other.
Like continuity, commonality is a qualitative assessment,
based on a relative comparison of resistance between other
points in a circuit. It is an important concept to grasp,
because there are certain facts regarding voltage in relation
to electrically common points that are valuable in circuit
analysis and troubleshooting, the first one being that there
will never be substantial voltage dropped between points
that are electrically common to each other.
Select a 10,000 ohm (10 kQ) resistor from your parts
assortment. This resistance value is indicated by a series of
color bands: Brown, Black, Orange, and then another color
representing the precision of the resistor, Gold (+/- 5%) or
Silver (+/- 10%). Some resistors have no color for precision,
which marks them as +/- 20%. Other resistors use five color
bands to denote their value and precision, in which case the
colors for a 10 kQ resistor will be Brown, Black, Black, Red,
and a fifth color for precision.
Connect the meter's test probes across the resistor as such,
and note its indication on the resistance scale:
ra ? Resistor
If the needle points very close to zero, you need to select a
lower resistance range on the meter, just as you needed to
select an appropriate voltage range when reading the
voltage of a battery.
If you are using a digital multimeter, you should see a
numerical figure close to 10 shown on the display, with a
small "k" symbol on the right-hand side denoting the metric
prefix for "kilo" (thousand). Some digital meters are
manually-ranged, and require appropriate range selection
just as the analog meter. If yours is like this, experiment with
different range switch positions and see which one gives you
the best indication.
Try reversing the test probe connections on the resistor.
Does this change the meter's indication at all? What does
this tell us about the resistance of a resistor? What happens
when you only touch one probe to the resistor? What does
this tell us about the nature of resistance, and how it is
measured? How does this compare with voltage
measurement, and what happened when we tried to
measure battery voltage by touching only one probe to the
battery?
When you touch the meter probes to the resistor terminals,
try not to touch both probe tips to your fingers. If you do,
you will be measuring the parallel combination of the
resistor and your own body, which will tend to make the
meter indication lower than it should be! When measuring a
10 kQ resistor, this error will be minimal, but it may be more
severe when measuring other values of resistor.
You may safely measure the resistance of your own body by
holding one probe tip with the fingers of one hand, and the
other probe tip with the fingers of the other hand. Note: be
very careful with the probes, as they are often sharpened to
a needle-point. Hold the probe tips along their length, not at
the very points! You may need to adjust the meter range
again after measuring the 10 kQ resistor, as your body
resistance tends to be greater than 10,000 ohms hand-to-
hand. Try wetting your fingers with water and re-measuring
resistance with the meter. What impact does this have on
the indication? Try wetting your fingers with sa/twater
prepared using the glass of water and table salt, and re-
measuring resistance. What impact does this have on your
body's resistance as measured by the meter?
Resistance is the measure of friction to electron flow through
an object. The less resistance there is between two points,
the harder it is for electrons to move (flow) between those
two points. Given that electric shock is caused by a large
flow of electrons through a person's body, and increased
body resistance acts as a safeguard by making it more
difficult for electrons to flow through us, what can we
ascertain about electrical safety from the resistance
readings obtained with wet fingers? Does water increase or
decrease shock hazard to people?
Measure the resistance of a rectifying diode with an analog
meter. Try reversing the test probe connections to the diode
and re-measure resistance. What strikes you as being
remarkable about the diode, especially in contrast to the
resistor?
Take a piece of paper and draw a very heavy black mark on
it with a pencil (not a pen!). Measure resistance on the black
strip with your meter, placing the probe tips at each end of
the mark like this:
Mark made with
Paper
Move the probe tips closer together on the black mark and
note the change in resistance value. Does it increase or
decrease with decreased probe spacing? If the results are
inconsistent, you need to redraw the mark with more and
heavier pencil strokes, so that it is consistent in its density.
What does this teach you about resistance versus length of a
conductive material?
Connect your meter to the terminals of a cadmium-sulphide
(CdS) photocell and measure the change in resistance
created by differences in light exposure. Just as with the
light-emitting diode (LED) of the voltmeter experiment, you
may want to use alligator-clip jumper wires to make
connection with the component, leaving your hands free to
hold the photocell to a light source and/or change meter
ranges:
“Vo Photocell
Light source
Experiment with measuring the resistance of several
different types of materials, just be sure not to try measure
anything that produces substantial voltage, like a battery.
Suggestions for materials to measure are: fabric, plastic,
wood, metal, clean water, dirty water, salt water, glass,
diamond (on a diamond ring or other piece of jewelry),
paper, rubber, and oil.
A very simple circuit
PARTS AND MATERIALS
e 6-volt battery
e 6-volt incandescent lamp
e Jumper wires
e Breadboard
e Terminal strip
From this experiment on, a multimeter is assumed to be
necessary and will not be included in the required list of
parts and materials. In all subsequent illustrations, a digital
multimeter will be shown instead of an analog meter unless
there is some particular reason to use an analog meter. You
are encouraged to use both types of meters to gain
familiarity with the operation of each in these experiments.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 1: "Basic
Concepts of Electricity"
LEARNING OBJECTIVES
Essential configuration needed to make a circuit
Normal voltage drops in an operating circuit
Importance of continuity to a circuit
Working definitions of "open" and "short" circuits
Breadboard usage
Terminal strip usage
SCHEMATIC DIAGRAM
Battery — Lamp
ILLUSTRATION
Lamp
INSTRUCTIONS
This is the simplest complete circuit in this collection of
experiments: a battery and an incandescent lamp. Connect
the lamp to the battery as shown in the illustration, and the
lamp should light, assuming the battery and lamp are both
in good condition and they are matched to one another in
terms of voltage.
If there is a "break" (discontinuity) anywhere in the circuit,
the lamp will fail to light. It does not matter where such a
break occurs! Many students assume that because electrons
leave the negative (-) side of the battery and continue
through the circuit to the positive (+) side, that the wire
connecting the negative terminal of the battery to the lamp
iS more important to circuit operation than the other wire
providing a return path for electrons back to the battery.
This is not true!
No light!
break in circuit
break in circuit
| No light!
a x) Lamp
break in circuit ¥%
a
No light!
Battery
break in circuit
No light!
Using your multimeter set to the appropriate "DC volt"
range, measure voltage across the battery, across the lamp,
and across each jumper wire. Familiarize yourself with the
normal voltages in a functioning circuit.
Now, "break" the circuit at one point and re-measure voltage
between the same sets of points, additionally measuring
voltage across the break like this:
No light!
What voltages measure the same as before? What voltages
are different since introducing the break? How much voltage
is manifest, or dropped across the break? What is the
polarity of the voltage drop across the break, as indicated by
the meter?
Re-connect the jumper wire to the lamp, and break the
circuit in another place. Measure all voltage "drops" again,
familiarizing yourself with the voltages of an "open" circuit.
Construct the same circuit on a breadboard, taking care to
place the lamp and wires into the breadboard in such a way
that continuity will be maintained. The example shown here
is only that: an example, not the only way to build a circuit
on a breadboard:
Breadboard
Experiment with different configurations on the breadboard,
plugging the lamp into different holes. If you encounter a
situation where the lamp refuses to light up and the
connecting wires are getting warm, you probably have a
situation known as a Short circuit, where a lower-resistance
path than the lamp bypasses current around the lamp,
preventing enough voltage from being dropped across the
lamp to light it up. Here is an example of a short circuit
made on a breadboard:
No light!
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ooocoooooees?oesogseoeoeoeoeeee ee 8 8 8 eo
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Breadboard
Here is an example of an accidental short circuit of the type
typically made by students unfamiliar with breadboard
usage:
No light!
Breadboard
Here there is no "shorting" wire present on the breadboard,
yet there /s a short circuit, and the lamp refuses to light.
Based on your understanding of breadboard hole
connections, can you determine where the "short" is in this
circuit?
Short circuits are generally to be avoided, as they result in
very high rates of electron flow, causing wires to heat up
and battery power sources to deplete. If the power source is
substantial enough, a short circuit may cause heat of
explosive proportions to manifest, causing equipment
damage and hazard to nearby personnel. This is what
happens when a tree limb "shorts" across wires on a power
line: the limb -- being composed of wet wood -- acts as a low-
resistance path to electric current, resulting in heat and
Sparks.
You may also build the battery/lamp circuit on a terminal
Strip: a length of insulating material with metal bars and
screws to attach wires and component terminals to. Here is
an example of how this circuit might be constructed ona
terminal strip:
Terminal
strip |le
Ammeter usage
PARTS AND MATERIALS
e 6-volt battery
e 6-volt incandescent lamp
Basic circuit construction components such as breadboard,
terminal strip, and jumper wires are also assumed to be
available from now on, leaving only components and
materials unique to the project listed under "Parts and
Materials."
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 1: "Basic
Concepts of Electricity"
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
LEARNING OBJECTIVES
e How to measure current with a multimeter
e How to check a multimeter's internal fuse
e Selection of proper meter range
SCHEMATIC DIAGRAM
Ammeter
Battery — Lamp
ILLUSTRATION
INSTRUCTIONS
Current is the measure of the rate of electron "flow" ina
circuit. It is measured in the unit of the Ampere, simply
called "Amp," (A).
The most common way to measure current in a circuit is to
break the circuit open and insert an "ammeter" in series (in-
line) with the circuit so that all electrons flowing through the
circuit also have to go through the meter. Because
measuring current in this manner requires the meter be
made part of the circuit, it is a more difficult type of
measurement to make than either voltage or resistance.
Some digital meters, like the unit shown in the illustration,
have a separate jack to insert the red test lead plug when
measuring current. Other meters, like most inexpensive
analog meters, use the same jacks for measuring voltage,
resistance, and current. Consult your owner's manual on the
particular model of meter you own for details on measuring
Current.
When an ammeter is placed in series with a circuit, it ideally
drops no voltage as current goes through it. In other words,
it acts very much like a piece of wire, with very little
resistance from one test probe to the other. Consequently,
an ammeter will act as a short circuit if placed in parallel
(across the terminals of) a substantial source of voltage. If
this is done, a surge in current will result, potentially
damaging the meter:
»
SHORT CIRCUIT !
current >
Ammeters are generally protected from excessive current by
means of a small fuse located inside the meter housing. If
the ammeter is accidently connected across a substantial
voltage source, the resultant surge in current will "blow" the
fuse and render the meter incapable of measuring current
until the fuse is replaced. Be very careful to avoid this
scenario!
You may test the condition of a multimeter's fuse by
switching it to the resistance mode and measuring
continuity through the test leads (and through the fuse). On
a meter where the same test lead jacks are used for both
resistance and current measurement, simply leave the test
lead plugs where they are and touch the two probes
together. On a meter where different jacks are used, this is
how you insert the test lead plugs to check the fuse:
Low resistance _—-
indication = good fuse
High resistance
indication = "blown" fuse |
Internal
location of
fuse
touch probes together
Build the one-battery, one-lamp circuit using jumper wires to
connect the battery to the lamp, and verify that the lamp
lights up before connecting the meter in series with it. Then,
break the circuit open at any point and connect the meter's
test probes to the two points of the break to measure
current. As usual, if your meter is manually-ranged, begin by
selecting the highest range for current, then move the
selector switch to lower range positions until the strongest
indication is obtained on the meter display without over-
ranging it. If the meter indication is "backwards," (left
motion on analog needle, or negative reading on a digital
display), then reverse the test probe connections and try
again. When the ammeter indicates a normal reading (not
"backwards"), electrons are entering the black test lead and
exiting the red. This is how you determine direction of
Current using a meter.
For a 6-volt battery and a small lamp, the circuit current will
be in the range of thousandths of an amp, or mi/liamps.
Digital meters often show a small letter "m" in the right-
hand side of the display to indicate this metric prefix.
Try breaking the circuit at some other point and inserting
the meter there instead. What do you notice about the
amount of current measured? Why do you think this is?
Re-construct the circuit on a breadboard like this:
Breadboard
Students often get confused when connecting an ammeter
to a breadboard circuit. How can the meter be connected so
as to intercept all the circuit's current and not create a short
circuit? One easy method that guarantees success is this:
e Identify what wire or component terminal you wish to
measure current through.
e Pull that wire or terminal out of the breadboard hole.
Leave it hanging in mid-air.
Insert a spare piece of wire into the hole you just pulled
the other wire or terminal out of. Leave the other end of
this wire hanging in mid-air.
Connect the ammeter between the two unconnected
wire ends (the two that were hanging in mid-air). You are
now assured of measuring current through the wire or
terminal initially identified.
wire pulled
out of
breadboard
spare wire
Again, measure current through different wires in this
circuit, following the same connection procedure outlined
above. What do you notice about these current
measurements? The results in the breadboard circuit should
be the same as the results in the free-form (no breadboard)
circuit.
Building the same circuit on a terminal strip should also
yield similar results:
Terminal
strip
The current figure of 24.70 milliamps (24.70 mA) shown in
the illustrations is an arbitrary quantity, reasonable fora
small incandescent lamp. If the current for your circuit is a
different value, that is okay, so long as the lamp is
functioning when the meter is connected. If the lamp refuses
to light when the meter is connected to the circuit, and the
meter registers a much greater reading, you probably have a
short-circuit condition through the meter. If your lamp
refuses to light when the meter is connected in the circuit,
and the meter registers zero current, you've probably blown
the fuse inside the meter. Check the condition of your
meter's fuse as described previously in this section and
replace the fuse if necessary.
Ohm's Law
PARTS AND MATERIALS
e Calculator (or pencil and paper for doing arithmetic)
e 6-volt battery
e Assortment of resistors between 1 KQ and 100 kQ in
value
I'm purposely restricting the resistance values between 1 kO
and 100 kQ for the sake of obtaining accurate voltage and
current readings with your meter. With very low resistance
values, the internal resistance of the ammeter has a
significant impact on measurement accuracy. Very high
resistance values can cause problems for voltage
measurement, the internal resistance of the voltmeter
substantially changing circuit resistance when it is
connected in parallel with a high-value resistor.
At the recommended resistance values, there will still be a
small amount of measurement error due to the "impact" of
the meter, but not enough to cause serious disagreement
with calculated values.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 2: "Ohm's
Law"
LEARNING OBJECTIVES
Voltmeter use
Ammeter use
Ohmmeter use
Use of Ohm's Law
SCHEMATIC DIAGRAM
Ammeter
Battery —
ILLUSTRATION
Ammeter
Terminal
Voltmeter
INSTRUCTIONS
Select a resistor from the assortment, and measure its
resistance with your multimeter set to the appropriate
resistance range. Be sure not to hold the resistor terminals
when measuring resistance, or else your hand-to-hand body
resistance will influence the measurement! Record this
resistance value for future use.
Build a one-battery, one-resistor circuit. A terminal strip is
shown in the illustration, but any form of circuit construction
is okay. Set your multimeter to the appropriate voltage
range and measure voltage across the resistor as it is being
powered by the battery. Record this voltage value along with
the resistance value previously measured.
Set your multimeter to the highest current range available.
Break the circuit and connect the ammeter within that
break, so it becomes a part of the circuit, in series with the
battery and resistor. Select the best current range:
whichever one gives the strongest meter indication without
over-ranging the meter. If your multimeter is autoranging, of
course, you need not bother with setting ranges. Record this
current value along with the resistance and voltage values
previously recorded.
Taking the measured figures for voltage and resistance, use
the Ohm's Law equation to calculate circuit current.
Compare this calculated figure with the measured figure for
circuit Current:
Ohm’s Law
(solving for current)
E
[ =——
R
Where,
E = Voltage in volts
I = Current in amps
R = Resistance in ohms
Taking the measured figures for voltage and current, use the
Ohm's Law equation to calculate circuit resistance. Compare
this calculated figure with the measured figure for circuit
resistance:
Ohm’s Law
(solving for resistance)
bec
I
Finally, taking the measured figures for resistance and
current, use the Ohm's Law equation to calculate circuit
voltage. Compare this calculated figure with the measured
figure for circuit voltage:
Ohm’s Law
(solving for voltage)
E=IR
There should be close agreement between all measured and
all calculated figures. Any differences in respective
quantities of voltage, current, or resistance are most likely
due to meter inaccuracies. These differences should be
rather small, no more than several percent. Some meters, of
course, are more accurate than others!
Substitute different resistors in the circuit and re-take all
resistance, voltage, and current measurements. Re-calculate
these figures and check for agreement with the
experimental data (measured quantities). Also note the
simple mathematical relationship between changes in
resistor value and changes in circuit current. Voltage should
remain approximately the same for any resistor size inserted
into the circuit, because it is the nature of a battery to
maintain voltage at a constant level.
Nonlinear resistance
PARTS AND MATERIALS
e Calculator (or pencil and paper for doing arithmetic)
e 6-volt battery
e Low-voltage incandescent lamp (Radio Shack catalog #
272-1130 or equivalent)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 2: "Ohm's
Law"
LEARNING OBJECTIVES
Voltmeter use
Ammeter use
Ohmmeter use
Use of Ohm's Law
Realization that some resistances are unstable!
Scientific method
SCHEMATIC DIAGRAM
Ammeter
Battery —
ILLUSTRATION
Ammeter
Terminal
strip
INSTRUCTIONS
Measure the resistance of the lamp with your multimeter.
This resistance figure is due to the thin metal "filament"
inside the lamp. It has substantially more resistance than a
jumper wire, but less than any of the resistors from the last
experiment. Record this resistance value for future use.
Build a one-battery, one-lamp circuit. Set your multimeter to
the appropriate voltage range and measure voltage across
the lamp as it is energized (lit). Record this voltage value
along with the resistance value previously measured.
Set your multimeter to the highest current range available.
Break the circuit and connect the ammeter within that
break, so it becomes a part of the circuit, in series with the
battery and lamp. Select the best current range: whichever
one gives the strongest meter indication without over-
ranging the meter. If your multimeter is autoranging, of
course, you need not bother with setting ranges. Record this
current value along with the resistance and voltage values
previously recorded.
Taking the measured figures for voltage and resistance, use
the Ohm's Law equation to calculate circuit current.
Compare this calculated figure with the measured figure for
circuit Current:
Ohm’s Law
(solving for current)
po
R
Where,
E = Voltage in volts
[ = Current in amps
R = Resistance in ohms
What you should find is a marked difference between
measured current and calculated current: the calculated
figure is much greater. Why is this?
To make things more interesting, try measuring the lamp's
resistance again, this time using a different model of meter.
You will need to disconnect the lamp from the battery circuit
in order to obtain a resistance reading, because voltages
outside of the meter interfere with resistance measurement.
This is a general rule that should be remembered: measure
resistance only on an unpowered component!
Using a different ohmmeter, the lamp will probably register
as a different value of resistance. Usually, analog meters
give higher lamp resistance readings than digital meters.
This behavior is very different from that of the resistors in
the last experiment. Why? What factor(s) might influence
the resistance of the lamp filament, and how might those
factors be different between conditions of lit and unlit, or
between resistance measurements taken with different
types of meters?
This problem is a good test case for the application of
scientific method. Once you've thought of a possible reason
for the lamp's resistance changing between lit and unlit
conditions, try to duplicate that cause by some other means.
For example, if you think the lamp resistance might change
as it is exposed to light (its own light, when lit), and that this
accounts for the difference between the measured and
calculated circuit currents, try exposing the lamp to an
external source of light while measuring its resistance. If you
measure substantial resistance change as a result of light
exposure, then your hypothesis has some evidential support.
If not, then your hypothesis has been falsified, and another
cause must be responsible for the change in circuit Current.
Power dissipation
PARTS AND MATERIALS
e Calculator (or pencil and paper for doing arithmetic)
e 6 volt battery
e Two 1/4 watt resistors: 10 QO and 330 Q.
e Small thermometer
The resistor values need not be exact, but within five
percent of the figures specified (+/- 0.5 QO for the 10 O
resistor; +/- 16.5 Q for the 330 O resistor). Color codes for
5% tolerance 10 Q and 330 O resistors are as follows: Brown,
Black, Black, Gold (10, +/- 5%), and Orange, Orange, Brown,
Gold (330, +/- 5%).
Do not use any battery size other than 6 volts for this
experiment.
The thermometer should be as small as possible, to facilitate
rapid detection of heat produced by the resistor. |
recommend a medical thermometer, the type used to take
body temperature.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 2: "Ohm's
Law"
LEARNING OBJECTIVES
Voltmeter use
Ammeter use
Ohmmeter use
Use of Joule's Law
Importance of component power ratings
Significance of electrically common points
SCHEMATIC DIAGRAM
ILLUSTRATION
Thermometer
Caution: do not hold resistor with
your fingers while powered!
INSTRUCTIONS
Measure each resistor's resistance with your ohmmeter,
noting the exact values on a piece of paper for later
reference.
Connect the 330 Q resistor to the 6 volt battery using a pair
of jumper wires as shown in the illustration. Connect the
jumper wires to the resistor terminals before connecting the
other ends to the battery. This will ensure your fingers are
not touching the resistor when battery power is applied.
You might be wondering why | advise no bodily contact with
the powered resistor. This is because it will become hot when
powered by the battery. You will use the thermometer to
measure the temperature of each resistor when powered.
With the 330 Q resistor connected to the battery, measure
voltage with a voltmeter. In measuring voltage, there is more
than one way to obtain a proper reading. Voltage may be
measured directly across the battery, or directly across the
resistor. Battery voltage is the same as resistor voltage in
this circuit, since those two components share the same set
of electrically common points: one side of the resistor is
directly connected to one side of the battery, and the other
side of the resistor is directly connected to the other side of
the battery.
All points of contact along the upper wire in the illustration
(colored red) are electrically common to each other. All
points of contact along the lower wire (colored black) are
likewise electrically common to each other. Voltage
measured between any point on the upper wire and any
point on the lower wire should be the same. Voltage
measured between any two common points, however,
should be zero.
Using an ammeter, measure current through the circuit.
Again, there is no one "correct" way to measure current, so
long as the ammeter is placed within the flow-path of
electrons through the resistor and not across a source of
voltage. To do this, make a break in the circuit, and place the
ammeter within that break: connect the two test probes to
the two wire or terminal ends left open from the break. One
viable option is shown in the following illustration:
YY
[con®
Now that you've measured and recorded resistor resistance,
circuit voltage, and circuit current, you are ready to
calculate power dissipation. Whereas voltage is the measure
of electrical "push" motivating electrons to move through a
circuit, and current is the measure of electron flow rate,
power is the measure of work-rate: how fast work is being
done in the circuit. It takes a certain amount of work to push
electrons through a resistance, and power is a description of
how rapidly that work is taking place. In mathematical
equations, power is symbolized by the letter "P" and
measured in the unit of the Watt (W).
Power may be calculated by any one of three equations --
collectively referred to as Joule's Law -- given any two out of
three quantities of voltage, current, and resistance:
Joule’s Law
(solving for power)
P=IE
P=IR
R
Try calculating power in this circuit, using the three
measured values of voltage, current, and resistance. Any
way you calculate it, the power dissipation figure should be
roughly the same. Assuming a battery with 6.000 volts and a
resistor of exactly 330 QO, the power dissipation will be
0.1090909 watts, or 109.0909 milli-watts (mW), to use a
metric prefix. Since the resistor has a power rating of 1/4
watt (0.25 watts, or 250 mW), it is more than capable of
sustaining this level of power dissipation. Because the
actual power level is almost half the rated power, the
resistor should become noticeably warm but it should not
overheat. Touch the thermometer end to the middle of the
resistor and see how warm it gets.
The power rating of any electrical component does not tell
us how much power it wi// dissipate, but simply how much
power it may dissipate without sustaining damage. If the
actual amount of dissipated power exceeds a component's
power rating, that component will increase temperature to
the point of damage.
To illustrate, disconnect the 330 Q resistor and replace it
with the 10 Q resistor. Again, avoid touching the resistor
once the circuit is complete, as it will heat up rapidly. The
safest way to do this is to disconnect one jumper wire from a
battery terminal, then disconnect the 330 O resistor from
the two alligator clips, then connect the 10 Q resistor
between the two clips, and finally reconnect the jumper wire
back to the battery terminal.
Caution: keep the 10 QO resistor away from any
flammable materials when it is powered by the
battery!
You may not have enough time to take voltage and current
measurements before the resistor begins to smoke. At the
first sign of distress, disconnect one of the jumper wires from
a battery terminal to interrupt circuit current, and give the
resistor a few moments to cool down. With power still
disconnected, measure the resistor's resistance with an
ohmmeter and note any substantial deviation from its
Original value. If the resistor still measures within +/- 5% of
its advertised value (between 9.5 and 10.5 Q), re-connect
the jumper wire and let it smoke a bit more.
What trend do you notice with the resistor's value as it is
damaged more and more by overpowering? It is typical of
resistors to fail with a greater-than-normal resistance when
overheated. This is often a self-protective mode of failure, as
an increased resistance results in less current and
(generally) less power dissipation, cooling it down again.
However, the resistor's normal resistance value will not
return if sufficiently damaged.
Performing some Joule's Law calculations for resistor power
again, we find that a 10 O resistor connected to a 6 volt
battery dissipates about 3.6 watts of power, about 14.4
times its rated power dissipation. Little wonder it smokes so
quickly after connection to the battery!
Circuit with a switch
PARTS AND MATERIALS
6-volt battery
Low-voltage incandescent lamp (Radio Shack catalog #
272-1130 or equivalent)
Long lengths of wire, 22-gauge or larger
Household light switch (these are readily available at
any hardware store)
Household light switches are a bargain for students of basic
electricity. They are readily available, very inexpensive, and
almost impossible to damage with battery power. Do not get
"dimmer" switches, just the simple on-off "toggle" variety
used for ordinary household wall-mounted light controls.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 1: "Basic
Concepts of Electricity"
LEARNING OBJECTIVES
e Switch behavior
e Using an ohmmeter to check switch action
SCHEMATIC DIAGRAM
Switch
ILLUSTRATION
Switch
INSTRUCTIONS
Build a one-battery, one-switch, one-lamp circuit as shown in
the schematic diagram and in the illustration. This circuit is
most impressive when the wires are /ong, as it shows how
the switch is able to control circuit current no matter how
physically large the circuit may be.
Measure voltage across the battery, across the switch
(measure from one screw terminal to another with the
voltmeter), and across the lamp with the switch in both
positions. When the switch is turned off, it is said to be open,
and the lamp will go out just the same as if a wire were
pulled loose from a terminal. As before, any break in the
circuit at any location causes the lamp to immediately de-
energize (darken).
Electromagnetism
PARTS AND MATERIALS
e 6-volt battery
e Magnetic compass
e Small permanent magnet
e Spool of 28-gauge magnet wire
e Large bolt, nail, or steel rod
e Electrical tape
Magnet wire is a term for thin-gauge copper wire with
enamel insulation instead of rubber or plastic insulation. Its
small size and very thin insulation allow for many "turns" to
be wound in a compact coil. You will need enough magnet
wire to wrap hundreds of turns around the bolt, nail, or other
rod-shaped steel form.
Be sure to select a bolt, nail, or rod that is magnetic.
Stainless steel, for example, is non-magnetic and will not
function for the purpose of an electromagnet coil! The ideal
material for this experiment is soft iron, but any commonly
available steel will suffice.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 14:
"Magnetism and Electromagnetism"
LEARNING OBJECTIVES
e Application of the left-hand rule
e Electromagnet construction
SCHEMATIC DIAGRAM
ILLUSTRATION
Electromagnet
(wire coil wrapped
Compass around steel bar)
INSTRUCTIONS
Wrap a single layer of electrical tape around the steel bar (or
bolt, or mail) to protect the wire from abrasion. Proceed to
wrap several hundred turns of wire around the steel bar,
making the coil as even as possible. It is okay to overlap
wire, and it is okay to wrap in the same style that a fishing
reel wraps line around the spool. The only rule you must
follow is that all turns must be wrapped around the bar in
the same direction (no reversing from clockwise to counter-
clockwise!). | find that a drill press works as a great tool for
coil winding: clamp the rod in the drill's chuck as if it were a
drill bit, then turn the drill motor on at a slow speed and let
it do the wrapping! This allows you to feed wire onto the rod
in a very steady, even manner.
After you've wrapped several hundred turns of wire around
the rod, wrap a layer or two of electrical tape over the wire
coil to secure the wire in place. Scrape the enamel insulation
off the ends of the coil wires for connection to jumper leads,
then connect the coil to a battery.
When electric current goes through the coil, it will produce a
strong magnetic field: one "pole" at each end of the rod. This
phenomenon is known as e/ectromagnetism. The magnetic
compass is used to identify the "North" and "South" poles of
the electromagnet.
With the electromagnet energized (connected to the
battery), place a permanent magnet near one pole and note
whether there is an attractive or repulsive force. Reverse the
orientation of the permanent magnet and note the
difference in force.
Electromagnetism has many applications, including relays,
electric motors, solenoids, doorbells, buzzers, computer
printer mechanisms, and magnetic media "write" heads
(tape recorders, disk drives).
You might notice a significant spark whenever the battery is
disconnected from the electromagnet coil: much greater
than the spark produced if the battery is simply short-
circuited. This spark is the result of a high-voltage surge
created whenever current is suddenly interrupted through
the coil. The effect is known as inductive "kickback" and is
capable of delivering a small but harmless electric shock! To
avoid receiving this shock, do not place your body across
the break in the circuit when de-energizing! Use one hand at
a time when un-powering the coil and you'll be perfectly
safe. This phenomenon will be explored in greater detail in
the next chapter (DC Circuits).
Electromagnetic induction
PARTS AND MATERIALS
e Electromagnet from previous experiment
e Permanent magnet
See previous experiment for instructions on electromagnet
construction.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 14:
"Magnetism and Electromagnetism"
LEARNING OBJECTIVES
e Relationship between magnetic field strength and
induced voltage
SCHEMATIC DIAGRAM
+
Voltmeter (Vv)
ILLUSTRATION
Electromagnet
INSTRUCTIONS
Electromagnetic induction is the complementary
phenomenon to electromagnetism. Instead of producing a
magnetic field from electricity, we produce electricity from a
magnetic field. There is one important difference, though:
whereas electromagnetism produces a steady magnetic field
from a steady electric current, electromagnetic induction
requires motion between the magnet and the coil to produce
a voltage.
Connect the multimeter to the coil, and set it to the most
sensitive DC voltage range available. Move the magnet
slowly to and from one end of the electromagnet, noting the
polarity and magnitude of the induced voltage. Experiment
with moving the magnet, and discover for yourself what
factor(s) determine the amount of voltage induced. Try the
other end of the coil and compare results. Try the other end
of the permanent magnet and compare.
If using an analog multimeter, be sure to use long jumper
wires and locate the meter far away from the coil, as the
magnetic field from the permanent magnet may affect the
meter's operation and produce false readings. Digital meters
are unaffected by magnetic fields.
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4]
—/ | 4]
Lessons In Electric Circuits
-- Volume VI
Chapter 3
DC CIRCUITS
Introduction
Series batteries
Parallel batteries
e Voltage divider
e Current divider
e Potentiometer as a voltage divider
e Potentiometer as a rheostat
e Precision potentiometer
e Rheostat range limiting
e Thermoelectricity
e Make your own multimeter
e Sensitive voltage detector
e Potentiometric voltmeter
e 4-wire resistance measurement
e Avery simple computer
e Potato battery
e Capacitor charging_and discharging
e Rate-of-change indicator
Introduction
"DC" stands for Direct Current, which can refer to either
voltage or current in a constant polarity or direction,
respectively. These experiments are designed to introduce
you to several important concepts of electricity related to DC
Circuits.
Series batteries
PARTS AND MATERIALS
e Two 6-volt batteries
e One 9-volt battery
Actually, any size batteries will suffice for this experiment,
but it is recommended to have at least two different
voltages available to make it more interesting.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 5: "Series and
Parallel Circuits"
Lessons In Electric Circuits, Volume 1, chapter 11: "Batteries
and Power Systems"
LEARNING OBJECTIVES
e How to connect batteries to obtain different voltage
levels
SCHEMATIC DIAGRAM
IF
+
Voltmeter
lun
ILLUSTRATION
INSTRUCTIONS
Connecting components in series means to connect them in-
line with each other, so that there is but a single path for
electrons to flow through them all. If you connect batteries
so that the positive of one connects to the negative of the
other, you will find that their respective voltages add.
Measure the voltage across each battery individually as they
are connected, then measure the total voltage across them
both, like this:
6.00
Measuring
total voltage viel
=
A*/ Measuring (A)
second battery
Try connecting batteries of different sizes in series with each
other, for instance a 6-volt battery with a 9-volt battery.
What is the total voltage in this case? Try reversing the
terminal connections of just one of these batteries, so that
they are opposing each other like this:
opposing
al
| T +
Series- | Voltmeter
+1
7
How does the total voltage compare in this situation to the
previous one with both batteries "aiding?" Note the polarity
of the total voltage as indicated by the voltmeter indication
and test probe orientation. Remember, if the meter's digital
indication is a positive number, the red probe is positive (+)
and the black probe negative (-); if the indication is a
negative number, the polarity is "backward" (red=negative,
black=positive). Analog meters simply will not read properly
if reverse-connected, because the needle tries to move the
wrong direction (left instead of right). Can you predict what
the overall voltage polarity will be, knowing the polarities of
the individual batteries and their respective strengths?
Parallel batteries
PARTS AND MATERIALS
e Four 6-volt batteries
e 12-volt light bulb, 25 or 50 watt
e Lamp socket
High-wattage 12-volt lamps may be purchased from
recreational vehicle (RV) and boating supply stores.
Common sizes are 25 watt and 50 watt. This lamp will be
used asa "heavy" load for your batteries (heavy load = one
that draws substantial current).
A regular household (120 volt) lamp socket will work just
fine for these low-voltage "RV" lamps.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 5: "Series and
Parallel Circuits"
Lessons In Electric Circuits, Volume 1, chapter 11: "Batteries
and Power Systems"
LEARNING OBJECTIVES
e Voltage source regulation
e Boosting current capacity through parallel connections
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
Begin this experiment by connecting one 6-volt battery to
the lamp. The lamp, designed to operate on 12 volts, should
glow dimly when powered by the 6-volt battery. Use your
voltmeter to read voltage across the lamp like this:
The voltmeter should register a voltage lower than the usual
voltage of the battery. If you use your voltmeter to read the
voltage directly at the battery terminals, you will measure a
low voltage there as well. Why is this? The large current
drawn by the high-power lamp causes the voltage at the
battery terminals to "sag" or "droop," due to voltage
dropped across resistance internal to the battery.
We may overcome this problem by connecting batteries in
paralle/ with each other, so that each battery only has to
supply a fraction of the total current demanded by the lamp.
Parallel connections involve making all the positive (+)
battery terminals electrically common to each other by
connection through jumper wires, and all negative (-)
terminals common to each other as well. Add one battery at
a time in parallel, noting the lamp voltage with the addition
of each new, parallel-connected battery:
There should also be a noticeable difference in light
intensity as the voltage "sag" is improved.
Try measuring the current of one battery and comparing it to
the total current (light bulb current). Shown here is the
easiest way to measure single-battery current:
By breaking the circuit for just one battery, and inserting our
ammeter within that break, we intercept the current of that
one battery and are therefore able to measure it. Measuring
total current involves a similar procedure: make a break
somewhere in the path that total current must take, then
insert the ammeter within than break:
Note the difference in current between the single-battery
and total measurements.
To obtain maximum brightness from the light bulb, a series-
parallel connection is required. Two 6-volt batteries
connected series-aiding will provide 12 volts. Connecting
two of these series-connected battery pairs in parallel
improves their current-sourcing ability for minimum voltage
Sag:
Voltage divider
PARTS AND MATERIALS
e Calculator (or pencil and paper for doing arithmetic)
e 6-volt battery
e Assortment of resistors between 1 KQ and 100 kQ in
value
I'm purposely restricting the resistance values between 1 kO
and 100 kQ for the sake of obtaining accurate voltage and
current readings with your meter. With very low resistance
values, the internal resistance of the ammeter has a
significant impact on measurement accuracy. Very high
resistance values may cause problems for voltage
measurement, the internal resistance of the voltmeter
substantially changing circuit resistance when it is
connected in parallel with a high-value resistor.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 6: "Divider
Circuits and Kirchhoff's Laws"
LEARNING OBJECTIVES
Voltmeter use
Ammeter use
Ohmmeter use
Use of Ohm's Law
Use of Kirchhoff's Voltage Law ("KVL")
Voltage divider design
SCHEMATIC DIAGRAM
4
Voltmeter
ILLUSTRATION
Breadboard
" construction
"Free-form
INSTRUCTIONS
Shown here are three different methods of circuit
construction: on a breadboard, on a terminal strip, and "free-
form." Try building the same circuit each way to familiarize
yourself with the different construction techniques and their
respective merits. The "free-form" method -- where all
components are connected together with "alligator-" style
jumper wires -- is the least professional, but appropriate for a
simple experiment such as this. Breadboard construction is
versatile and allows for high component density (many parts
in a small space), but is quite temporary. Terminal strips offer
a much more permanent form of construction at the cost of
low component density.
Select three resistors from your resistor assortment and
measure the resistance of each one with an ohmmeter. Note
these resistance values with pen and paper, for reference in
your circuit calculations.
Connect the three resistors in series, and to the 6-volt
battery, as shown in the illustrations. Measure battery
voltage with a voltmeter after the resistors have been
connected to it, noting this voltage figure on paper as well.
It is advisable to measure battery voltage while its powering
the resistor circuit because this voltage may differ slightly
from a no-load condition. We saw this effect exaggerated in
the "parallel battery" experiment while powering a high-
wattage lamp: battery voltage tends to "sag" or "droop"
under load. Although this three-resistor circuit should not
present a heavy enough load (not enough current drawn) to
cause Significant voltage "sag," measuring battery voltage
under load is a good scientific practice because it provides
more realistic data.
Use Ohm's Law (I=E/R) to calculate circuit current, then
verify this calculated value by measuring current with an
ammeter like this ("terminal strip" version of the circuit
shown as an arbitrary choice in construction method):
If your resistor values are indeed between 1 kO and 100 kQ,
and the battery voltage approximately 6 volts, the current
should be a very small value, in the milliamp (mA) or
microamp (uA) range. When you measure current with a
digital meter, the meter may show the appropriate metric
prefix symbol (m or uw) in some corner of the display. These
metric prefix telltales are easy to overlook when reading the
display of a digital meter, so pay close attention!
The measured value of current should agree closely with
your Ohm's Law calculation. Now, take that calculated value
for current and multiply it by the respective resistances of
each resistor to predict their voltage drops (E=IR). Switch
you multimeter to the "voltage" mode and measure the
voltage dropped across each resistor, verifying the accuracy
of your predictions. Again, there should be close agreement
between the calculated and measured voltage figures.
Each resistor voltage drop will be some fraction or
percentage of the total voltage, hence the name vo/tage
divider given to this circuit. This fractional value is
determined by the resistance of the particular resistor and
the total resistance. If a resistor drops 50% of the total
battery voltage in a voltage divider circuit, that proportion
of 50% will remain the same as long as the resistor values
are not altered. So, if the total voltage is 6 volts, the voltage
across that resistor will be 50% of 6, or 3 volts. If the total
voltage is 20 volts, that resistor will drop 10 volts, or 50% of
20 volts.
The next part of this experiment is a validation of Kirchhoff's
Voltage Law. For this, you need to identify each unique point
in the circuit with a number. Points that are electrically
common (directly connected to each other with insignificant
resistance between) must bear the same number. An
example using the numbers 0 through 3 is shown here in
both illustrative and schematic form. In the illustration, |
show how points in the circuit may be labeled with small
pieces of tape, numbers written on the tape:
@
Terminal strip
0 0
Using a digita/ voltmeter (this is important!), measure
voltage drops around the loop formed by the points 0-1-2-3-
0. Write on paper each of these voltages, along with its
respective sign as indicated by the meter. In other words, if
the voltmeter registers a negative voltage such as -1.325
volts, you should write that figure as a negative number. Do
not reverse the meter probe connections with the circuit to
make the number read "correctly." Mathematical sign is very
significant in this phase of the experiment! Here is a
sequence of illustrations showing how to "step around" the
circuit loop, starting and ending at point 0:
Measure voltage from
Measure voltage from
Measure voltage from
3 to 2
Measure voltage from
0 to 3
Using the voltmeter to "step" around the circuit in this
manner yields three positive voltage figures and one
negative:
3 3
These figures, algebraically added ("algebraically" =
respecting the signs of the numbers), should equal zero.
This is the fundamental principle of Kirchhoff's Voltage Law:
that the algebraic sum of all voltage drops in a "loop" add to
zero.
It is important to realize that the "loop" stepped around does
not have to be the same path that current takes in the
circuit, or even a legitimate current path at all. The loop in
which we tally voltage drops can be any collection of points,
so long as it begins and ends with the same point. For
example, we may measure and add the voltages in the loop
1-2-3-1, and they will form a sum of zero as well:
Measure voltage from
3 to 2
Measure voltage from
| to 3
oY),
Q)/ (ALARA
Try stepping between any set of points, in any order, around
your circuit and see for yourself that the algebraic sum
always equals zero. This Law holds true no matter what the
configuration of the circuit: series, parallel, series-parallel, or
even an irreducible network.
Kirchhoff's Voltage Law is a powerful concept, allowing us to
predict the magnitude and polarity of voltages in a circuit by
developing mathematical equations for analysis based on
the truth of all voltages in a loop adding up to zero. This
experiment is intended to give empirical evidence for and a
deep understanding of Kirchhoff's Voltage Law as a general
principle.
COMPUTER SIMULATION
Netlist (make a text file containing the following text,
verbatim):
Voltage divider
vl 3 0
rl 3 2 5k
r2 2 1 3k
r3 1 0 2k
.dc vl 661
* Voltages around 0-1-2-3-0 se algebraically add to zero:
.print dc v(1,0) v(2,1) v(3,2) v(0,3)
* Voltages around 1-2-3-1 Loop algebraically add to zero:
.print dc v(2,1) v(3,2) v(1,3)
end
This computer simulation is based on the point numbers
shown in the previous diagrams for illustrating Kirchhoff's
Voltage Law (points 0 through 3). Resistor values were
chosen to provide 50%, 30%, and 20% proportions of total
voltage across Rj, R3, and R3, respectively. Feel free to
modify the voltage source value (in the ".dc" line, shown
here as 6 volts), and/or the resistor values.
When run, SPICE will print a line of text containing four
voltage figures, then another line of text containing three
voltage figures, along with lots of other text lines describing
the analysis process. Add the voltage figures in each line to
see that the sum is zero.
Current divider
PARTS AND MATERIALS
e Calculator (or pencil and paper for doing arithmetic)
e 6-volt battery
e Assortment of resistors between 1 KQ and 100 kQ in
value
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 6: "Divider
Circuits and Kirchhoff's Laws"
LEARNING OBJECTIVES
e Voltmeter use
e Ammeter use
e Ohmmeter use
e Use of Ohm's Law
e Use of Kirchhoff's Current Law (KCL)
e Current divider design
SCHEMATIC DIAGRAM
Ammeter
ILLUSTRATION
ooo oo ooocoeoees%oss 0
Breadboard
Terminal
Normally, it is considered improper to secure more than two
wires under a single terminal strip screw. In this illustration, |
show three wires joining at the top screw of the rightmost
lug used on this strip. This is done for the ease of proving a
concept (of current summing at a circuit node), and does not
represent professional assembly technique.
Piece of stiff wire serves
an as a connection point
"Free-form" construction
The non-professional nature of the "free-form" construction
method merits no further comment.
INSTRUCTIONS
Once again, | show different methods of constructing the
Same circuit: breadboard, terminal strip, and "free-form."
Experiment with all these construction formats and become
familiar with their respective advantages and
disadvantages.
Select three resistors from your resistor assortment and
measure the resistance of each one with an ohmmeter. Note
these resistance values with pen and paper, for reference in
your circuit calculations.
Connect the three resistors in parallel to and each other, and
with the 6-volt battery, as shown in the illustrations.
Measure battery voltage with a voltmeter after the resistors
have been connected to it, noting this voltage figure on
paper as well. It is advisable to measure battery voltage
while its powering the resistor circuit because this voltage
may differ slightly from a no-load condition.
Measure voltage across each of the three resistors. What do
you notice? In a series circuit, current is equal through all
components at any given time. In a parallel circuit, vo/tage
is the common variable between all components.
Use Ohm's Law (I=E/R) to calculate current through each
resistor, then verify this calculated value by measuring
current with a digital ammeter. Place the red probe of the
ammeter at the point where the positive (+) ends of the
resistors connect to each other and lift one resistor wire at a
time, connecting the meter's black probe to the lifted wire.
In this manner, measure each resistor current, noting both
the magnitude of the current and the polarity. In these
illustrations, | show an ammeter used to measure the current
through Rj:
Breadboard
Measure current for each of the three resistors, comparing
with the current figures calculated previously. With the
digital ammeter connected as shown, all three indications
should be positive, not negative.
Now, measure total circuit current, keeping the ammeter's
red probe on the same point of the circuit, but disconnecting
the wire leading to the positive (+) side of the battery and
touching the black probe to it:
Breadboard
Note both the magnitude and the sign of the current as
indicated by the ammeter. Add this figure (algebraically) to
the three resistor currents. What do you notice about the
result that is similar to the Kirchhoff's Voltage Law
experiment? Kirchhoff's Current Law is to currents
"Summing" at a point (node) in a circuit, just as Kirchhoff's
Voltage Law is to voltages adding in a series loop: in both
cases, the algebraic sum is equal to zero.
This Law is also very useful in the mathematical analysis of
circuits. Along with Kirchhoff's Voltage Law, it allows us to
generate equations describing several variables in a circuit,
which may then be solved using a variety of mathematical
techniques.
Now consider the four current measurements as all positive
numbers: the first three representing the current through
each resistor, and the fourth representing total circuit
current as a positive sum of the three "branch" currents.
Each resistor (branch) current is a fraction, or percentage, of
the total current. This is why a parallel resistor circuit is
often called a current divider.
Disconnect the battery from the rest of the circuit, and
measure resistance across the parallel resistors. You may
read total resistance across any of the individual resistors’
terminals and obtain the same indication: it will be a value
less than any of the individual resistor values. This is often
surprising to new students of electricity, that you read the
exact same (total) resistance figure when connecting an
ohmmeter across any one of a set of parallel-connected
resistors. It makes sense, though, if you consider the points
in a parallel circuit in terms of electrical commonality. All
parallel components are connected between two sets of
electrically common points. Since the meter cannot
distinguish between points common to each other by way of
direct connection, to read resistance across one resistor is to
read the resistance of them all. The same is true for voltage,
which is why battery voltage could be read across any one of
the resistors as easily as it could be read across the battery
terminals directly.
If you divide the battery voltage (previously measured) by
this total resistance figure, you should obtain a figure for
total current (I=E/R) closely matching the measured figure.
The ratio of resistor current to total current is the same as
the ratio of total resistance to individual resistance. For
example, if a 10 kQ resistor is part of a current divider circuit
with a total resistance of 1 kQ, that resistor will conduct 1/10
of the total current, whatever value that current total
happens to be.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
V
Ammeters in SPICE simulations are actually zero-voltage
sources inserted in the paths of electron flow. You will notice
the voltage sources V;,1, Vix, and V;-3 are set to 0 volts in the
netlist. When electrons enter the negative side of one of
these "dummy" batteries and out the positive, the battery's
Current indication will be a positive number. In other words,
these 0-volt sources are to be regarded as ammeters with
the red probe on the long-line side of the battery symbol
and the black probe on the short-line side.
Netlist (make a text file containing the following text,
verbatim):
Current divider
vl 10
rl 3 0 2k
r2 4 0 3k
r3 5 0 5k
vitotal 2 1 dc 0
virl 2 3 dc 0
vir2 2 4 dc 0
vir3 2 5 dc 0
.dc vl 661
.print dc i(vitotal) i(virl) i(vir2) i(vir3)
end
When run, SPICE will print a line of text containing four
current figures, the first current representing the total asa
negative quantity, and the other three representing currents
for resistors Rj, Ro, and R3. When algebraically added, the
one negative figure and the three positive figures will form a
sum of zero, as described by Kirchhoff's Current Law.
Potentiometer as a voltage divider
PARTS AND MATERIALS
Two 6-volt batteries
Carbon pencil "lead" for a mechanical-style pencil
Potentiometer, single turn, 5 kQ to 50 kQ, linear taper
(Radio Shack catalog # 271-1714 through 271-1716)
e Potentiometer, multi turn, 1 kKQ to 20 kQ, (Radio Shack
catalog # 271-342, 271-343, 900-8583, or 900-8587
through 900-8590)
Potentiometers are variable voltage dividers with a shaft or
slide control for setting the division ratio. They are
manufactured in panel-mount as well as breadboard
(printed-circuit board) mount versions. Any style of
potentiometer will suffice for this experiment.
If you salvage a potentiometer from an old radio or other
audio device, you will likely be getting what is called an
audio taper potentiometer. These potentiometers exhibit a
logarithmic relationship between division ratio and shaft
position. By contrast, a /inear potentiometer exhibits a direct
correlation between shaft position and voltage division ratio.
| highly recommend a linear potentiometer for this
experiment, and for most experiments in general.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 6: "Divider
Circuits and Kirchhoff's Laws"
LEARNING OBJECTIVES
e Voltmeter use
e Ohmmeter use
e Voltage divider design and function
e How voltages add in series
SCHEMATIC DIAGRAM
Potentiometer
ILLUSTRATION
Pencil "lead"
f
i
RP
Potentiometer
Potentiometer
INSTRUCTIONS
Begin this experiment with the pencil "lead" circuit. Pencils
use a rod made of a graphite-clay mixture, not lead (the
metal), to make black marks on paper. Graphite, being a
mediocre electrical conductor, acts as a resistor connected
across the battery by the two alligator-clip jumper wires.
Connect the voltmeter as shown and touch the red test
probe to the graphite rod. Move the red probe along the
length of the rod and notice the voltmeter's indication
change. What probe position gives the greatest voltage
indication?
Essentially, the rod acts as a pair of resistors, the ratio
between the two resistances established by the position of
the red test probe along the rod's length:
Pencil "lead"
equivalent to
Now, change the voltmeter connection to the circuit so as to
measure voltage across the "upper resistor" of the pencil
lead, like this:
Move the black test probe position along the length of the
rod, noting the voltmeter indication. Which position gives
the greatest voltage drop for the meter to measure? Does
this differ from the previous arrangement? Why?
Manufactured potentiometers enclose a resistive strip inside
a metal or plastic housing, and provide some kind of
mechanism for moving a "wiper" across the length of that
resistive strip. Here is an illustration of a rotary
potentiometer's construction:
Terminals
f\\
Rotary potentiometer
construction
Wiper
Resistive strip
Some rotary potentiometers have a Spiral resistive strip, and
a wiper that moves axially as it rotates, so as to require
multiple turns of the shaft to drive the wiper from one end of
the potentiometer's range to the other. Multi-turn
potentiometers are used in applications where precise
setting is important.
Linear potentiometers also contain a resistive strip, the only
difference being the wiper's direction of travel. Some linear
potentiometers use a slide mechanism to move the wiper,
while others a screw, to facilitate multiple-turn operation:
Linear potentiometer construction
Wiper as
Resistive strip
/
Terminals
It should be noted that not all linear potentiometers have
the same pin assignments. On some, the middle pin is the
wiper.
Set up a circuit using a manufactured potentiometer, not the
"home-made" one made from a pencil lead. You may use any
form of construction that is convenient.
Measure battery voltage while powering the potentiometer,
and make note of this voltage figure on paper. Measure
voltage between the wiper and the potentiometer end
connected to the negative (-) side of the battery. Adjust the
potentiometer mechanism until the voltmeter registers
exactly 1/3 of total voltage. For a 6-volt battery, this will be
approximately 2 volts.
Now, connect two batteries in a series-aiding configuration,
to provide approximately 12 volts across the potentiometer.
Measure the total battery voltage, and then measure the
voltage between the same two points on the potentiometer
(wiper and negative side). Divide the potentiometer's
measured output voltage by the measured total voltage. The
quotient should be 1/3, the same voltage division ratio as
was Set previously:
Voltmeter measuring output
of potentiometer.
Potentiometer as a rheostat
PARTS AND MATERIALS
e 6 volt battery
e Potentiometer, single turn, 5 kQ, linear taper (Radio
Shack catalog # 271-1714)
e Small "hobby" motor, permanent-magnet type (Radio
Shack catalog # 273-223 or equivalent)
For this experiment, you will need a relatively low-value
potentiometer, certainly not more than 5 kQ.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 2: "Ohm's
Law"
LEARNING OBJECTIVES
e Rheostat use
e Wiring a potentiometer as a rheostat
e Simple motor speed control
e Use of voltmeter over ammeter to verify a continuous
circuit
SCHEMATIC DIAGRAM
ILLUSTRATION
Potentiometer
INSTRUCTIONS
Potentiometers find their most sophisticated application as
voltage dividers, where shaft position determines a specific
voltage division ratio. However, there are applications where
we don't necessarily need a variable voltage divider, but
merely a variable resistor: a two-terminal device.
Technically, a variable resistor is known as a rheostat, but
potentiometers can be made to function as rheostats quite
easily.
In its simplest configuration, a potentiometer may be used
as a rheostat by simply using the wiper terminal and one of
the other terminals, the third terminal left unconnected and
unused:
Motor
} Potentiometer
Moving the potentiometer control in the direction that
brings the wiper closest to the other used terminal results in
a lower resistance. The direction of motion required to
increase or decrease resistance may be changed by using a
different set of terminals:
Less resistance when turned clockwise More resistance when turned clockwise
Wiper Wiper
Resistive strip Resistive strip
Be careful, though, that you don't use the two outer
terminals, as this will result in no change in resistance as the
potentiometer shaft is turned. In other words, it will no
longer function as a variable resistance:
No resistance change when wiper moves!
f \
Build the circuit as shown in the schematic and illustration,
using just two terminals on the potentiometer, and see how
motor speed may be controlled by adjusting shaft position.
Experiment with different terminal connections on the
potentiometer, noting the changes in motor speed control. If
your potentiometer has a high resistance (as measured
between the two outer terminals), the motor might not move
at all until the wiper is brought very close to the connected
outer terminal.
As you Can see, motor speed may be made variable using a
series-connected rheostat to change total circuit resistance
and limit total current. This simple method of motor speed
control, however, is inefficient, as it results in substantial
amounts of power being dissipated (wasted) by the rheostat.
A much more efficient means of motor control relies on fast
"pulsing" of power to the motor, using a high-speed
switching device such as a transistor. A similar method of
power control is used in household light "dimmer" switches.
Unfortunately, these techniques are much too sophisticated
to explore at this point in the experiments.
When a potentiometer is used as a rheostat, the "Unused"
terminal is often connected to the wiper terminal, like this:
At first, this seems rather pointless, as it has no impact on
resistance control. You may verify this fact for yourself by
inserting another wire in your circuit and comparing motor
behavior before and after the change:
~«— add wire
Potentiometer
If the potentiometer is in good working order, this additional
wire makes no difference whatsoever. However, if the wiper
ever loses contact with the resistive strip inside the
potentiometer, this connection ensures the circuit does not
completely open: that there will still be a resistive path for
current through the motor. In some applications, this may be
an important. Old potentiometers tend to suffer from
intermittent losses of contact between the wiper and the
resistive strip, and if a circuit cannot tolerate the complete
loss of continuity (infinite resistance) created by this
condition, that "extra" wire provides a measure of protection
by maintaining circuit continuity.
You may simulate such a wiper contact "failure" by
disconnecting the potentiometer's middle terminal from the
terminal strip, measuring voltage across the motor to ensure
there is still power getting to it, however small:
(+ | [cue
It would have been valid to measure circuit current instead
of motor voltage to verify a completed circuit, but this isa
safer method because it does not involve breaking the
circuit to insert an ammeter in series. Whenever an ammeter
is used, there is risk of causing a short circuit by connecting
it across a substantial voltage source, possibly resulting in
instrument damage or personal injury. Voltmeters lack this
inherent safety risk, and so whenever a voltage
measurement may be made instead of a current
measurement to verify the same thing, it is the wiser choice.
Precision potentiometer
PARTS AND MATERIALS
e Two single-turn, linear-taper potentiometers, 5 kO each
(Radio Shack catalog # 271-1714)
e One single-turn, linear-taper potentiometer, 50 kO
(Radio Shack catalog # 271-1716)
e Plastic or metal mounting box
e Three "banana" jack style binding posts, or other
terminal hardware, for connection to potentiometer
circuit (Radio Shack catalog # 274-662 or equivalent)
This is a project useful to those who want a precision
potentiometer without spending a lot of money. Ordinarily,
multi-turn potentiometers are used to obtain precise voltage
division ratios, but a cheaper alternative exists using
multiple, single-turn (sometimes called "3/4-turn")
potentiometers connected together in a compound divider
network.
Because this is a useful project, | recommend building it in
permanent form using some form of project enclosure.
Suppliers such as Radio Shack offer nice project boxes, but
boxes purchased at a general hardware store are much less
expensive, if a bit ugly. The ultimate in low cost for a new
box are the plastic boxes sold as light switch and receptacle
boxes for household electrical wiring.
"Banana" jacks allow for the temporary connection of test
leads and jumper wires equipped with matching "banana"
plug ends. Most multimeter test leads have this style of plug
for insertion into the meter jacks. Banana plugs are so
named because of their oblong appearance formed by
spring steel strips, which maintain firm contact with the jack
walls when inserted. Some banana jacks are called binding
posts because they also allow plain wires to be firmly
attached. Binding posts have screw-on sleeves that fit over a
metal post. The sleeve is used as a nut to secure a wire
wrapped around the post, or inserted through a
perpendicular hole drilled through the post. A brief
inspection of any binding post will clarify this verbal
description.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 6: "Divider
Circuits and Kirchhoff's Laws"
LEARNING OBJECTIVES
e Soldering practice
e Potentiometer function and operation
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
It is essential that the connecting wires be so/dered to the
potentiometer terminals, not twisted or taped. Since
potentiometer action is dependent on resistance, the
resistance of all wiring connections must be carefully
controlled to a bare minimum. Soldering ensures a condition
of low resistance between joined conductors, and also
provides very good mechanical strength for the connections.
When the circuit is assembled, connect a 6-volt battery to
the outer two binding posts. Connect a voltmeter between
the "wiper" post and the battery's negative (-) terminal. This
voltmeter will measure the "output" of the circuit.
The circuit works on the principle of compressed range: the
voltage output range of this circuit available by adjusting
potentiometer R3 is restricted between the limits set by
potentiometers R, and R>. In other words, if R; and R> were
set to output 5 volts and 3 volts, respectively, from a 6-volt
battery, the range of output voltages obtainable by
adjusting R3 would be restricted from 3 to 5 volts for the full
rotation of that potentiometer. If only a single potentiometer
were used instead of this three-potentiometer circuit, full
rotation would produce an output voltage from 0 volts to full
battery voltage. The "range compression" afforded by this
circuit allows for more precise voltage adjustment than
would be normally obtainable using a single potentiometer.
Operating this potentiometer network is more complex than
using a single potentiometer. To begin, turn the R3
potentiometer fully clockwise, so that its wiper is in the full
"up" position as referenced to the schematic diagram
(electrically "closest" to R;'s wiper terminal). Adjust
potentiometer R, until the upper voltage limit is reached, as
indicated by the voltmeter.
Turn the R3 potentiometer fully counter-clockwise, so that its
wiper is in the full "down" position as referenced to the
schematic diagram (electrically "closest" to R's wiper
terminal). Adjust potentiometer R> until the lower voltage
limit is reached, as indicated by the voltmeter.
When either the R; or the Ry potentiometer is adjusted, it
interferes with the prior setting of the other. In other words,
if R, is initially adjusted to provide an upper voltage limit of
5.000 volts from a 6 volt battery, and then R> is adjusted to
provide some lower limit voltage different from what it was
before, R, will no longer be set to 5.000 volts.
To obtain precise upper and lower voltage limits, turn R3
fully clockwise to read and adjust the voltage of R;, then
turn R3 fully counter-clockwise to read and adjust the
voltage of R>, repeating as necessary.
Technically, this phenomenon of one adjustment affecting
the other is known as interaction, and it is usually
undesirable due to the extra effort required to set and re-set
the adjustments. The reason that R,; and R> were specified
as 10 times less resistance than R3 is to minimize this effect.
If all three potentiometers were of equal resistance value,
the interaction between R, and R> would be more severe,
though manageable with patience. Bear in mind that the
upper and lower voltage limits need not be set precisely in
order for this circuit to achieve its goal of increased
precision. So long as R3's adjustment range is compressed to
some lesser value than full battery voltage, we will enjoy
greater precision than a single potentiometer could provide.
Once the upper and lower voltage limits have been set,
potentiometer R3 may be adjusted to produce an output
voltage anywhere between those limits.
Rheostat range limiting
PARTS AND MATERIALS
e Several 10 kQ resistors
e One 10 kQ potentiometer, linear taper (Radio Shack
catalog # 271-1715)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 5: "Series and
Parallel Circuits"
Lessons In Electric Circuits, Volume 1, chapter 7: "Series-
Parallel Combination Circuits"
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
LEARNING OBJECTIVES
e Series-parallel resistances
e Calibration theory and practice
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
This experiment explores the different resistance ranges
obtainable from combining fixed-value resistors with a
potentiometer connected as a rheostat. To begin, connect a
10 kQ potentiometer as a rheostat with no other resistors
connected. Adjusting the potentiometer through its full
range of travel should result in a resistance that varies
smoothly from 0 QO to 10,000 Q:
Suppose we wanted to elevate the lower end of this
resistance range so that we had an adjustable range from 10
kQ to 20 kQ with a full sweep of the potentiometer's
adjustment. This could be easily accomplished by adding a
10 kQ resistor in series with the potentiometer. Add one to
the circuit as shown and re-measure total resistance while
adjusting the potentiometer:
A shift in the low end of an adjustment range is called a zero
calibration, in metrological terms. With the addition of a
series 10 kQO resistor, the "zero point" was shifted upward by
10,000 Q. The difference between high and low ends of a
range -- called the span of the circuit -- has not changed,
though: a range of 10 kQ to 20 kQ has the same 10,000 Q
span as a range of 0 Oto 10 kQ. If we wish to shift the span
of this rheostat circuit as well, we must change the range of
the potentiometer itself. We could replace the potentiometer
with one of another value, or we could simulate a lower-
value potentiometer by placing a resistor in para//e/ with it,
diminishing its maximum obtainable resistance. This will
decrease the span of the circuit from 10 kQ to something
less.
Add a 10 kQ resistor in parallel with the potentiometer, to
reduce the span to one-half of its former value: from 10 KO
to 5 kQ. Now the calibrated resistance range of this circuit
will be 10 kQO to 15 kQ:
There is nothing we can do to /ncrease the span of this
rheostat circuit, short of replacing the potentiometer with
another of greater total resistance. Adding resistors in
parallel can only decrease the span. However, there is no
such restriction with calibrating the zero point of this circuit,
as it began at 0 OQ and may be made as great as we wish by
adding resistance in series.
A multitude of resistance ranges may be obtained using only
10 KQ fixed-value resistors, if we are creative with series-
parallel combinations of them. For instance, we can create a
range of 7.5 kQ to 10 kQ by building the following circuit:
All resistors = 10 kQ
Creating a custom resistance range from fixed-value
resistors and a potentiometer is a very useful technique for
producing precise resistances required for certain circuits,
especially meter circuits. In many electrical instruments --
multimeters especially -- resistance is the determining factor
for the instrument's range of measurement. If an
instrument's internal resistance values are not precise,
neither will its indications be. Finding a fixed-value resistor
of just the right resistance for placement in an instrument
circuit design is unlikely, so custom resistance "networks"
may need to be built to provide the desired resistance.
Having a potentiometer as part of the resistor network
provides a means of correction if the network's resistance
should "drift" from its original value. Designing the network
for minimum span ensures that the potentiometer's effect
will be small, so that precise adjustment is possible and so
that accidental movement of its mechanism will not result in
severe calibration errors.
Experiment with different resistor "networks" and note the
effects on total resistance range.
Thermoelectricity
PARTS AND MATERIALS
e Length of bare (uninsulated) copper wire
e Length of bare (uninsulated) iron wire
e Candle
e Ice cubes
lron wire may be obtained from a hardware store. If some
cannot be found, aluminum wire also works.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 9: "Electrical
Instrumentation Signals"
LEARNING OBJECTIVES
e Thermocouple function and purpose
SCHEMATIC DIAGRAM
+
ILLUSTRATION
iron wire
=
copper wire
Candle
INSTRUCTIONS
Twist one end of the iron wire together with one end of the
copper wire. Connect the free ends of these wires to
respective terminals on a terminal strip. Set your voltmeter
to its most sensitive range and connect it to the terminals
where the wires attach. The meter should indicate nearly
zero voltage.
What you have just constructed is a thermocouple: a device
which generates a small voltage proportional to the
temperature difference between the tip and the meter
connection points. When the tip is at a temperature equal to
the terminal strip, there will be no voltage produced, and
thus no indication seen on the voltmeter.
Light a candle and insert the twisted-wire tip into the flame.
You should notice an indication on your voltmeter. Remove
the thermocouple tip from the flame and let cool until the
voltmeter indication is nearly zero again. Now, touch the
thermocouple tip to an ice cube and note the voltage
indicated by the meter. Is it a greater or lesser magnitude
than the indication obtained with the flame? How does the
polarity of this voltage compare with that generated by the
flame?
After touching the thermocouple tip to the ice cube, warm it
by holding it between your fingers. It may take a short while
to reach body temperature, so be patient while observing
the voltmeter's indication.
A thermocouple is an application of the Seebeck effect: the
production of a small voltage proportional to a temperature
gradient along the length of a wire. This voltage is
dependent upon the magnitude of the temperature
difference and the type of wire. Directly measuring the
Seebeck voltage produced along a length of continuous wire
from a temperature gradient is quite difficult, and so will not
be attempted in this experiment.
Thermocouples, being made of two dissimilar metals joined
at one end, produce a voltage proportional to the
temperature of the junction. The temperature gradient along
both wires resulting from a constant temperature at the
junction produces different Seebeck voltages along those
wires' lengths, because the wires are made of different
metals. The resultant voltage between the two free wire
ends is the difference between the two Seebeck voltages:
iron wire voltage
f \ ~— Resultant
HOT COOL 1. — voltage
‘ VA
copper wire voltage
Thermocouples are widely used as temperature-sensing
devices because the mathematical relationship between
temperature difference and resultant voltage is both
repeatable and fairly linear. By measuring voltage, it is
possible to infer temperature. Different ranges of
temperature measurement are possible by selecting
different metal pairs to be joined together.
Make your own multimeter
PARTS AND MATERIALS
Sensitive meter movement (Radio Shack catalog # 22-
410)
Selector switch, single-pole, multi-throw, break-before-
make (Radio Shack catalog # 275-1386 is a 2-pole, 6-
position unit that works well)
Multi-turn potentiometers, PCB mount (Radio Shack
catalog # 271-342 and 271-343 are 15-turn, 1 kQ and
10 kQ "trimmer" units, respectively)
Assorted resistors, preferably high-precision metal film
or wire-wound types (Radio Shack catalog # 271-309 is
an assortment of metal-film resistors, +/- 1% tolerance)
Plastic or metal mounting box
Three "banana" jack style binding posts, or other
terminal hardware, for connection to potentiometer
circuit (Radio Shack catalog # 274-662 or equivalent)
The most important and expensive component in a meter is
the movement. the actual needle-and-scale mechanism
whose task it is to translate an electrical current into
mechanical displacement where it may be visually
interpreted. The ideal meter movement is physically large
(for ease of viewing) and as sensitive as possible (requires
minimal current to produce full-scale deflection of the
needle). High-quality meter movements are expensive, but
Radio Shack carries some of acceptable quality that are
reasonably priced. The model recommended in the parts list
is sold as a voltmeter with a 0-15 volt range, but is actually
a milliammeter with a range ("multiplier") resistor included
separately.
It may be cheaper to purchase an inexpensive analog meter
and disassemble it for the meter movement alone. Although
the thought of destroying a working multimeter in order to
have parts to make your own may sound counter-productive,
the goal here is /earning, not meter function.
| cannot specify resistor values for this experiment, as these
depend on the particular meter movement and
measurement ranges chosen. Be sure to use high-precision
fixed-value resistors rather than carbon-composition
resistors. Even if you happen to find carbon-composition
resistors of just the right value(s), those values will change
or "drift" over time due to aging and temperature
fluctuations. Of course, if you don't care about the long-term
stability of this meter but are building it just for the learning
experience, resistor precision matters little.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
LEARNING OBJECTIVES
e Voltmeter design and use
e Ammeter design and use
e Rheostat range limiting
e Calibration theory and practice
e Soldering practice
SCHEMATIC DIAGRAM
Movement
R
shunt
"Common"
jack
Riautipier TSistors are actually rheostat networks
R puttiplier
WV —
ILLUSTRATION
Meter
movement
Common
=|
VimA
eeeto
switc
}Y
INSTRUCTIONS
First, you need to determine the characteristics of your
meter movement. Most important is to know the ful/ scale
deflection in milliamps or microamps. To determine this,
connect the meter movement, a potentiometer, battery, and
digital ammeter in series. Adjust the potentiometer until the
meter movement is deflected exactly to full-scale. Read the
ammeter's display to find the full-scale current value:
Meter
movement
Be very careful not to apply too much current to the meter
movement, as movements are very sensitive devices and
easily damaged by overcurrent. Most meter movements
have full-scale deflection current ratings of 1 mA or less, so
choose a potentiometer value high enough to limit current
appropriately, and begin testing with the potentiometer
turned to maximum resistance. The lower the full-scale
current rating of a movement, the more sensitive it is.
After determining the full-scale current rating of your meter
movement, you must accurately measure its internal
resistance. To do this, disconnect all components from the
previous testing circuit and connect your digital onhmmeter
across the meter movement terminals. Record this resistance
figure along with the full-scale current figure obtained in the
last procedure.
Perhaps the most challenging portion of this project is
determining the proper range resistance values and
implementing those values in the form of rheostat networks.
The calculations are outlined in chapter 8 of volume 1
("Metering Circuits"), but an example is given here. Suppose
your meter movement had a full-scale rating of 1 mA and an
internal resistance of 400 Q. If we wanted to determine the
necessary range resistance ("Rmultiplier') to give this
movement a range of 0 to 15 volts, we would have to divide
15 volts (total applied voltage) by 1 mA (full-scale current)
to obtain the total probe-to-probe resistance of the
voltmeter (R=E/l). For this example, that total resistance is
15 kQ. From this total resistance figure, we subtract the
movement's internal resistance, leaving 14.6 kO for the
range resistor value. A simple rheostat network to produce
14.6 kQ (adjustable) would be a 10 kQ potentiometer in
parallel with a 10 kQ fixed resistor, all in series with another
10 kQ fixed resistor:
= 15 kQ, adjustable
10 kQ me
10 kQ
One position of the selector switch directly connects the
meter movement between the black Common binding post
and the red V/mA binding post. In this position, the meter is
a sensitive ammeter with a range equal to the full-scale
current rating of the meter movement. The far clockwise
position of the switch disconnects the positive (+) terminal
of the movement from either red binding post and shorts it
directly to the negative (-) terminal. This protects the meter
from electrical damage by isolating it from the red test
probe, and it "dampens" the needle mechanism to further
guard against mechanical shock.
The shunt resistor (Repunt) necessary for a high-current
ammeter function needs to be a low-resistance unit with a
high power dissipation. You will definitely not be using any
1/4 watt resistors for this, unless you form a resistance
network with several smaller resistors in parallel
combination. If you plan on having an ammeter range in
excess of 1 amp, | recommend using a thick piece of wire or
even a Skinny piece of sheet metal as the "resistor," suitably
filed or notched to provide just the right amount of
resistance.
To calibrate a home-made shunt resistor, you will need to
connect the your multimeter assembly to a calibrated source
of high current, or a high-current source in series with a
digital ammeter for reference. Use a small metal file to shave
off shunt wire thickness or to notch the sheet metal strip in
small, careful amounts. The resistance of your shunt will
increase with every stroke of the file, causing the meter
movement to deflect more strongly. Remember that you can
always approach the exact value in slower and slower steps
(file strokes), but you cannot go "backward" and decrease
the shunt resistance!
Build the multimeter circuit on a breadboard first while
determining proper range resistance values, and perform all
calibration adjustments there. For final construction, solder
the components on to a printed-circuit board. Radio Shack
sells printed circuit boards that have the same layout as a
breadboard, for convenience (catalog # 276-170). Feel free
to alter the component layout from what is shown.
| strongly recommend that you mount the circuit board and
all components in a sturdy box, so that the meter is durably
finished. Despite the limitations of this multimeter (no
resistance function, inability to measure alternating current,
and lower precision than most purchased analog
multimeters), it is an excellent project to assist learning
fundamental instrument principles and circuit function. A far
more accurate and versatile multimeter may be constructed
using many of the same parts if an amplifier circuit is added
to it, so save the parts and pieces for a later experiment!
Sensitive voltage detector
PARTS AND MATERIALS
e High-quality "closed-cup" audio headphones
e Headphone jack: female receptacle for headphone plug
(Radio Shack catalog # 274-312)
Small step-down power transformer (Radio Shack
catalog # 273-1365 or equivalent, using the 6-volt
secondary winding tap)
e Two 1N4001 rectifying diodes (Radio Shack catalog #
276-1101)
1 kQ resistor
100 kQ potentiometer (Radio Shack catalog # 271-092)
Two "banana" jack style binding posts, or other terminal
hardware, for connection to potentiometer circuit (Radio
Shack catalog # 274-662 or equivalent)
e Plastic or metal mounting box
Regarding the headphones, the higher the "sensitivity"
rating in decibels (dB), the better, but listening is believing:
if you're serious about building a detector with maximum
sensitivity for small electrical signals, you should try a few
different headphone models at a high-quality audio store
and "listen" for which ones produce an audible sound for the
lowest volume setting on a radio or CD player. Beware, as
you could spend hundreds of dollars on a pair of
headphones to get the absolute best sensitivity! Take heart,
though: I've used an o/d pair of Radio Shack "Realistic"
brand headphones with perfectly adequate results, so you
don't need to buy the best.
A transformer is a device normally used with alternating
current ("AC") circuits, used to convert high-voltage AC
power into low-voltage AC power, and for many other
purposes. It is not important that you understand its
intended function in this experiment, other than it makes
the headphones become more sensitive to low-current
electrical signals.
Normally, the transformer used in this type of application
(audio speaker impedance matching) is called an "audio
transformer," with its primary and secondary windings
represented by impedance values (1000 ©: 8 Q) instead of
voltages. An audio transformer will work, but I've found
small step-down power transformers of 120/6 volt ratio to be
perfectly adequate for the task, cheaper (especially when
taken from an old thrift-store alarm clock radio), and far
more rugged.
The tolerance (precision) rating for the 1 kQ resistor is
irrelevant. The 100 kQ potentiometer is a recommended
option for incorporation into this project, as it gives the user
control over the loudness for any given signal. Even though
an audio-taper potentiometer would be appropriate for this
application, it is not necessary. A /inear-taper potentiometer
works quite well.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
Lessons In Electric Circuits, Volume 1, chapter 10: "DC
Network Analysis" (in regard to the Maximum Power Transfer
Theorem)
Lessons In Electric Circuits, Volume 2, chapter 9:
"Transformers"
Lessons In Electric Circuits, Volume 2, chapter 12: "AC
Metering Circuits"
LEARNING OBJECTIVES
e Soldering practice
e Detection of extremely small electrical signals
e Using a potentiometer as a voltage divider/signal
attenuator
e Using diodes to "clip" voltage at some maximum level
SCHEMATIC DIAGRAM
headphones
test lead
transformer
1 kQ jack plug
diodes
——<X
test lead
ILLUSTRATION
headphones
resistor
Binding
posts
transformer jack plug
INSTRUCTIONS
The headphones, most likely being stereo units (Separate
left and right speakers) will have a three-contact plug. You
will be connecting to only two of those three contact points.
If you only have a "mono" headphone set with a two-contact
plug, just connect to those two contact points. You may
either connect the two stereo speakers in series or in
parallel. I've found the series connection to work best, that
is, to produce the most sound from a small signal:
To transformer To transformer
| if tf
common right left common right left
Speakers in series Speakers in parallel
Solder all wire connections well. This detector system is
extremely sensitive, and any loose wire connections in the
circuit will add unwanted noise to the sounds produced by
the measured voltage signal. The two diodes (arrow-like
component symbols) connected in parallel with the
transformer's primary winding, along with the series-
connected 1 kQ resistor, work together to prevent any more
than about 0.7 volts from being dropped across the primary
coil of the transformer. This does one thing and one thing
only: limit the amount of sound the headphones can
produce. The system will work without the diodes and
resistor in place, but there will be no limit to sound volume
in the circuit, and the resulting sound caused by accidently
connecting the test leads across a substantial voltage source
(like a battery) can be deafening!
Binding posts provide points of connection for a pair of test
probes with banana-style plugs, once the detector
components are mounted inside a box. You may use ordinary
multimeter probes, or make your own probes with alligator
clips at the ends for secure connection to a circuit.
Detectors are intended to be used for balancing bridge
measurement circuits, potentiometric (null-balance)
voltmeter circuits, and detect extremely low-amplitude AC
("alternating current") signals in the audio frequency range.
It is a valuable piece of test equipment, especially for the
low-budget experimenter without an oscilloscope. It is also
valuable in that it allows you to use a different bodily sense
in interpreting the behavior of a circuit.
For connection across any non-trivial source of voltage (1
volt and greater), the detector's extremely high sensitivity
should be attenuated. This may be accomplished by
connecting a voltage divider to the "front" of the circuit:
SCHEMATIC DIAGRAM
test lead
1 kQ
100
kQ —<
test lead
ILLUSTRATION
@ : , : 2 ;
Adjust the 100 kQ voltage divider potentiometer to about
mid-range when initially sensing a voltage signal of
unknown magnitude. If the sound is too loud, turn the
potentiometer down and try again. If too soft, turn it up and
try again. The detector produces a "click" sound whenever
the test leads make or break contact with the voltage source
under test. With my cheap headphones, I've been able to
detect currents of less than 1/10 of a microamp (< 0.1 WA).
A good demonstration of the detector's sensitivity is to
touch both test leads to the end of your tongue, with the
sensitivity adjustment set to maximum. The voltage
produced by metal-to-electrolyte contact (called ga/vanic
voltage) is very small, but enough to produce soft "clicking"
sounds every time the leads make and break contact on the
wet skin of your tongue.
Try unplugged the headphone plug from the jack
(receptacle) and similarly touching it to the end of your
tongue. You should still hear soft clicking sounds, but they
will be much smaller in amplitude. Headphone speakers are
“low impedance" devices: they require low voltage and
"high" current to deliver substantial sound power.
Impedance is a measure of opposition to any and all forms of
electric current, including alternating current (AC).
Resistance, by comparison, is a strictly measure of
opposition to direct current (DC). Like resistance, impedance
is measured in the unit of the Ohm (Q), but it is symbolized
in equations by the capital letter "Z" rather than the capital
letter "R". We use the term "impedance" to describe the
headphone's opposition to current because it is primarily AC
signals that headphones are normally subjected to, not DC.
Most small signal sources have high internal impedances,
some much higher than the nominal 8 Q of the headphone
speakers. This is a technical way of saying that they are
incapable of supplying substantial amounts of current. As
the Maximum Power Transfer Theorem predicts, maximum
sound power will be delivered by the headphone speakers
when their impedance is "matched" to the impedance of the
voltage source. The transformer does this. The transformer
also helps aid the detection of small DC signals by producing
inductive "kickback" every time the test lead circuit is
broken, thus "amplifying" the signal by magnetically storing
up electrical energy and suddenly releasing it to the
headphone speakers.
| recommend building this detector in a permanent fashion
(mounting all components inside of a box, and providing
nice test lead wires) so it may be easily used in the future.
Constructed as such, it might look something like this:
headphones
Ce) Sensitivity plug
Potentiometric voltmeter
PARTS AND MATERIALS
e Two 6 volt batteries
e One potentiometer, single turn, 10 kQ, linear taper
(Radio Shack catalog # 271-1715)
e Two high-value resistors (at least 1 MO each)
e Sensitive voltage detector (from previous experiment)
e Analog voltmeter (from previous experiment)
The potentiometer value is not critical: anything from 1 kQ
to 100 kQ is acceptable. If you have built the "precision
potentiometer" described earlier in this chapter, it is
recommended that you use it in this experiment.
Likewise, the actual values of the resistors are not critical. In
this particular experiment, the greater the value, the better
the results. They need not be precisely equal value, either.
If you have not yet built the sensitive voltage detector, it is
recommended that you build one before proceeding with
this experiment! It is a very useful, yet simple, piece of test
equipment that you should not be without. You can use a
digital multimeter set to the "DC millivolt" (DC mV) range in
lieu of a voltage detector, but the headphone-based voltage
detector is more appropriate because it demonstrates how
you can make precise voltage measurements without using
expensive or advanced meter equipment. | recommend
using your home-made multimeter for the same reason,
although any voltmeter will suffice for this experiment.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
LEARNING OBJECTIVES
e Voltmeter loading: its causes and its solution
e Using a potentiometer as a source of variable voltage
e Potentiometric method of voltage measurement
SCHEMATIC DIAGRAM
1MQ
6VvV —
6V
measure voltage Test
1MQ hes fest poles probes
Potentiometric voltmeter
ILLUSTRATION
Test circuit
Test probes
headphones
INSTRUCTIONS
Build the two-resistor voltage divider circuit shown on the
left of the schematic diagram and of the illustration. If the
two high-value resistors are of equal value, the battery's
voltage should be split in half, with approximately 3 volts
dropped across each resistor.
Measure the battery voltage directly with a voltmeter, then
measure each resistor's voltage drop. Do you notice
anything unusual about the voltmeter's readings? Normally,
series voltage drops add to equal the total applied voltage,
but in this case you will notice a serious discrepancy. Is
Kirchhoff's Voltage Law untrue? Is this an exception to one of
the most fundamental laws of electric circuits? No! What is
happening is this: when you connect a voltmeter across
either resistor, the voltmeter itself a/ters the circuit so that
the voltage is not the same as with no meter connected.
| like to use the analogy of an air pressure gauge used to
check the pressure of a pneumatic tire. When a gauge is
connected to the tire's fill valve, it releases some air out of
the tire. This affects the pressure in the tire, and so the
gauge reads a Slightly lower pressure than what was in the
tire before the gauge was connected. In other words, the act
of measuring tire pressure a/ters the tire's pressure.
Hopefully, though, there is so little air released from the tire
during the act of measurement that the reduction in
pressure is negligible. Voltmeters similarly impact the
voltage they measure, by bypassing some current around
the component whose voltage drop is being measured. This
affects the voltage drop, but the effect is so slight that you
usually don't notice it.
In this circuit, though, the effect is very pronounced. Why is
this? Try replacing the two high-value resistors with two of
100 kQ value each and repeat the experiment. Replace
those resistors with two 10 KQ units and repeat. What do
you notice about the voltage readings with lower-value
resistors? What does this tell you about voltmeter "impact"
on a circuit in relation to that circuit's resistance? Replace
any low-value resistors with the original, high-value (>= 1
MQ) resistors before proceeding.
Try measuring voltage across the two high-value resistors --
one at a time -- with a digital voltmeter instead of an analog
voltmeter. What do you notice about the digital meter's
readings versus the analog meter's? Digital voltmeters
typically have greater internal (probe-to-probe) resistance,
meaning they draw less current than a comparable analog
voltmeter when measuring the same voltage source. An
ideal voltmeter would draw zero current from the circuit
under test, and thus suffer no voltage "impact" problems.
If you happen to have two voltmeters, try this: connect one
voltmeter across one resistor, and the other voltmeter across
the other resistor. The voltage readings you get will add up
to the total voltage this time, no matter what the resistor
values are, even though they're different from the readings
obtained from a single meter used twice. Unfortunately,
though, it is unlikely that the voltage readings obtained this
way are equal to the true voltage drops with no meters
connected, and so it is not a practical solution to the
problem.
Is there any way to make a "perfect" voltmeter: one that has
infinite resistance and draws no current from the circuit
under test? Modern laboratory voltmeters approach this goal
by using semiconductor "amplifier" circuits, but this method
is too technologically advanced for the student or hobbyist
to duplicate. A much simpler and much older technique is
called the potentiometric or null-balance method. This
involves using an adjustable voltage source to "balance" the
measured voltage. When the two voltages are equal, as
indicated by a very sensitive nu// detector, the adjustable
voltage source is measured with an ordinary voltmeter.
Because the two voltage sources are equal to each other,
measuring the adjustable source is the same as measuring
across the test circuit, except that there is no "impact" error
because the adjustable source provides any current needed
by the voltmeter. Consequently, the circuit under test
remains unaffected, allowing measurement of its true
voltage drop.
Examine the following schematic to see how the
potentiometric voltmeter method is implemented:
Potentiometric voltmeter
Test circuit
The circle symbol with the word "null" written inside
represents the null detector. This can be any arbitrarily
sensitive meter movement or voltage indicator. Its sole
purpose in this circuit is to indicate when there is zero
voltage: when the adjustable voltage source (potentiometer)
is precisely equal to the voltage drop in the circuit under
test. The more sensitive this null detector is, the more
precisely the adjustable source may be adjusted to equal
the voltage under test, and the more precisely that test
voltage may be measured.
Build this circuit as shown in the illustration and test its
operation measuring the voltage drop across one of the
high-value resistors in the test circuit. It may be easier to
use a regular multimeter as a null detector at first, until you
become familiar with the process of adjusting the
potentiometer for a "null" indication, then reading the
voltmeter connected across the potentiometer.
If you are using the headphone-based voltage detector as
your null meter, you will need to intermittently make and
break contact with the circuit under test and listen for
"clicking" sounds. Do this by firmly securing one of the test
probes to the test circuit and momentarily touching the
other test probe to the other point in the test circuit again
and again, listening for sounds in the headphones indicating
a difference of voltage between the test circuit and the
potentiometer. Adjust the potentiometer until no clicking
sounds can be heard from the headphones. This indicates a
"null" or "balanced" condition, and you may read the
voltmeter indication to see how much voltage is dropped
across the test circuit resistor. Unfortunately, the
headphone-based null detector provides no indication of
whether the potentiometer voltage is greater than, or less
than the test circuit voltage, so you will have to listen for
decreasing "click" intensity while turning the potentiometer
to determine if you need to adjust the voltage higher or
lower.
You may find that a single-turn ("3/4 turn") potentiometer is
too coarse of an adjustment device to accurately "null" the
measurement circuit. A multi-turn potentiometer may be
used instead of the single-turn unit for greater adjustment
precision, or the "precision potentiometer" circuit described
in an earlier experiment may be used.
Prior to the advent of amplified voltmeter technology, the
potentiometric method was the on/y method for making
highly accurate voltage measurements. Even now, electrical
standards laboratories make use of this technique along
with the latest meter technology to minimize meter "impact"
errors and maximize measurement accuracy. Although the
potentiometric method requires more skill to use than
simply connecting a modern digital voltmeter across a
component, and is considered obsolete for all but the most
precise measurement applications, it is still a valuable
learning process for the new student of electronics, and a
useful technique for the hobbyist who may lack expensive
instrumentation in their home laboratory.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Netlist (make a text file containing the following text,
verbatim):
Potentiometric voltmeter
vl 10 dc 6
v2 3 0
rl 12 1meg
r2 2 0 1lmeg
rnull 2 3 10k
rmeter 3 0 50k
.dc v2 0 6 0.5
»print dc v(2,0) v(2,3) v(3,0)
.end
This SPICE simulation shows the actual voltage across R> of
the test circuit, the null detector's voltage, and the voltage
across the adjustable voltage source, as that source is
adjusted from 0 volts to 6 volts in 0.5 volt steps. In the
output of this simulation, you will notice that the voltage
across R> /s impacted significantly when the measurement
circuit is unbalanced, returning to its true voltage only when
there is practically zero voltage across the null detector. At
that point, of course, the adjustable voltage source is ata
value of 3.000 volts: precisely equal to the (unaffected) test
circuit voltage drop.
What is the lesson to be learned from this simulation? That a
potentiometric voltmeter avoids impacting the test circuit
only when it is in a condition of perfect balance ("null") with
the test circuit!
4-wire resistance measurement
PARTS AND MATERIALS
e 6-volt battery
e Electromagnet made from experiment in previous
chapter, or a large spool of wire
It would be ideal in this experiment to have two meters: one
voltmeter and one ammeter. For experimenters on a budget,
this may not be possible. Whatever ammeter is used should
be capable measuring at least a few amps of current. A 6-
volt "lantern" battery essentially short-circuited by a long
piece of wire may produce currents of this magnitude, and
your ammeter needs to be capable of measuring it without
blowing a fuse or sustaining other damage. Make sure the
highest current range on the meter is at least 5 amps!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
LEARNING OBJECTIVES
e Operating principle of Kelvin (4-wire) resistance
measurement
e How to measure low resistances with common test
equipment
SCHEMATIC DIAGRAM
unknown
ILLUSTRATION
INSTRUCTIONS
Although this experiment is best performed with two meters,
and indeed is shown as such in the schematic diagram and
illustration, one multimeter is sufficient.
Most ohmmeters operate on the principle of applying a small
voltage across an unknown resistance (Rynknown) and
inferring resistance from the amount of current drawn by it.
Except in special cases such as the megger, both the
voltage and current quantities employed by the meter are
quite small.
This presents a problem for measurement of low resistances,
as a low resistance specimen may be of much smaller
resistance value than the meter circuitry itself. Imagine
trying to measure the diameter of a cotton thread with a
yardstick, or measuring the weight of a coin with a scale
built for weighing freight trucks, and you will appreciate the
problem at hand.
One of the many sources of error in measuring small
resistances with an ordinary ohmmeter is the resistance of
the ohmmeter's own test leads. Being part of the
measurement circuit, the test leads may contain more
resistance than the resistance of the test specimen,
incurring significant measurement error by their presence:
Lead resistance:
0.25 Q
Lead resistance:
0.25 Q
One solution is called the Ke/vin, or 4-wire, resistance
measurement method. It involves the use of an ammeter
and voltmeter, determining specimen resistance by Ohm's
Law calculation. A current is passed through the unknown
resistance and measured. The voltage dropped across the
resistance is measured by the voltmeter, and resistance
calculated using Ohm's Law (R=E/I). Very small resistances
may be measured easily by using large current, providing a
more easily measured voltage drop from which to infer
resistance than if a small current were used.
Because only the voltage dropped by the unknown
resistance is factored into the calculation -- not the voltage
dropped across the ammeter's test leads or any other
connecting wires carrying the main current -- errors
otherwise caused by these stray resistances are completely
eliminated.
First, select a suitably low resistance specimen to use in this
experiment. | suggest the electromagnet coil specified in the
last chapter, or a spool of wire where both ends may be
accessed. Connect a 6-volt battery to this specimen, with an
ammeter connected in series. WARNING: the ammeter used
should be capable of measuring at least 5 amps of current,
so that it will not be damaged by the (possibly) high current
generated in this near-short circuit condition. If you havea
second meter, use it to measure voltage across the
specimen's connection points, as shown in the illustration,
and record both meters’ indications.
If you have only one meter, use it to measure current first,
recording its indication as quickly as possible, then
immediately opening (breaking) the circuit. Switch the
meter to its voltage mode, connect it across the specimen's
connection points, and re-connect the battery, quickly
noting the voltage indication. You don't want to leave the
battery connected to the specimen for any longer than
necessary for obtaining meter measurements, as it will begin
to rapidly discharge due to the high circuit current, thus
compromising measurement accuracy when the meter is re-
configured and the circuit closed once more for the next
measurement. When two meters are used, this is not as
significant an issue, because the current and voltage
indications may be recorded simultaneously.
Take the voltage measurement and divide it by the current
measurement. The quotient will be equal to the specimen's
resistance in ohms.
A very simple computer
PARTS AND MATERIALS
e Three batteries, each one with a different voltage
e Three equal-value resistors, between 10 kQ and 47 kO
each
When selecting resistors, measure each one with an
ohmmeter and choose three that are the closest in value to
each other. Precision is very important for this experiment!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 10: "DC
Network Analysis"
LEARNING OBJECTIVES
e How a resistor network can function as a voltage signal
averager
e Application of Millman's Theorem
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
This deceptively crude circuit performs the function of
mathematically averaging three voltage signals together,
and so fulfills a specialized computational role. In other
words, it is a computer that can only do one mathematical
operation: averaging three quantities together.
Build this circuit as shown and measure all battery voltages
with a voltmeter. Write these voltage figures on paper and
average them together (E, + E> + E3, divided by three).
When you measure each battery voltage, keep the black test
probe connected to the "ground" point (the side of the
battery directly joined to the other batteries by jumper
wires), and touch the red probe to the other battery
terminal. Polarity is important here! You will notice one
battery in the schematic diagram connected "backward" to
the other two, negative side "up." This battery's voltage
should read as a negative quantity when measured by a
properly connected digital meter, the other batteries
measuring positive.
When the voltmeter is connected to the circuit at the point
shown in the schematic and illustrations, it should register
the algebraic average of the three batteries' voltages. If the
resistor values are chosen to match each other very closely,
the "output" voltage of this circuit should match the
calculated average very closely as well.
If one battery is disconnected, the output voltage will equal
the average voltage of the remaining batteries. If the jumper
wires formerly connecting the removed battery to the
averager circuit are connected to each other, the circuit will
average the two remaining voltages together with 0 volts,
producing a smaller output signal:
The sheer simplicity of this circuit deters most people from
calling it a "computer," but it undeniably performs the
mathematical function of averaging. Not only does it
perform this function, but it performs it much faster than
any modern digital computer can! Digital computers, such
as personal computers (PCs) and pushbutton calculators,
perform mathematical operations in a series of discrete
steps. Analog computers perform calculations in continuous
fashion, exploiting Ohm's and Kirchhoff's Laws for an
arithmetic purpose, the "answer" computed as fast as
voltage propagates through the circuit (ideally, at the speed
of light!).
With the addition of circuits called amplifiers, voltage
signals in analog computer networks may be boosted and re-
used in other networks to perform a wide variety of
mathematical functions. Such analog computers excel at
performing the calculus operations of numerical
differentiation and integration, and as such may be used to
simulate the behavior of complex mechanical, electrical, and
even chemical systems. At one time, analog computers were
the ultimate tool for engineering research, but since then
have been largely supplanted by digital computer
technology. Digital computers enjoy the advantage of
performing mathematical operations with much better
precision than analog computers, albeit at much slower
theoretical speeds.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
4 4 4 4
Netlist (make a text file containing the following text,
verbatim):
Voltage averager
v1 10
v2 0 2 dc 9
v3 3 @ dc 1.5
rl 14 10k
r2 2 4 10k
r3 3 4 10k
.dc vl 6 61
.print dc v(4,0)
.end
With this SPICE netlist, we can force a digital computer to
simulate and analog computer, which averages three
numbers together. Obviously, we aren't doing this for the
practical task of averaging numbers, but rather to learn
more about circuits and more about computer simulation of
circuits!
Potato battery
PARTS AND MATERIALS
e One large potato
e One lemon (optional)
e Strip of zinc, or galvanized metal
e Piece of thick copper wire
The basic experiment is based on the use of a potato, but
many fruits and vegetables work as potential batteries!
For the zinc electrode, a large galvanized nail works well.
Nails with a thick, rough zinc texture are preferable to
galvanized nails that are smooth.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 11: "Batteries
and Power Systems"
LEARNING OBJECTIVES
e The importance of chemical activity in battery operation
e How electrode surface area affects battery operation
ILLUSTRATION
Galvanized
Copper
nail
wire
INSTRUCTIONS
Push both the nail and the wire deep into the potato.
Measure voltage output by the potato battery with a
voltmeter. Now, wasn't that easy?
Seriously, though, experiment with different metals,
electrode depths, and electrode spacings to obtain the
greatest voltage possible from the potato. Try other
vegetables or fruits and compare voltage output with the
same electrode metals.
It can be difficult to power a load with a single "potato"
battery, so don't expect to light up an incandescent lamp or
power a hobby motor or do anything like that. Even if the
voltage output is adequate, a potato battery has a fairly
high internal resistance which causes its voltage to "sag"
badly under even a light load. With multiple potato batteries
connected in series, parallel, or series-parallel arrangement,
though, it is possible to obtain enough voltage and current
Capacity to power a small load.
Capacitor charging and discharging
PARTS AND MATERIALS
e 6 volt battery
e Two large electrolytic capacitors, 1000 UF minimum
(Radio Shack catalog # 272-1019, 272-1032, or
equivalent)
e Two 1 kOQ resistors
e One toggle switch, SPST ("Single-Pole, Single-Throw")
Large-value capacitors are required for this experiment to
produce time constants slow enough to track with a
voltmeter and stopwatch. Be warned that most large
Capacitors are of the "electrolytic" type, and they are
polarity sensitive! One terminal of each capacitor should be
marked with a definite polarity sign. Usually capacitors of
the size specified have a negative (-) marking or series of
negative markings pointing toward the negative terminal.
Very large capacitors are often polarity-labeled by a positive
(+) marking next to one terminal. Failure to heed proper
polarity will almost surely result in capacitor failure, even
with a source voltage as low as 6 volts. When electrolytic
Capacitors fail, they typically explode, spewing caustic
chemicals and emitting foul odors. Please, try to avoid this!
| recommend a household light switch for the "SPST toggle
switch" specified in the parts list.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 13:
"Capacitors"
Lessons In Electric Circuits, Volume 1, chapter 16: "RC and
L/R Time Constants"
LEARNING OBJECTIVES
Capacitor charging action
Capacitor discharging action
Time constant calculation
Series and parallel capacitance
SCHEMATIC DIAGRAM
Charging circuit
Discharging circuit
ILLUSTRATION
Charging circuit
®
Discharging circuit
INSTRUCTIONS
Build the "charging" circuit and measure voltage across the
capacitor when the switch is closed. Notice how it increases
slowly over time, rather than suddenly as would be the case
with a resistor. You can "reset" the capacitor back toa
voltage of zero by shorting across its terminals with a piece
of wire.
The "time constant" (Tt) of a resistor capacitor circuit is
calculated by taking the circuit resistance and multiplying it
by the circuit capacitance. Fora 1 kQ resistor and a 1000 UF
Capacitor, the time constant should be 1 second. This is the
amount of time it takes for the capacitor voltage to increase
approximately 63.2% from its present value to its final
value: the voltage of the battery.
It is educational to plot the voltage of a charging capacitor
over time on a sheet of graph paper, to see how the inverse
exponential curve develops. In order to plot the action of
this circuit, though, we must find a way of slowing it down. A
one-second time constant doesn't provide much time to take
voltmeter readings!
We can increase this circuit's time constant two different
ways: changing the total circuit resistance, and/or changing
the total circuit capacitance. Given a pair of identical
resistors and a pair of identical capacitors, experiment with
various series and parallel combinations to obtain the
slowest charging action. You should already know by now
how multiple resistors need to be connected to form a
greater total resistance, but what about capacitors? This
circuit will demonstrate to you how capacitance changes
with series and parallel capacitor connections. Just be sure
that you insert the capacitor(s) in the proper direction: with
the ends labeled negative (-) electrically "closest" to the
battery's negative terminal!
The discharging circuit provides the same kind of changing
capacitor voltage, except this time the voltage jumps to full
battery voltage when the switch closes and slowly falls when
the switch is opened. Experiment once again with different
combinations of resistors and capacitors, making sure as
always that the capacitor's polarity is correct.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Netlist (make a text file containing the following text,
verbatim):
Capacitor charging circuit
vl 10 dc 6
rl 12 1k
cl 2 0 1000u ic=0
.tran 0.1 5 uic
.plot tran v(2,0)
.end
Rate-of-change indicator
PARTS AND MATERIALS
Two 6 volt batteries
Capacitor, 0.1 uF (Radio Shack catalog # 272-135)
1 MQ resistor
Potentiometer, single turn, 5 kQ, linear taper (Radio
Shack catalog # 271-1714)
The potentiometer value is not especially critical, although
lower-resistance units will, in theory, work better for this
experiment than high-resistance units. I've used a 10 kQ
potentiometer for this circuit with excellent results.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 13:
"Capacitors"
LEARNING OBJECTIVES
e How to build a differentiator circuit
e Obtain an empirical understanding of the derivative
calculus function
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
Measure voltage between the potentiometer's wiper
terminal and the "ground" point shown in the schematic
diagram (the negative terminal of the lower 6-volt battery).
This is the input voltage for the circuit, and you can see how
it smoothly varies between zero and 12 volts as the
potentiometer control is turned full-range. Since the
potentiometer is used here as a voltage divider, this
behavior should be unsurprising to you.
Now, measure voltage across the 1 MQ resistor while moving
the potentiometer control. A digital voltmeter is highly
recommended, and | advise setting it to a very sensitive
(millivolt) range to obtain the strongest indications. What
does the voltmeter indicate while the potentiometer is not
being moved? Turn the potentiometer slowly clockwise and
note the voltmeter's indication. Turn the potentiometer
slowly counter-clockwise and note the voltmeter's
indication. What difference do you see between the two
different directions of potentiometer control motion?
Try moving the potentiometer in such a way that the
voltmeter gives a steady, small indication. What kind of
potentiometer motion provides the steadiest voltage across
the 1 MQ resistor?
In calculus, a function representing the rate of change of
one variable as compared to another is called the derivative.
This simple circuit illustrates the concept of the derivative
by producing an output voltage proportional to the input
voltage's rate of change over time. Because this circuit
performs the calculus function of differentiation with respect
to time (outputting the time-derivative of an incoming
Signal), it is called a differentiator circuit.
Like the averager circuit shown earlier in this chapter, the
differentiator circuit is a kind of analog computer.
Differentiation is a far more complex mathematical function
than averaging, especially when implemented in a digital
computer, so this circuit is an excellent demonstration of the
elegance of analog circuitry in performing mathematical
computations.
More accurate differentiator circuits may be built by
combining resistor-capacitor networks with electronic
amplifier circuits. For more detail on computational circuitry,
go to the "Analog Integrated Circuits" chapter in this
Experiments volume.
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
|| 4] l_—
—/ | 4]
Lessons In Electric Circuits
-- Volume VI
Chapter 4
AC CIRCUITS
e Introduction
Transformer -- power supply
Build a transformer
Variable inductor
Sensitive audio detector
Sensing AC magnetic fields
Sensing AC electric fields
Automotive alternator
Induction motor
Induction motor, large
Phase shift
Sound cancellation
Musical keyboard as a signal generator
PC Oscilloscope
Waveform analysis
Inductor-capacitor "tank" circuit
Signal coupling
Introduction
"AC" stands for Alternating Current, which can refer to either
voltage or current that alternates in polarity or direction,
respectively. These experiments are designed to introduce
you to several important concepts specific to AC.
A convenient source of AC voltage is household wall-socket
power, which presents significant shock hazard. In order to
minimize this hazard while taking advantage of the
convenience of this source of AC, a small power supply will
be the first project, consisting of a transformer that steps
the hazardous voltage (110 to 120 volts AC, RMS) down to
12 volts or less. The title of "power supply" is somewhat
misleading. This device does not really act as a source or
supply of power, but rather as a power converter, to reduce
the hazardous voltage of wall-socket power to a much safer
level.
Transformer -- power supply
PARTS AND MATERIALS
e Power transformer, 120VAC step-down to 12VAC, with
center-tapped secondary winding (Radio Shack catalog
# 273-1365, 273-1352, or 273-1511).
Terminal strip with at least three terminals.
Household wall-socket power plug and cord.
Line cord switch.
Box (optional).
Fuse and fuse holder (optional).
Power transformers may be obtained from old radios, which
can usually be obtained from a thrift store for a few dollars
(or less!). The radio would also provide the power cord and
plug necessary for this project. Line cord switches may be
obtained from a hardware store. If you want to be absolutely
sure what kind of transformer you're getting, though, you
should purchase one from an electronics supply store.
If you decide to equip your power supply with a fuse, be sure
to get a slow-acting, or slow-blow fuse. Transformers may
draw high "surge" currents when initially connected to an AC
source, and these transient currents will blow a fast-acting
fuse. Determine the proper current rating of the fuse by
dividing the transformer's "VA" rating by 120 volts: in other
words, calculate the full allowable primary winding current
and size the fuse accordingly.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 1: "Basic AC
Theory"
Lessons In Electric Circuits, Volume 2, chapter 9:
"Transformers"
LEARNING OBJECTIVES
e Transformer voltage step-down behavior.
e Purpose of tapped windings.
e Safe wiring techniques for power cords.
SCHEMATIC DIAGRAM
Box (optional)
Transformer
ILLUSTRATION
Terminal
strip
eo wires soldered and taped
~— with electrical tape
12/6 volts CT
120 volts
\ wires soldered and taped
with electrical tape
2-conductor
"Zip" cord "Zip" cord split into
separate wires
INSTRUCTIONS
Warning! This project involves the use of dangerous
voltages. You must make sure all high-voltage (120 volt
household power) conductors are safely insulated from
accidental contact. No bare wires should be seen anywhere
on the "primary" side of the transformer circuit. Be sure to
solder all wire connections so that they're secure, and use
real electrical tape (not duct tape, scotch tape, packing
tape, or any other kind!) to insulate your soldered
connections.
If you wish to enclose the transformer inside of a box, you
may use an electrical "junction" box, obtained from a
hardware store or electrical supply house. If the enclosure
used is metal rather than plastic, a three-prong plug should
be used, with the "ground" prong (the longest one on the
plug) connected directly to the metal case for maximum
Safety.
Before plugging the plug into a wall socket, do a safety
check with an ohmmeter. With the line switch in the "on"
position, measure resistance between either plug prong and
the transformer case. There should be infinite (maximum)
resistance. If the meter registers continuity (some resistance
value less than infinity), then you have a "short" between
one of the power conductors and the case, which is
dangerous!
Next, check the transformer windings themselves for
continuity. With the line switch in the "on" position, there
should be a small amount of resistance between the two
plug prongs. When the switch is turned "off," the resistance
indication should increase to infinity (open circuit -- no
continuity). Measure resistance between pairs of wires on
the secondary side. These secondary windings should
register much lower resistances than the primary. Why is
this?
Plug the cord into a wall socket and turn the switch on. You
should be able to measure AC voltage at the secondary side
of the transformer, between pairs of terminals. Between two
of these terminals, you should measure about 12 volts.
Between either of these two terminals and the third
terminal, you should measure half that. This third wire is the
“center-tap" wire of the secondary winding.
It would be advisable to keep this project assembled for use
In powering other experiments shown in this book. From
here on, | will designate this "low-voltage AC power supply"
using this illustration:
Low-voltage
AC power supply
b-o-o-5-0
COMPUTER SIMULATION
Schematic with SPICE node numbers:
1 2
Rioadt
120 V Ly :
Riad?
0 0
Netlist (make a text file containing the following text,
verbatim):
transformer with center-tap secondary
v1 10 ac 120 sin
rbogusl 1 2 le-3
ll 2 0 10
12 5 4 0.025
13 4 3 0.025
k1 11 12 0.999
k2 12 13 0.999
kK3 ll 13 0.999
rbogus2 3 0 1lel2
rload1l 5 4 1k
rload2 4 3 1k
* Sets up AC analysis at 60 Hz:
.ac lin 1 60 60
* Prints primary voltage between nodes 2 and 0:
.print ac v(2,0)
* Prints (top) secondary voltage between nodes 5 and 4:
.print ac v(5,4)
* Prints (bottom) secondary voltage between nodes 4 and 3:
.print ac v(4,3)
* Prints (total) secondary voltage between nodes 5 and 3:
.print ac v(5,3)
.end
Build a transformer
PARTS AND MATERIALS
e Steel flatbar, 4 pieces
e Miscellaneous bolts, nuts, washers
e 28 gauge "magnet" wire
e Low-voltage AC power supply
“Magnet wire" is small-gauge wire insulated with a thin
enamel coating. It is intended to be used to make
electromagnets, because many "turns" of wire may be
wrapped in a relatively small-diameter coil. Any gauge of
wire will work, but 28 gauge is recommended so as to make
a coil with as many turns as possible in a small diameter.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 9:
"Transformers"
LEARNING OBJECTIVES
e Effects of electromagnetism.
e Effects of electromagnetic induction.
e Effects of magnetic coupling on voltage regulation.
e Effects of winding turns on "step" ratio.
SCHEMATIC DIAGRAM
Transformer
s|lé
ILLUSTRATION
bolt
wire coil ~~ wire coil
steel "flatbar"
INSTRUCTIONS
Wrap two, equal-length bars of steel with a thin layer of
electrically-insulating tape. Wrap several hundred turns of
magnet wire around these two bars. You may make these
windings with an equal or unequal number of turns,
depending on whether or not you want the transformer to be
able to "step" voltage up or down. | recommend equal turns
to begin with, then experiment later with coils of unequal
turn count.
Join those bars together in a rectangle with two other,
shorter, bars of steel. Use bolts to secure the bars together
(it is recommended that you drill bolt holes through the bars
before you wrap wire around them).
Check for shorted windings (ohmmeter reading between
wire ends and steel bar) after you're finished wrapping the
windings. There should be no continuity (infinite resistance)
between the winding and the steel bar. Check for continuity
between winding ends to ensure that the wire isn't broken
open somewhere within the coil. If either resistance
measurements indicate a problem, the winding must be re-
made.
Power your transformer with the low-voltage output of the
"power supply" described at the beginning of this chapter.
Do not power your transformer directly from wall-socket
voltage (120 volts), as your home-made windings really
aren't rated for any significant voltage!
Measure the output voltage (Secondary winding) of your
transformer with an AC voltmeter. Connect a load of some
kind (light bulbs are good!) to the secondary winding and
re-measure voltage. Note the degree of voltage "sag" at the
secondary winding as load current is increased.
Loosen or remove the connecting bolts from one of the short
bar pieces, thus increasing the re/uctance (analogous to
resistance) of the magnetic "circuit" coupling the two
windings together. Note the effect on output voltage and
voltage "sag" under load.
If you've made your transformer with unequal-turn windings.
try it in step-up versus step-down mode, powering different
AC loads.
Variable inductor
PARTS AND MATERIALS
e Paper tube, from a toilet-paper roll
e Bar of iron or steel, large enough to almost fill diameter
of paper tube
28 gauge "magnet" wire
Low-voltage AC power supply
Incandescent lamp, rated for power supply voltage
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 14:
"Magnetism and Electromagnetism"
Lessons In Electric Circuits, Volume 1, chapter 15:
"Inductors"
Lessons In Electric Circuits, Volume 2, chapter 3: "Reactance
and Impedance -- Inductive"
LEARNING OBJECTIVES
e Effects of magnetic permeability on inductance.
e How inductive reactance can control current in an AC
circuit.
SCHEMATIC DIAGRAM
Variable inductor
Lamp
ILLUSTRATION
Low-voltage
AC power supply
12
INSTRUCTIONS
Wrap hundreds of turns of magnet wire around the paper
tube. Connect this home-made inductor in series with an AC
power supply and lamp to form a circuit. When the tube is
empty, the lamp should glow brightly. When the steel bar is
inserted in the tube, the lamp dims from increased
inductance (L) and consequently increased inductive
reactance (X,).
Try using bars of different materials, such as copper and
stainless steel, if available. Not all metals have the same
effect, due to differences in magnetic permeability.
Sensitive audio detector
PARTS AND MATERIALS
e High-quality "closed-cup" audio headphones
e Headphone jack: female receptacle for headphone plug
(Radio Shack catalog # 274-312)
Small step-down power transformer (Radio Shack
catalog # 273-1365 or equivalent, using the 6-volt
secondary winding tap)
e Two 1N4001 rectifying diodes (Radio Shack catalog #
276-1101)
1 kQ resistor
100 kQ potentiometer (Radio Shack catalog # 271-092)
Two "banana" jack style binding posts, or other terminal
hardware, for connection to potentiometer circuit (Radio
Shack catalog # 274-662 or equivalent)
e Plastic or metal mounting box
Regarding the headphones, the higher the "sensitivity"
rating in decibels (dB), the better, but listening is believing:
if you're serious about building a detector with maximum
sensitivity for small electrical signals, you should try a few
different headphone models at a high-quality audio store
and "listen" for which ones produce an audible sound for the
lowest volume setting on a radio or CD player. Beware, as
you could spend hundreds of dollars on a pair of
headphones to get the absolute best sensitivity! Take heart,
though: I've used an o/d pair of Radio Shack "Realistic"
brand headphones with perfectly adequate results, so you
don't need to buy the best.
Normally, the transformer used in this type of application
(audio speaker impedance matching) is called an "audio
transformer," with its primary and secondary windings
represented by impedance values (1000 ©: 8 Q) instead of
voltages. An audio transformer will work, but I've found
small step-down power transformers of 120/6 volt ratio to be
perfectly adequate for the task, cheaper (especially when
taken from an old thrift-store alarm clock radio), and far
more rugged.
The tolerance (precision) rating for the 1 kQ resistor is
irrelevant. The 100 kQ potentiometer is a recommended
option for incorporation into this project, as it gives the user
control over the loudness for any given signal. Even though
an audio-taper potentiometer would be appropriate for this
application, it is not necessary. A /inear-taper potentiometer
works quite well.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 8: "DC
Metering Circuits"
Lessons In Electric Circuits, Volume 2, chapter 9:
"Transformers"
Lessons In Electric Circuits, Volume 2, chapter 12: "AC
Metering Circuits"
LEARNING OBJECTIVES
Soldering practice
Use of a transformer for impedance matching
Detection of extremely small electrical signals
Using diodes to "clip" voltage at some maximum level
SCHEMATIC DIAGRAM
headphones
test lead transformer
diodes
1kQ jack plug
test lead
ILLUSTRATION
headphones
resistor
Binding
posts
transformer jack plug
INSTRUCTIONS
This experiment is identical in construction to the "Sensitive
Voltage Detector" described in the DC experiments chapter.
If you've already built this detector, you may skip this
experiment.
The headphones, most likely being stereo units (Separate
left and right speakers) will have a three-contact plug. You
will be connecting to only two of those three contact points.
If you only have a "mono" headphone set with a two-contact
plug, just connect to those two contact points. You may
either connect the two stereo speakers in series or in
parallel. I've found the series connection to work best, that
is, to produce the most sound from a small signal:
To transformer To transformer
i if ae
common right left common right left
Speakers in series Speakers in parallel
Solder all wire connections well. This detector system is
extremely sensitive, and any loose wire connections in the
circuit will add unwanted noise to the sounds produced by
the measured voltage signal. The two diodes connected in
parallel with the transformer's primary winding, along with
the series-connected 1 kQ resistor, work together to "clip"
the input voltage to a maximum of about 0.7 volts. This does
one thing and one thing only: limit the amount of sound the
headphones can produce. The system will work without the
diodes and resistor in place, but there will be no limit to
sound volume in the circuit, and the resulting sound caused
by accidentally connecting the test leads across a
substantial voltage source (like a battery) can be deafening!
Binding posts provide points of connection for a pair of test
probes with banana-style plugs, once the detector
components are mounted inside a box. You may use ordinary
multimeter probes, or make your own probes with alligator
clips at the ends for secure connection to a circuit.
Detectors are intended to be used for balancing bridge
measurement circuits, potentiometric (null-balance)
voltmeter circuits, and detect extremely low-amplitude AC
("alternating current") signals in the audio frequency range.
It is a valuable piece of test equipment, especially for the
low-budget experimenter without an oscilloscope. It is also
valuable in that it allows you to use a different bodily sense
in interpreting the behavior of a circuit.
For connection across any non-trivial source of voltage (1
volt and greater), the detector's extremely high sensitivity
should be attenuated. This may be accomplished by
connecting a voltage divider to the "front" of the circuit:
SCHEMATIC DIAGRAM
test lead
1 kQ2
100
kQ —“,
test lead
ILLUSTRATION
potentiometer
Adjust the 100 kQ voltage divider potentiometer to about
mid-range when initially sensing a voltage signal of
unknown magnitude. If the sound is too loud, turn the
potentiometer down and try again. If too soft, turn it up and
try again. This detector even senses DC and radio-frequency
signals (frequencies below and above the audio range,
respectively), a "click" being heard whenever the test leads
make or break contact with the source under test. With my
cheap headphones, I've been able to detect currents of less
than 1/10 of a microamp (< 0.1 WA) DC, and similarly low-
magnitude RF signals up to 2 MHz.
A good demonstration of the detector's sensitivity is to
touch both test leads to the end of your tongue, with the
sensitivity adjustment set to maximum. The voltage
produced by metal-to-electrolyte contact (called ga/vanic
voltage) is very small, but enough to produce soft "clicking"
sounds every time the leads make and break contact on the
wet skin of your tongue.
Try unplugging the headphone plug from the jack
(receptacle) and similarly touching it to the end of your
tongue. You should still hear soft clicking sounds, but they
will be much smaller in amplitude. Headphone speakers are
“low impedance" devices: they require low voltage and
"high" current to deliver substantial sound power.
Impedance is a measure of opposition to any and all forms of
electric current, including alternating current (AC).
Resistance, by comparison, is a strictly measure of
opposition to direct current (DC). Like resistance, impedance
is measured in the unit of the Ohm (Q), but it is symbolized
in equations by the capital letter "Z" rather than the capital
letter "R". We use the term "impedance" to describe the
headphone's opposition to current because it is primarily AC
signals that headphones are normally subjected to, not DC.
Most small signal sources have high internal impedances,
some much higher than the nominal 8 Q of the headphone
speakers. This is a technical way of saying that they are
incapable of supplying substantial amounts of current. As
the Maximum Power Transfer Theorem predicts, maximum
sound power will be delivered by the headphone speakers
when their impedance is "matched" to the impedance of the
voltage source. The transformer does this. The transformer
also helps aid the detection of small DC signals by producing
inductive "kickback" every time the test lead circuit is
broken, thus "amplifying" the signal by magnetically storing
up electrical energy and suddenly releasing it to the
headphone speakers.
As with the low-voltage AC power supply experiment, |
recommend building this detector in a permanent fashion
(mounting all components inside of a box, and providing
nice test lead wires) so it can be easily used in the future.
Constructed as such, it might look something like this:
headphones
fe) Sensitivity plug
Sensing AC magnetic fields
PARTS AND MATERIALS
e Audio detector with headphones
e Electromagnet coil from relay or solenoid
What is needed for an electromagnet coil is a coil with many
turns of wire, so as to produce the most voltage possible
from induction with stray magnetic fields. The coil taken
from an old relay or solenoid works well for this purpose.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 7: "Mixed-
Frequency AC Signals"
LEARNING OBJECTIVES
e Effects of electromagnetic induction.
e Electromagnetic shielding techniques.
SCHEMATIC DIAGRAM
sensing
coil
ILLUSTRATION
headphones
sensing 6s)
coil ¢€
Ce) Sersitivity plug
INSTRUCTIONS
Using the audio detector circuit explained earlier to detect
AC voltage in the audio frequencies, a coil of wire may serve
as sensor of AC magnetic fields. The voltages produced by
the coil will be quite small, so it is advisable to adjust the
detector's sensitivity control to "maximum."
There are many sources of AC magnetic fields to be found in
the average home. Try, for instance, holding the coil close to
a television screen or circuit-breaker box. The coil's
orientation is every bit as important as its proximity to the
source, as you will soon discover on your own! If you want to
listen to more interesting tones, try holding the coil close to
the motherboard of an operating computer (be careful not to
"short" any connections together on the computer's circuit
board with any exposed metal parts on the sensing coil!), or
to its hard drive while a read/write operation is taking place.
One very strong source of AC magnetic fields is the home-
made transformer project described earlier. Try
experimenting with various degrees of "coupling" between
the coils (the steel bars tightly fastened together, versus
loosely fastened, versus dismantled). Another source is the
variable inductor and lamp circuit described in another
section of this chapter.
Note that physical contact with a magnetic field source is
unnecessary: magnetic fields extend through space quite
easily. You may also want to try "shielding" the coil from a
strong source using various materials. Try aluminum foil,
paper, sheet steel, plastic, or whatever other materials you
can think of. What materials work best? Why? What angles
(orientations) of coil position minimize magnetic coupling
(result in a minimum of detected signal)? What does this tell
us regarding inductor positioning if inter-circuit interference
from other inductors is a bad thing?
Whether or not stray magnetic fields like these pose any
health hazard to the human body is a hotly debated subject.
One thing is clear: in today's modern society, low-level
magnetic fields of all frequencies are easy to find!
Sensing AC electric fields
PARTS AND MATERIALS
e Audio detector with headphones
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 7: "Mixed-
Frequency AC Signals"
LEARNING OBJECTIVES
e Effects of electrostatic (capacitive) coupling.
e Electrostatic shielding techniques.
SCHEMATIC DIAGRAM
sensing
wire
1kQ
100
kQ —<
ILLUSTRATION
headphones
sensing
wire
plug
connection to
water pipe
INSTRUCTIONS
"Ground" one lead of the detector to a metal object in
contact with the earth (dirt). Most any water pipe or faucet
in a house will suffice. Take the other lead and hold it close
to an electrical appliance or lamp fixture. Do not try to
make contact with the appliance or with any
conductors within! Any AC electric fields produced by the
appliance will be heard in the headphones as a buzzing
tone.
Try holding the wire in different positions next to a good,
strong source of electric fields. Try using a piece of
aluminum foil clipped to the wire's end to maximize
Capacitance (and therefore its ability to intercept an electric
field). Try using different types of material to "shield" the
wire from an electric field source. What material(s) work
best? How does this compare with the AC magnetic field
experiment?
As with magnetic fields, there is controversy whether or not
stray electric fields like these pose any health hazard to the
human body.
Automotive alternator
PARTS AND MATERIALS
e Automotive alternator (one required, but two
recommended)
Old alternators may be obtained for low prices at automobile
wrecking yards. Many yards have alternators already
removed from the automobile, for your convenience. | do not
recommend paying full price for a new alternator, as used
units cost far less money and function just as well for the
purposes of this experiment.
| highly recommend using a Delco-Remy brand of alternator.
This is the type used on General Motors (GMC, Chevrolet,
Cadillac, Buick, Oldsmobile) vehicles. One particular model
has been produced by Delco-Remy since the early 1960's
with little design change. It is a very common unit to locate
in a wrecking yard, and very easy to work with.
If you obtain two alternators, you may use one as a
generator and the other as a motor. The steps needed to
prepare an alternator as a three-phase generator and as a
three-phase motor are the same.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 14:
"Magnetism and Electromagnetism"
Lessons In Electric Circuits, Volume 2, chapter 10:
"Polyphase AC Circuits"
LEARNING OBJECTIVES
Effects of electromagnetism
Effects of electromagnetic induction
Construction of real electromagnetic machines
Construction and application of three-phase windings
SCHEMATIC DIAGRAM
Typical alternator
meses "battery"
field terminal
terminals
shaft
An automotive alternator is a three-phase generator with a
built-in rectifier circuit consisting of six diodes. As the
sheave (most people call it a "pulley") is rotated by a belt
connected to the automobile engine's crankshaft, a magnet
is soun past a stationary set of three-phase windings (called
the stator), usually connected in a Y configuration. The
spinning magnet is actually an electromagnet, not a
permanent magnet. Alternators are designed this way so
that the magnetic field strength can be controlled, in order
that output voltage may be controlled independently of
rotor speed. This rotor magnet coil (called the fie/d coil, or
simply field) is energized by battery power, so that it takes a
small amount of electrical power input to the alternator to
get it to generate a lot of output power.
Electrical power is conducted to the rotating field coil
through a pair of copper "slip rings" mounted concentrically
on the shaft, contacted by stationary carbon "brushes." The
brushes are held in firm contact with the slip rings by spring
pressure.
Many modern alternators are equipped with built-in
"regulator" circuits that automatically switch battery power
on and off to the rotor coil to regulate output voltage. This
circuit, if present in the alternator you choose for the
experiment, is unnecessary and will only impede your study
if left in place. Feel free to "surgically remove" it, just make
sure you leave access to the brush terminals so that you can
power the field coil with the alternator fully assembled.
ILLUSTRATION
sy
INSTRUCTIONS
First, consult an automotive repair manual on the specific
details of your alternator. The documentation provided in
the book you're reading now is as general as possible to
accommodate different brands of alternators. You may need
more specific information, and a service manual is the best
place to obtain it.
For this experiment, you'll be connecting wires to the coils
inside the alternator and extending them outside the
alternator case, for easy connection to test equipment and
circuits. Unfortunately, the connection terminals provided
by the manufacturer won't suit our needs here, so you need
to make your own connections.
Disassemble the unit and locate terminals for connecting to
the two carbon brushes. Solder a pair of wires to these
terminals (at least 20 gauge in size) and extend these wires
through vent holes in the alternator case, making sure they
won't get snagged on the spinning rotor when the alternator
is re-assembled and used.
Locate the three-phase line connections coming from the
stator windings and connect wires to them as well,
extending these wires outside the alternator case through
some vent holes. Use the largest gauge wire that is
convenient to work with for these wires, as they may be
carrying substantial current. As with the field wires, route
them in such a way that the rotor will turn freely with the
alternator reassembled. The stator winding line terminals are
easy to locate: the three of them connect to three terminals
on the diode assembly, usually with "ring-lug" terminals
soldered to the ends of the wires.
Interior view of alternator,
rotor removed
stator
add these
wires
| recommend that you solder ring-lug terminals to your
wires, and attach them underneath the terminal nuts along
with the stator wire ends, so that each diode block terminal
is securing two ring lugs.
Re-assemble the alternator, taking care to secure the carbon
brushes in a retracted position so that the rotor doesn't
damage them upon re-insertion. On Delco-Remy alternators,
a small hole is provided on the back case half, and also at
the front of the brush holder assembly, through which a
paper clip or thin-gauge wire may be inserted to hold the
brushes back against their spring pressure. Consult the
service manual for more details on alternator assembly.
When the alternator has been assembled, try spinning the
shaft and listen for any sounds indicative of colliding parts
or snagged wires. If there is any such trouble, take it apart
again and correct whatever is wrong.
If and when it spins freely as it should, connect the two
"field" wires to a 6-volt battery. Connect an voltmeter to any
two of the three-phase line connections:
With the multimeter set to the "DC volts" function, slowly
rotate the alternator shaft. The voltmeter reading should
alternate between positive and negative as the shaft it
turned: a demonstration of very slow alternating voltage (AC
voltage) being generated. If this test is successful, switch
the multimeter to the "AC volts" setting and try again. Try
spinning the shaft slow and fast, comparing voltmeter
readings between the two conditions.
Short-circuit any two of the three-phase line wires and try
spinning the alternator. What you should notice is that the
alternator shaft becomes more difficult to spin. The heavy
electrical load you've created via the short circuit causes a
heavy mechanical load on the alternator, as mechanical
energy is converted into electrical energy.
Now, try connecting 12 volts DC to the field wires. Repeat
the DC voltmeter, AC voltmeter, and short-circuit tests
described above. What difference(s) do you notice?
Find some sort of polarity-insensitive 6 or 12 volts loads,
such as small incandescent lamps, and connect them to the
three-phase line wires. Wrap a thin rope or heavy string
around the groove of the sheave ("pulley") and spin the
alternator rapidly, and the loads should function.
If you have a second alternator, modify it as you modified
the first one, connecting five of your own wires to the field
brushes and stator line terminals, respectively. You can then
use it as a three-phase motor, powered by the first
alternator.
Connect each of the three-phase line wires of the first
alternator to the respective wires of the second alternator.
Connect the field wires of one alternator to a 6 volt battery.
This alternator will be the generator. Wrap rope around the
sheave in preparation to spin it. Take the two field wires of
the second alternator and short them together. This
alternator will be the motor:
Spin the generator shaft while watching the motor shaft's
rotation. Try reversing any two of the three-phase line
connections between the two units and spin the generator
again. What is different this time?
Connect the field wires of the motor unit to the a 6 volt
battery (you may parallel-connect this field with the field of
the generator unit, across the same battery terminals, if the
battery is strong enough to deliver the several amps of
current both coils will draw together). This will magnetize
the rotor of the motor. Try spinning the generator again and
note any differences in operation.
In the first motor setup, where the field wires were simple
shorted together, the motor was functioning as an /nduction
motor. In the second setup, where the motor's rotor was
magnetized, it functioned as a synchronous motor.
If you are feeling particularly ambitious and are skilled in
metal fabrication techniques, you may make your own high-
power generator platform by connecting the modified
alternator to a bicycle. I've built an arrangement that looks
like this:
The rear wheel drives the generator sheave with a /ong v-
belt. This belt also supports the rear of the bicycle,
maintaining a constant tension when a rider is pedaling the
bicycle. The generator hangs from a steel support structure
(1 used welded 2-inch square tubing, but a frame could be
made out of lumber). Not only is this machine practical, but
it is reliable enough to be used as an exercise machine, and
it is inexpensive to make:
You can see a bank of three 12-volt "RV" light bulbs behind
the bicycle unit (in the lower-left corner of the photograph),
which I| use for a load when riding the bicycle as an exercise
machine. A set of three switches is mounted at the front of
the bicycle, where | can turn loads on and off while riding.
By rectifying the three-phase AC power produced, it is
possible to have the alternator power its own field coil with
DC voltage, eliminating the need for a battery. However,
some independent source of DC voltage will still be
necessary for start-up, as the field coil must be energized
before any AC power can be produced.
Induction motor
PARTS AND MATERIALS
AC power source: 120VAC
Capacitor, 3.3 UF (or 2.2 uF) L20VAC or 350VDC, non-
polarized
e 15 to 25 watt incandescent lamp or 8200 25 watt
resistors
#32 AWG magnet wire
wooden board approx. 5 in. square.
AC line cord with plug
1.75 inch dia. cardboard tubing (toilet paper roll)
lamp socket
e AC power source: 220VAC
Capacitor, 1.5 uF 240VAC or 680VDC, non-polarized
e 25 to 40 watt incandescent lamp or 8200 25 watt
resistors
#32 AWG magnet wire
wooden board approx. 15 cm. square.
AC line cord with plug
4.5 to 5 cm. dia. cardboard tubing.
lamp socket
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 13: "AC
motors", "Single Phase induction motors","Permanent split-
Capacitor motor".
LEARNING OBJECTIVES
e To build an AC permanent capacitor split-phase
induction motor.
e To illustrate the simplicity of the AC induction motor.
INSTRUCTIONS
There are two parts lists to choose from depending upon the
availability of L2OVAC or 220VAC. Choose the one for your
location. This set of instructions is for the 120VAC version.
This is a simplified version of a "permanent capacitor split-
phase induction motor". By simplified, we mean the coils
only requires a few hundred turns of wire instead of a few
thousand. This is easier to wind. Though, the larger few
thousand turns model is impressive. There are two stator
coils as shown in the illustration above. Approximately 440
turns of #32 AWG (American wire gauge) enameled magnet
wire are wound over a one inch length of a slightly longer
section of 1.75 inch diameter toilet paper tube. To avoid
counting the turns, close-wind four layers of magnet wire
over a one inch width of the tube. See (b) above. Leave a
few inches of magnet wire for the leads. Tape the beginning
lead near the end of the tube so that the windings will cover
and anchor the tape. Do not cut the final width of the
cardboard tube until the winding is finished. Close wind a
single layer. Tape or cement the first layer to prevent
unwinding before proceeding to the second layer. Though it
is possible to wind additional layers directly over existing
layers, consider applying tape or paper between the layers
as shown in schematic (b). After four layers are wound, glue
the windings in place.
If close winding four layers of magnet wire it too difficult,
scramble wind 440 turns of the magnet wire over the end of
the cardboard tube. However, the close-wound style coil
mounts more easily to the baseboard. Keep the windings
within a one inch length.
Cut the finished winding from the end of the cardboard tube
with a razor knife allowing the form to extend a little beyond
the winding. Strip the enamel from an inch off the ends of
the pair of lead wires with sandpaper. Splice the bare ends
to heavier gauge insulated hook-up wire. Solder the splice.
Insulate with tape or heat-shrink tubing. Secure the splice to
the coil body. Then proceed with a second identical coil.
Refer to both the schematic diagram and the illustration for
assembly. Note that the coils are mounted at right angles.
They may be cemented to an insulating baseboard like
wood. The 25 watt lamp is wired in series with one coil. This
limits the current flowing through the coil. The lamp is a
substitute for an 820 Q power resistor. The capacitor is wired
in series with the other coil. It also limits the current through
the coil. In addition, it provides a leading phase shift of the
current with respect to voltage. The schematic and
illustration show no power switch or fuse. Add these if
desired.
The rotor must be made of a ferromagnetic material like a
steel can lid or bottle cap. The illustration below shows how
to make the rotor. Select a circular rotor either smaller than
the coil forms or a little larger. Use geometry to locate and
mark the center. The center needs to be dimpled. Select an
eighth inch diameter (a few mm) nail (a) and file or grind
the point round as shown at (b). Place the rotor atop a piece
of soft wood (c) and hammer the rounded point into the
center (d). Practice on a piece of similar scrap metal. Take
care not to pierce the rotor. A dished rotor (f) or a lid (g)
balance better than the flat rotor (e). The pivot point (e) may
be a straight pin driven through a movable wooden
pedestal, or through the main board. The tip of a ball-point
pen also works. If the rotor does not balance atop the pivot,
remove metal from the heavy side.
od
Double check the wiring. Check that any bare wire has been
insulated. The circuit may be powered-up without the rotor.
The lamp should light. Both coils will warm within a few
minutes. Excessive heating means that a lower wattage
(higher resistance) lamp and a lower value capacitor should
be substituted in series with the respective coils.
Place the rotor atop the pivot and move it between both
coils. It should spin. The closer it is, the faster it should spin.
Both coils should be warm, indicating power. Try different
size and style rotors. Try a small rotor on the opposite side of
the coils compared to the illustration.
For lack of #32 AWG magnet wire try 440 turns of slightly a
larger diameter (lesser AWG number) wire. This will require
more than 4 layers for the required turns. A night-light
fixture might be less expensive than the full-size lamp
socket illustrated. Though night-light bulbs are too low a
wattage at 3 or 7 watts, 15 watt bulbs fit the socket.
Induction motor, large
PARTS AND MATERIALS
AC power source: 120VAC
Capacitor, 3.3 uF L2Z0VAC or 350VDC, non-polarized
#33 AWG magnet wire, 2 pounds
wooden board approx. 6 to 12 in. square.
AC line cord with plug
5.1 inch dia. plastic 3 liter soda bottle
discarded ballpoint pen
misc. small wood blocks
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 13: "AC
motors", "Single Phase induction motors","Permanent split-
Capacitor motor".
LEARNING OBJECTIVES
e To build a large exhibit size AC permanent split-
Capacitor induction motor.
e To illustrate the simplicity of the AC induction motor.
SCHEMATIC DIAGRAM
ri
yee
120 L?
Vac 3200 turns
5.1 in.
Cl
3.3 LF
(a) (b)
FEN
ILLUSTRATION
INSTRUCTIONS
This is a larger version of a "permanent capacitor split-phase
induction motor". There are two different stator coils. The 1.0
inch wide 3200 turn L2 winding is shown in the illustration
above (b), wound over a section of 5.1 inch diameter plastic
3-liter soda bottle. L1 is approximately 3800 turns of #33
AWG (American wire gauge) enameled magnet wire wound
over a 1.25 width of a section of soda bottle, wider than
shown at (b). Mark a 1.25 inch wide cylinder with 0.25 inch
margins on each end. The wire will be wound on the 1.25
inch zone. The form is cut from the bottle on the outside
edges of the margin. Cuts of 0.25 inch from the margin to
winding zone are spaced at 1 inch intervals around the
circumference of both ends so that the margin may be bent
up at 90° to hold the wire on the form. To avoid counting the
3800 turns, scramble wind a 1/8 inch thickness of magnet
wire over the one inch width of the form. Else, count the
turns. Scrape the enamel from 1-inch on the free end, and
scrape only a small section from the lead to the spool. Do
NOT cut the lead to the spool. Measure the resistance, and
estimate how much more wire to wind to achieve 894 Q.
Apply enamel, nail polish, tape, or other insulation to the
bare spot on the spool lead. Continue winding, and recheck
the resistance. Once the approximate 894 Q is achieved,
leave a few inches of magnet wire for the lead. Cut the lead
from the spool. Secure the windings to the form with lacing
twine or other means.
The L1 winding of 3200 turns is approximately 744 QO and is
wound on a 1.0 inch wide form as shown at (b) in a manner
similar to the previous L2 winding.
Strip the enamel off 1-inch of the ends of magnet wire leads
if not already done. Splice the bare ends to heavier gauge
insulated hook-up wire. Solder the splice. Insulate with tape
or heat-shrink tubing. Secure the splice to the coil body.
Then proceed with the second coil. The coils may be
mounted in one corner of the wooden base. Alternatively, for
more flexability in use, they may be mounted to movable
pallets.
Refer to both the schematic diagram and the illustration for
assembly. Note that the coils are mounted at right angles.
L2, the smaller coil is wired to both sides of the 120 Vac line.
The capacitor is wired in series with the wider coil Ll. The
Capacitor provides a leading phase shift of the current with
respect to voltage. The schematic and illustration show no
power switch or fuse. Add these additions are recommended.
If this device is intended for use by non-technicians as an
unsupervised exhibit, all exposed bare terminations like the
Capacitor must be made finger safe by covering with shields.
The switch and fuse mentioned above are necessary. Finally,
the enamel on the coils only provides a single layer of
insulation. For safety, a second layer such as an insulating
wrapping, Plexiglas box, or other means is called for. Replace
all wooden components with Plexiglas for superior fire safety
in an unsupervised exhibit.
The rotor must be made of a ferromagnetic material like a
steel vegetable can, fruitcake can, etc. A too long vegetable
can may be cut in half. The illustration for the previous small
induction motor shows rotor dimpled bearing and pivot
details. The rotor may be smaller than the coil forms as in
the case of a cut down vegetable can. It can even be as
small as the can lid rotor used with the previous small motor.
It is also possible to drive a rotor larger than the coils, which
is the case with the fruitcake can. Locate and mark the
center of the rotor. The center needs to be dimpled. Select
an eighth inch diameter (a few mm) nail (a) and file or grind
the point round. Use this and a block of wood to dimple the
rotor as shown in the previous small motor A fairly long can
balances better than a flat rotor due to the lower center of
gravity. The tip of a ball point pen works well as a pivot for
larger rotors. Mount the pivot to a movable wooden
pedestal.
Double check the wiring. Check that any bare wire has been
insulated. The circuit may be powered-up without the rotor.
Excessive heating in L2 indicates that more turns are
required. Excessive heat in L1 calls for a reduction in the
capacitance of Cl. No heat at all indicates indicates an open
circuit to the affected coil.
Place the rotor atop the pivot and move it between both
energized coils. It should spin. The closer it is, the faster it
should spin. Both coils should be warm, indicating power. Try
different size and style rotors. Try a small rotor on the
opposite side of the coils compared to the illustration.
Three models of this motor have been built using #33 AWG
magnet wire because a large spool was on hand. AWG #32
magnet wire is probably easier to get. It should work.
Although the current will be higher due to the lower
resistance of the larger diameter # 32 wire. If a 3.3UF
Capacitor is not available, use somenting close as long as it
has an adequate voltage rating. A discarded AC motor run
Capacitor (bath tub shaped) was used by the author. Do no
use a motor start capacitor (black cylinder). These are only
usable for a few seconds of motor starting, and may explode
if used longer than that.
Try this: It is possible to simultaneously spin more than one
rotor. For example, in addition to the main rotor inside the
right angle formed by the coils, place a second smaller rotor
(can or bottle lid) near the pair of coils outside the right
angle at the vertex.
It is possible to reverse the direction of rotation by reversing
one of the coils. If the coils are mounted to movable pallets,
rotate one coil 180°. Another method, especially usefull with
fixed coils, is to wire one of the coils to a DPDT polarity
reversing switch. For example, disconnect L2 and wire it to
the wipers (center contacts) of the DPDT switch. The top
contacts go to the 120 Vac. The top contacts also go to the
the bottom contacts in an X-crossover pattern.
Phase shift
PARTS AND MATERIALS
e Low-voltage AC power supply
e Two Capacitors, 0.1 uF each, non-polarized (Radio Shack
catalog # 272-135)
e Two 27 kQ resistors
| recommend ceramic disk capacitors, because they are
insensitive to polarity (non-polarized), inexpensive, and
durable. Avoid capacitors with any kind of polarity marking,
as these will be destroyed when powered by AC!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 1: "Basic AC
Theory"
Lessons In Electric Circuits, Volume 2, chapter 4: "Reactance
and Impedance -- Capacitive"
LEARNING OBJECTIVES
e How out-of-phase AC voltages do not add algebraically,
but according to vector (phasor) arithmetic
SCHEMATIC DIAGRAM
27 kQ
27 kQ
12 V
RMS
ILLUSTRATION
Low-voltage
AC power supply
INSTRUCTIONS
Build the circuit and measure voltage drops across each
component with an AC voltmeter. Measure total (Supply)
voltage with the same voltmeter. You will discover that the
voltage drops do not add up to equal the total voltage. This
is due to phase shifts in the circuit: voltage dropped across
the capacitors is out-of-phase with voltage dropped across
the resistors, and thus the voltage drop figures do not add
up as one might expect. Taking phase angle into
consideration, they do add up to equal the total, but a
voltmeter doesn't provide phase angle measurements, only
amplitude.
Try measuring voltage dropped across both resistors at once.
This voltage drop wi// equal the sum of the voltage drops
measured across each resistor separately. This tells you that
both the resistors' voltage drop waveforms are in-phase with
each other, since they add simply and directly.
Measure voltage dropped across both capacitors at once.
This voltage drop, like the drop measured across the two
resistors, wi// equal the sum of the voltage drops measured
across each capacitor separately. Likewise, this tells you that
both the capacitors' voltage drop waveforms are in-phase
with each other.
Given that the power supply frequency is 60 Hz (household
power frequency in the United States), calculate
impedances for all components and determine all voltage
drops using Ohm's Law (E=IZ ; I=E/Z ; Z=E/l). The polar
magnitudes of the results should closely agree with your
voltmeter readings.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
The two large-value resistors Rpogus1 ANd Rpogus1 are
connected across the capacitors to provide a DC path to
ground in order that SPICE will work. This is a "fix" for one of
SPICE's quirks, to avoid it from seeing the capacitors as open
circuits in its analysis. These two resistors are entirely
unnecessary in the real circuit.
Netlist (make a text file containing the following text,
verbatim):
phase shift
v1 10 ac 12 sin
rl 1.2: 27k
r2 2 3 27k
cl 3 4 0.1u
c2 4 0 @.1u
rbogus1 3 4 1le9
rbogus2 4 0 1le9
.ac lin 1 60 60
* Voltage across each component:
.print ac v(1,2) v(2,3) v(3,4) v(4,0)
* Voltage across pairs of similar components
.print ac v(1,3) v(3,0)
.end
Sound cancellation
PARTS AND MATERIALS
e Low-voltage AC power supply
e Two audio speakers
e Two 220 OQ resistors
Large, low-frequency ("woofer") speakers are most
appropriate for this experiment. For optimum results, the
speakers should be identical and mounted in enclosures.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 1: "Basic AC
Theory"
LEARNING OBJECTIVES
e How phase shift can cause waves to either reinforce or
interfere with each other
e The importance of speaker "phasing" in stereo systems
SCHEMATIC DIAGRAM
220 2
ILLUSTRATION
Low-voltage
AC power supply
Speaker Speaker
INSTRUCTIONS
Connect each speaker to the low-voltage AC power supply
through a 220 OQ resistor. The resistor limits the amount of
power delivered to each speaker by the power supply. A low-
pitched, 60-Hertz tone should be heard from the speakers. If
the tone sounds too loud, use higher-value resistors.
With both speakers connected and producing sound,
position them so that they are only a foot or two away,
facing toward each other. Listen to the volume of the 60-
Hertz tone. Now, reverse the connections (the "polarity") of
just one of the speakers and note the volume again. Try
switching the polarity of one speaker back and forth from
original to reversed, comparing volume levels each way.
What do you notice?
By reversing wire connections to one speaker, you are
reversing the phase of that speaker's sound wave in
reference to the other speaker. In one mode, the sound
waves will reinforce one another for a strong volume. In the
other mode, the sound waves will destructively interfere,
resulting in diminished volume. This phenomenon is
common to a// wave events: sound waves, electrical signals
(voltage "waves"), waves in water, and even light waves!
Multiple speakers in a stereo sound system must be properly
"ohased" so that their respective sound waves don't cancel
each other, leaving less total sound level for the listener(s)
to hear. So, even in an AC system where there really is no
such thing as constant "polarity," the sequence of wire
connections may make a significant difference in system
performance.
This principle of volume reduction by destructive
interference may be exploited for noise cancellation. Such
systems sample the waveform of the ambient noise, then
produce an identical sound signal 180° out of phase with the
noise. When the two sound signals meet, they cancel each
other out, ideally eliminating all the noise. As one might
guess, this is much easier accomplished with noise sources
of steady frequency and amplitude. Cancellation of random,
broad-spectrum noise is very difficult, as some sort of signal-
processing circuit must sample the noise and generate
precisely the right amount of cancellation sound at just the
right time in order to be effective.
Musical keyboard as a Signal
generator
PARTS AND MATERIALS
e Electronic "keyboard" (musical)
e "Mono" (not stereo) headphone-type plug
e Impedance matching transformer (Lk Q to 8 Q ratio;
Radio Shack catalog # 273-1380)
e 10 kQ resistor
In this experiment, you'll learn how to use an electronic
musical keyboard as a source of variable-frequency AC
voltage signals. You need not purchase an expensive
keyboard for this -- but one with at least a few dozen "voice"
selections (piano, flute, harp, etc.) would be good. The
“mono” plug will be plugged into the headphone jack of the
musical keyboard, so get a plug that's the correct size for
the keyboard.
The "impedance matching transformer" is a small-size
transformer easily obtained from an electronics supply store.
One may be scavenged from a small, junk radio: it connects
between the speaker and the circuit board (amplifier), so is
easily identifiable by location. The primary winding is rated
in ohms of impedance (1000 Q), and is usually center-
tapped. The secondary winding is 8 Q and not center-
tapped. These impedance figures are not the same as DC
resistance, so don't expect to read 1000 QO and 8 QO with your
ohmmeter -- however, the 1000 ©O winding will read more
resistance than the 8 QO winding, because it has more turns.
If such a transformer cannot be obtained for the experiment,
a regular 120V/6V step-down power transformer works fairly
well, too.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 1: "Basic AC
Theory"
Lessons In Electric Circuits, Volume 2, chapter 7: "Mixed-
Frequency AC Signals"
LEARNING OBJECTIVES
e Difference between amplitude and frequency
e Measuring AC voltage, current with a meter
e Transformer operation, step-up
SCHEMATIC DIAGRAM
Keyboard
8 Q 1 kQ
plug
ILLUSTRATION
Bw?
"Voice" selection 7
npoooooo000000000 C) Volume
AMA
INSTRUCTIONS
Normally, a student of electronics in a school would have
access to a device called a signal generator, or function
generator, used to make variable-frequency voltage
waveforms to power AC circuits. An inexpensive electronic
keyboard is a cheaper alternative to a regular signal
generator, and provides features that most signal generators
cannot match, such as producing mixed-frequency waves.
To "tap in" to the AC voltage produced by the keyboard,
you'll need to insert a plug into the headphone jack
(sometimes just labeled "phone" on the keyboard) complete
with two wires for connection to circuits of your own design.
When you insert the plug into the jack, the normal speaker
built in to the keyboard will be disconnected (assuming the
keyboard is equipped with one), and the signal that used to
power that speaker will be available at the plug wires. In this
particular experiment, | recommend using the keyboard to
power the 8 Q side of an audio "output" transformer to step
up voltage to a higher level. If using a power transformer
instead of an audio output transformer, connect the
keyboard to the low-voltage winding so that it operates asa
step-up device. Keyboards produce very low voltage signals,
so there is no shock hazard in this experiment.
Using an inexpensive Yamaha keyboard, | have found that
the "panflute" voice setting produces the truest sine-wave
waveform. This waveform, or something close to it (flute, for
example), is recommended to start experimenting with since
it is relatively free of harmonics (many waveforms mixed
together, of integer-multiple frequency). Being composed of
just one frequency, it is a less complex waveform for your
multimeter to measure. Make sure the keyboard is set toa
mode where the note will be sustained as any key is held
down -- otherwise, the amplitude (voltage) of the waveform
will be constantly changing (high when the key is first
pressed, then decaying rapidly to zero).
Using an AC voltmeter, read the voltage direct from the
headphone plug. Then, read the voltage as stepped up by
the transformer, noting the step ratio. If your multimeter has
a "frequency" function, use it to measure the frequency of
the waveform produced by the keyboard. Try different notes
on the keyboard and record their frequencies. Do you notice
a pattern in frequency as you activate different notes,
especially keys that are similar to each other (notice the 12-
key black-and-white pattern repeated on the keyboard from
left to right)? If you don't mind making marks on your
keyboard, write the frequencies in Hertz in black ink on the
white keys, near the tops where fingers are less likely to rub
the numbers off.
Ideally, there should be no change in signal amplitude
(voltage) as different frequencies (notes on the keyboard)
are tried. If you adjust the volume up and down, you should
discover that changes in amplitude should have little or no
impact on frequency measurement. Amplitude and
frequency are two completely independent aspects of an AC
signal.
Try connecting the keyboard output to a 10 kQ load
resistance (through the headphone plug), and measure AC
current with your multimeter. If your multimeter has a
frequency function, you can measure the frequency of this
current as well. It should be the same as for the voltage for
any given note (keyboard key).
PC Oscilloscope
PARTS AND MATERIALS
e IBM-compatible personal computer with sound card,
running Windows 3.1 or better
Winscope software, downloaded free from internet
Electronic "keyboard" (musical)
"Mono" (not stereo) headphone-type plug for keyboard
"Mono" (not stereo) headphone-type plug for computer
sound card microphone input
e 10 kQ potentiometer
The Winscope program I've used was written by Dr.
Constantin Zeldovich, for free personal and academic use. It
plots waveforms on the computer screen in response to AC
voltage signals interpreted by the sound card microphone
input. A similar program, called Oscope, is made for the
Linux operating system. If you don't have access to either
software, you may use the "sound recorder" utility that
comes stock with most versions of Microsoft Windows to
display crude waveshapes.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 7: "Mixed-
Frequency AC Signals"
Lessons In Electric Circuits, Volume 2, chapter 12: "AC
Metering Circuits"
LEARNING OBJECTIVES
e Computer use
e Basic oscilloscope function
SCHEMATIC DIAGRAM
Keyboard Computer
plug plug
10kQ
ILLUSTRATION
plug
ooo al ooo €) Volume Be
WT Monitor
INSTRUCTIONS
The oscilloscope is an indispensable test instrument for the
electronics student and professional. No serious electronics
lab should be without one (or two!). Unfortunately,
commercial oscilloscopes tend to be expensive, and it is
almost impossible to design and build your own without
another oscilloscope to troubleshoot it! However, the sound
card of a personal computer is capable of "digitizing" low-
voltage AC signals from a range of a few hundred Hertz to
several thousand Hertz with respectable resolution, and free
software is available for displaying these signals in
oscilloscope form on the computer screen. Since most
people either have a personal computer or can obtain one
for less cost than an oscilloscope, this becomes a viable
alternative for the experimenter on a budget.
One word of caution: you can cause significant
hardware damage to your computer if signals of
excessive voltage are connected to the sound card's
microphone input! The AC voltages produced by a musical
keyboard are too low to cause damage to your computer
through the sound card, but other voltage sources might be
hazardous to your computer's health. Use this "oscilloscope"
at your own risk!
Using the keyboard and plug arrangement described in the
previous experiment, connect the keyboard output to the
outer terminals of a 10 kKO potentiometer. Solder two wires to
the connection points on the sound card microphone input
plug, so that you have a set of "test leads" for the
"oscilloscope." Connect these test leads to the
potentiometer: between the middle terminal (the wiper) and
either of the outer terminals.
Start the Winscope program and click on the "arrow" icon in
the upper-left corner (it looks like the "play" arrow seen on
tape player and CD player control buttons). If you press a
key on the musical keyboard, you should see some kind of
waveform displayed on the screen. Choose the "panflute" or
some other flute-like voice on the musical keyboard for the
best sine-wave shape. If the computer displays a waveform
that looks kind of like a Square wave, you need to adjust the
potentiometer for a lower-amplitude signal. Almost any
waveshape will be "clipped" to look like a square wave if it
exceeds the amplitude limit of the sound card.
Test different instrument "voices" on the musical keyboard
and note the different waveshapes. Note how complex some
of the waveshapes are, compared to the panflute voice.
Experiment with the different controls in the Winscope
window, noting how they change the appearance of the
waveform.
As a test instrument, this "oscilloscope" is quite poor. It has
almost no capability to make precision measurements of
voltage, although its frequency precision is surprisingly
good. It is very limited in the ranges of voltage and
frequency it can display, relegating it to the analysis of low-
and mid-range audio tones. | have had very little success
getting the "oscilloscope" to display good square waves,
presumably because of its limited frequency response. Also,
the coupling capacitor found in sound card microphone
input circuits prevents it from measuring DC voltage: it is as
though the "AC coupling" feature of a normal oscilloscope
were stuck "on."
Despite these shortcomings, it is useful as a demonstration
tool, and for initial explorations into waveform analysis for
the beginning student of electronics. For those who are
interested, there are several professional-quality
oscilloscope adapter devices manufactured for personal
computers whose performance is far beyond that of a sound
card, and they are typically sold at less cost than a complete
stand-alone oscilloscope (around $400, year 2002). Radio
Shack sells one made by Velleman, catalog # 910-3914.
Having a computer serve as the display medium brings
many advantages, not the least of which is the ability to
easily store waveform pictures as digital files.
Waveform analysis
PARTS AND MATERIALS
e IBM-compatible personal computer with sound card,
running Windows 3.1 or better
Winscope software, downloaded free from internet
Electronic "keyboard" (musical)
"Mono" (not stereo) headphone-type plug for keyboard
"Mono" (not stereo) headphone-type plug for computer
sound card microphone input, with wires for connecting
to voltage sources
e 10 kQ potentiometer
Parts and equipment for this experiment are identical to
those required for the "PC oscilloscope" experiment.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 7: "Mixed-
Frequency AC Signals"
LEARNING OBJECTIVES
e Understand the difference between time-domain and
frequency-domain plots
e Develop a qualitative sense of Fourier analysis
SCHEMATIC DIAGRAM
Keyboard Computer
plug plug
10kQ
ILLUSTRATION
Computer
"Voice" selecti fii
oooo goo googG ooo €) Volume
WM Monitor
INSTRUCTIONS
The Winscope program comes with another feature other
than the typical "time-domain" oscilloscope display:
"frequency-domain" display, which plots amplitude (vertical)
over frequency (horizontal). An oscilloscope's "time-domain"
display plots amplitude (vertical) over time (horizontal),
which is fine for displaying waveshape. However, when it is
desirable to see the harmonic constituency of a complex
wave, a frequency-domain plot is the best tool.
If using Winscope, click on the "rainbow" icon to switch to
frequency-domain mode. Generate a sine-wave signal using
the musical keyboard (panflute or flute voice), and you
should see a single "spike" on the display, corresponding to
the amplitude of the single-frequency signal. Moving the
mouse cursor beneath the peak should result in the
frequency being displayed numerically at the bottom of the
screen.
If two notes are activated on the musical keyboard, the plot
should show two distinct peaks, each one corresponding to a
particular note (frequency). Basic chords (three notes)
produce three spikes on the frequency-domain plot, and so
on. Contrast this with normal oscilloscope (time-domain) plot
by clicking once again on the "rainbow" icon. A musical
chord displayed in time-domain format is a very complex
waveform, but is quite simple to resolve into constituent
notes (frequencies) on a frequency-domain display.
Experiment with different instrument "voices" on the
musical keyboard, correlating the time-domain plot with the
frequency-domain plot. Waveforms that are symmetrical
above and below their centerlines contain only odd-
numbered harmonics (odd-integer multiples of the base, or
fundamental frequency), while nonsymmetrical waveforms
contain even-numbered harmonics as well. Use the cursor to
locate the specific frequency of each peak on the plot, anda
calculator to determine whether each peak is even- or odd-
numbered.
Inductor-capacitor "tank" circuit
PARTS AND MATERIALS
e Oscilloscope
Assortment of non-polarized capacitors (0.1 UF to 10 YF)
Step-down power transformer (120V / 6 V)
10 kQ resistors
Six-volt battery
The power transformer is used simply as an inductor, with
only one winding connected. The unused winding should be
left open. A simple iron core, single-winding inductor
(Sometimes known as a choke) may also be used, but such
inductors are more difficult to obtain than power
transformers.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 6:
"Resonance"
LEARNING OBJECTIVES
e How to build a resonant circuit
e Effects of capacitor size on resonant frequency
e How to produce antiresonance
SCHEMATIC DIAGRAM
ILLUSTRATION
(transformer used
as an inductor)
INSTRUCTIONS
If an inductor and a capacitor are connected in parallel with
each other, and then briefly energized by connection toa
DC voltage source, oscillations will ensue as energy is
exchanged from the capacitor to inductor and vice versa.
These oscillations may be viewed with an oscilloscope
connected in parallel with the inductor/capacitor circuit.
Parallel inductor/capacitor circuits are commonly known as
tank circuits.
Important note: | recommend against using a PC/sound
card as an oscilloscope for this experiment, because very
high voltages can be generated by the inductor when the
battery is disconnected (inductive "kickback"). These high
voltages will surely damage the sound card's input, and
perhaps other portions of the computer as well.
A tank circuit's natural frequency, called the resonant
frequency, is determined by the size of the inductor and the
size of the capacitor, according to the following equation:
l
resonant —
2x \V LC
Many small power transformers have primary (120 volt)
winding inductances of approximately 1 H. Use this figure as
a rough estimate of inductance for your circuit to calculate
expected oscillation frequency.
f
Ideally, the oscillations produced by a tank circuit continue
indefinitely. Realistically, oscillations will decay in amplitude
over the course of several cycles due to the resistive and
magnetic losses of the inductor. Inductors with a high "Q"
rating will, of course, produce longer-lasting oscillations than
low-Q inductors.
Try changing capacitor values and noting the effect on
oscillation frequency. You might notice changes in the
duration of oscillations as well, due to capacitor size. Since
you know how to calculate resonant frequency from
inductance and capacitance, can you figure out a way to
calculate inductor inductance from known values of circuit
Capacitance (as measured by a capacitance meter) and
resonant frequency (as measured by an oscilloscope)?
Resistance may be intentionally added to the circuit -- either
in series or parallel -- for the express purpose of dampening
oscillations. This effect of resistance dampening tank circuit
oscillation is known as antiresonance. |t is analogous to the
action of a shock absorber in dampening the bouncing of a
car after striking a bump in the road.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Ravay
1 F 2
L, C
0 0
Retray IS placed in the circuit to dampen oscillations and
produce a more realistic simulation. A lower Retray value
causes longer-lived oscillations because less energy is
dissipated. Eliminating this resistor from the circuit results in
endless oscillation.
Netlist (make a text file containing the following text,
verbatim)
tank circuit with loss
l1 101 ic=0
rstray 1 2 1000
cl 2 0 0.1lu ic=6
.tran 0.1m 20m uic
.plot tran v(1,0)
end
Signal coupling
PARTS AND MATERIALS
e 6 volt battery
e One capacitor, 0.22 uF (Radio Shack catalog # 272-
1070 or equivalent)
e One capacitor, 0.047 uF (Radio Shack catalog # 272-
134 or equivalent)
e Small "hobby" motor, permanent-magnet type (Radio
Shack catalog # 273-223 or equivalent)
Audio detector with headphones
Length of telephone cable, several feet long (Radio
Shack catalog # 278-87 2)
Telephone cable is also available from hardware stores. Any
unshielded multiconductor cable will suffice for this
experiment. Cables with thin conductors (telephone cable is
typically 24-gauge) produce a more pronounced effect.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 7: "Mixed-
Frequency AC Signals"
Lessons In Electric Circuits, Volume 2, chapter 8: "Filters"
LEARNING OBJECTIVES
e How to "couple" AC signals and block DC signals toa
measuring instrument
e How stray coupling happens in cables
e Techniques to minimize inter-cable coupling
SCHEMATIC DIAGRAM
Telephone cable
ILLUSTRATION
Telephone
cable
headphones
Cc
© >)
plug
INSTRUCTIONS
Connect the motor to the battery using two of the telephone
cable's four conductors. The motor should run, as expected.
Now, connect the audio signal detector across the motor
terminals, with the 0.047 uF capacitor in series, like this:
headphones
You should be able to hear a "buzz" or "whine" in the
headphones, representing the AC "noise" voltage produced
by the motor as the brushes make and break contact with
the rotating commutator bars. The purpose of the series
Capacitor is to act as a high-pass filter, so that the detector
only receives the AC voltage across the motor's terminals,
not any DC voltage. This is precisely how oscilloscopes
provide an "AC coupling" feature for measuring the AC
content of a signal without any DC bias voltage: a capacitor
is connected in series with one test probe.
Ideally, one would expect nothing but pure DC voltage at
the motor's terminals, because the motor is connected
directly in parallel with the battery. Since the motor's
terminals are electrically common with the respective
terminals of the battery, and the battery's nature is to
maintain a constant DC voltage, nothing but DC voltage
should appear at the motor terminals, right? Well, because
of resistance internal to the battery and along the conductor
lengths, current pulses drawn by the motor produce
oscillating voltage "dips" at the motor terminals, causing the
AC "noise" heard by the detector:
Battery
Use the audio detector to measure "noise" voltage directly
across the battery. Since the AC noise is produced in this
circuit by pulsating voltage drops along stray resistances,
the less resistance we measure across, the less noise voltage
we should detect:
headphones
You may also measure noise voltage dropped along either of
the telephone cable conductors supplying power to the
motor, by connecting the audio detector between both ends
of a single cable conductor. The noise detected here
originates from current pulses through the resistance of the
wire:
headphones
Now that we have established how AC noise is created and
distributed in this circuit, let's explore how it is coup/ed to
adjacent wires in the cable. Use the audio detector to
measure voltage between one of the motor terminals and
one of the unused wires in the telephone cable. The 0.047
UF capacitor is not needed in this exercise, because there is
no DC voltage between these points for the detector to
detect anyway:
headphones
plug
The noise voltage detected here is due to stray capacitance
between adjacent cable conductors creating an AC current
"path" between the wires. Remember that no current
actually goes through a capacitance, but the alternate
charging and discharging action of a capacitance, whether it
be intentional or unintentional, provides a/ternating current
a pathway of sorts.
If we were to try and conduct a voltage signal between one
of the unused wires and a point common with the motor,
that signal would become tainted with noise voltage from
the motor. This could be quite detrimental, depending on
how much noise was coupled between the two circuits and
how sensitive one circuit was to the other's noise. Since the
primary coupling phenomenon in this circuit is capacitive in
nature, higher-frequency noise voltages are more strongly
coupled than lower-frequency noise voltages.
If the additional signal was a DC signal, with no AC expected
in it, we could mitigate the problem of coupled noise by
“decoupling” the AC noise with a relatively large capacitor
connected across the DC signal's conductors. Use the 0.22
UF capacitor for this purpose, as shown:
"decoupling"
capacitor
The decoupling capacitor acts as a practical short-circuit to
any AC noise voltage, while not affecting DC voltage signals
between those two points at all. So long as the decoupling
capacitor value is significantly larger than the stray
"coupling" capacitance between the cable's conductors, the
AC noise voltage will be held to a minimum.
Another way of minimizing coupled noise in a cable is to
avoid having two circuits share a common conductor. To
illustrate, connect the audio detector between the two
unused wires and listen for a noise signal:
There should be far less noise detected between any two of
the unused conductors than between one unused conductor
and one used in the motor circuit. The reason for this drastic
reduction in noise is that stray capacitance between cable
conductors tends to couple the same noise voltage to both
of the unused conductors in approximately equal
proportions. Thus, when measuring voltage between those
two conductors, the detector only "sees" the difference
between two approximately identical noise signals.
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4]
—/ | 4]
Lessons In Electric Circuits
-- Volume VI
Chapter 5
DISCRETE
SEMICONDUCTOR
CIRCUITS
Introduction
Commutating diode
Half-wave rectifier
Full-wave center-tap rectifier
Full-wave bridge rectifier
Rectifier/filter circuit
Voltage regulator
Transistor as a switch
Static electricity sensor
Pulsed-light sensor
Voltage follower
Common-emitter amplifier
Multi-stage amplifier
Current mirror
JEET current regulator
Differential amplifier
Simple op-amp
Audio oscillator
Vacuum tube audio amplifier
Bibliography
Introduction
A semiconductor device is one made of silicon or any
number of other specially prepared materials designed to
exploit the unique properties of electrons in a crystal lattice,
where electrons are not as free to move as in a conductor,
but are far more mobile than in an insulator. A discrete
device is one contained in its own package, not built on a
common semiconductor substrate with other components,
as is the case with ICs, or integrated circuits. Thus, "discrete
semiconductor circuits" are circuits built out of individual
semiconductor components, connected together on some
kind of circuit board or terminal strip. These circuits employ
all the components and concepts explored in the previous
chapters, so a firm comprehension of DC and AC electricity is
essential before embarking on these experiments.
Just for fun, one circuit is included in this section using a
vacuum tube for amplification instead of a semiconductor
transistor. Before the advent of transistors, "vacuum tubes"
were the workhorses of the electronics industry: used to
make rectifiers, amplifiers, oscillators, and many other
circuits. Though now considered obsolete for most purposes,
there are still some applications for vacuum tubes, and it
can be fun building and operating circuits using these
devices.
Commutating diode
PARTS AND MATERIALS
e 6 volt battery
e Power transformer, 120VAC step-down to 12VAC (Radio
Shack catalog # 273-1365, 273-1352, or 273-1511).
e One 1N4001 rectifying diode (Radio Shack catalog #
276-1101)
e One neon lamp (Radio Shack catalog # 272-1102)
e Two toggle switches, SPST ("Single-Pole, Single-Throw")
A power transformer is specified, but any iron-core inductor
will suffice, even the home-made inductor or transformer
from the AC experiments chapter!
The diode need not be an exact model 1N4001. Any of the
"LN400X" series of rectifying diodes are suitable for the
task, and they are quite easy to obtain.
| recommend household light switches for their low cost and
durability.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 16: "RC and
L/R Time Constants"
Lessons In Electric Circuits, Volume 3, chapter 3: "Diodes
and Rectifiers"
LEARNING OBJECTIVES
e Review inductive "kickback"
e Learn how to suppress "kickback" using a diode
SCHEMATIC DIAGRAM
Switch
#1
Neon
Battery — Neor
ILLUSTRATION
Switch #1
|
r
breast
aoe
Switch #2
120 V
12V
INSTRUCTIONS
When assembling the circuit, be very careful of the diode's
orientation. The cathode end of the diode (the end marked
with a single band) must face the positive (+) side of the
battery. The diode should be reverse-biased and
nonconducting with switch #1 in the "on" position. Use the
high-voltage (120 V) winding of the transformer for the
inductor coil. The primary winding of a step-down
transformer has more inductance than the secondary
winding, and will give a greater lamp-flashing effect.
Set switch #2 to the "off" position. This disconnects the
diode from the circuit so that it has no effect. Quickly close
and open (turn "on" and then "off") switch #1. When that
switch is opened, the neon bulb will flash from the effect of
inductive "kickback." Rapid current decrease caused by the
switch's opening causes the inductor to create a large
voltage drop as it attempts to keep current at the same
magnitude and going in the same direction.
Inductive kickback is detrimental to switch contacts, as it
causes excessive arcing whenever they are opened. In this
circuit, the neon lamp actually diminishes the effect by
providing an alternate current path for the inductor's current
when the switch opens, dissipating the inductor's stored
energy harmlessly in the form of light and heat. However,
there is still a fairly high voltage dropped across the opening
contacts of switch #1, causing undue arcing and shortened
switch life.
If switch #2 is closed (turned "on"), the diode will now be a
part of the circuit. Quickly close and open switch #1 again,
noting the difference in circuit behavior. This time, the neon
lamp does not flash. Connect a voltmeter across the inductor
to verify that the inductor is still receiving full battery
voltage with switch #1 closed. If the voltmeter registers only
a small voltage with switch #1 "on," the diode is probably
connected backward, creating a short-circuit.
Half-wave rectifier
PARTS AND MATERIALS
e Low-voltage AC power supply (6 volt output)
6 volt battery
One 1N4001 rectifying diode (Radio Shack catalog #
276-1101)
e Small "hobby" motor, permanent-magnet type (Radio
Shack catalog # 273-223 or equivalent)
Audio detector with headphones
0.1 uF capacitor (Radio Shack catalog # 272-135 or
equivalent)
The diode need not be an exact model 1N4001. Any of the
"IN400X" series of rectifying diodes are suitable for the
task, and they are quite easy to obtain.
See the AC experiments chapter for detailed instructions on
building the "audio detector" listed here. If you haven't built
one already, you're missing a simple and valuable tool for
experimentation.
A 0.1 uF capacitor is specified for "coupling" the audio
detector to the circuit, so that only AC reaches the detector
circuit. This capacitor's value is not critical. I've used
Capacitors ranging from 0.27 uF to 0.015 UF with success.
Lower capacitor values attenuate low-frequency signals toa
greater degree, resulting in less sound intensity from the
headphones, so use a greater-value capacitor value if you
experience difficulty hearing the tone(s).
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 3: "Diodes
and Rectifiers"
LEARNING OBJECTIVES
e Function of a diode as a rectifier
e Permanent-magnet motor operation on AC versus DC
power
e Measuring "ripple" voltage with a voltmeter
SCHEMATIC DIAGRAM
Diode
AC
power
supply
ILLUSTRATION
Low-voltage
AC power supply
INSTRUCTIONS
Connect the motor to the low-voltage AC power supply
through the rectifying diode as shown. The diode only allows
current to pass through during one half-cycle of a full
positive-and-negative cycle of power supply voltage,
eliminating one half-cycle from ever reaching the motor. As a
result, the motor only "sees" current in one direction, albeit
a pulsating current, allowing it to spin in one direction.
Take a jumper wire and short past the diode momentarily,
noting the effect on the motor's operation:
Low-voltage
AC power supply
Temporary
As you Can see, permanent-magnet "DC" motors do not
function well on alternating current. Remove the temporary
jumper wire and reverse the diode's orientation in the
circuit. Note the effect on the motor.
Measure DC voltage across the motor like this:
Low-voltage
AC power supply
Then, measure AC voltage across the motor as well:
Low-voltage
AC power supply
Most digital multimeters do a good job of discriminating AC
from DC voltage, and these two measurements show the DC
average and AC "ripple" voltages, respectively of the power
"seen" by the motor. Ripple voltage is the varying portion of
the voltage, interpreted as an AC quantity by measurement
equipment although the voltage waveform never actually
reverses polarity. Ripple may be envisioned as an AC signal
superimposed on a steady DC "bias" or "offset" signal.
Compare these measurements of DC and AC with voltage
measurements taken across the motor while powered by a
battery:
Batteries give very "pure" DC power, and as a result there
should be very little AC voltage measured across the motor
in this circuit. Whatever AC voltage /s measured across the
motor is due to the motor's pulsating current draw as the
brushes make and break contact with the rotating
commutator bars. This pulsating current causes pulsating
voltages to be dropped across any stray resistances in the
circuit, resulting in pulsating voltage "dips" at the motor
terminals.
A qualitative assessment of ripple voltage may be obtained
by using the sensitive audio detector described in the AC
experiments chapter (the same device described as a
"sensitive voltage detector" in the DC experiments chapter).
Turn the detector's sensitivity down for low volume, and
connect it across the motor terminals through a small (0.1
UF) capacitor, like this:
headphones
Capacitor
The capacitor acts as a high-pass filter, blocking DC voltage
from reaching the detector and allowing easier "listening" of
the remaining AC voltage. This is the exact same technique
used in oscilloscope circuitry for "AC coupling," where DC
signals are blocked from viewing by a series-connected
capacitor. With a battery powering the motor, the ripple
should sound like a high-pitched "buzz" or "whine." Try
replacing the battery with the AC power supply and
rectifying diode, "listening" with the detector to the low-
pitched "buzz" of the half-wave rectified power:
headphones
Low-voltage
AC power supply
COMPUTER SIMULATION
Schematic with SPICE node numbers:
D,
load
Netlist (make a text file containing the following text,
verbatim):
Halfwave rectifier
v1 10 sin(0 8.485 60 0 0)
rload 2 0 10k
di 12 modi
.model modl d
.tran .5m 25m
.plot tran v(1,0) v(2,0)
.end
This simulation plots the input voltage as a sine wave and
the output voltage as a series of "humps" corresponding to
the positive half-cycles of the AC source voltage. The
dynamics of a DC motor are far too complex to be simulated
using SPICE, unfortunately.
AC source voltage is specified as 8.485 instead of 6 volts
because SPICE understands AC voltage in terms of peak
value only. A 6 volt RMS sine-wave voltage is actually 8.485
volts peak. In simulations where the distinction between
RMS and peak value isn't relevant, | will not bother with an
RMS-to-peak conversion like this. To be truthful, the
distinction is not terribly important in this simulation, but |
discuss it here for your edification.
Full-wave center-tap rectifier
PARTS AND MATERIALS
e Low-voltage AC power supply (6 volt output)
e Two 1N4001 rectifying diodes (Radio Shack catalog #
276-1101)
e Small "hobby" motor, permanent-magnet type (Radio
Shack catalog # 273-223 or equivalent)
e Audio detector with headphones
e 0.1 UF capacitor
e One toggle switch, SPST ("Single-Pole, Single-Throw")
It is essential for this experiment that the low-voltage AC
power supply be equipped with a center tap. A transformer
with a non-tapped secondary winding simply will not work
for this circuit.
The diodes need not be exact model 1N4001 units. Any of
the "1N400X" series of rectifying diodes are suitable for the
task, and they are quite easy to obtain.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 3: "Diodes
and Rectifiers"
LEARNING OBJECTIVES
e Design of a center-tap rectifier circuit
e Measuring "ripple" voltage with a voltmeter
SCHEMATIC DIAGRAM
ILLUSTRATION
Low-voltage
AC power supply
LI
ia
Terminal
strip
INSTRUCTIONS
This rectifier circuit is called full-wave because it makes use
of the entire waveform, both positive and negative half-
cycles, of the AC source voltage in powering the DC load. As
a result, there is less "ripple" voltage seen at the load. The
RMS (Root-Mean-Square) value of the rectifier's output is
also greater for this circuit than for the half-wave rectifier.
Use a voltmeter to measure both the DC and AC voltage
delivered to the motor. You should notice the advantages of
the full-wave rectifier immediately by the greater DC and
lower AC indications as compared to the last experiment.
An experimental advantage of this circuit is the ease of
which it may be "de-converted" to a half-wave rectifier:
simply disconnect the short jumper wire connecting the two
diodes' cathode ends together on the terminal strip. Better
yet, for quick comparison between half and full-wave
rectification, you may add a switch in the circuit to open and
close this connection at will:
(close for full-wave operation)
Low-voltage
AC power supply
With the ability to quickly switch between half- and full-
wave rectification, you may easily perform qualitative
comparisons between the two different operating modes.
Use the audio signal detector to "listen" to the ripple voltage
present between the motor terminals for half-wave and full-
wave rectification modes, noting both the intensity and the
quality of the tone. Remember to use a coupling capacitor in
series with the detector so that it only receives the AC
"ripple" voltage and not DC voltage:
headphones
Capacitor
0.1 pF OO is plug
Test ~ ¢
"probes"
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Vv, Road
Netlist (make a text file containing the following text,
verbatim):
Fullwave center-tap rectifier
v1 10 sin(0O 8.485 60 0 0)
v2 0 3 sin(0 8.485 60 0 0)
rload 2 0 10k
dil 1 2 modl
d2 3 2 modl
.model modl d
.tran .5m 25m
.plot tran v(1,0) v(2,0)
.end
Full-wave bridge rectifier
PARTS AND MATERIALS
e Low-voltage AC power supply (6 volt output)
e Four 1N4001 rectifying diodes (Radio Shack catalog #
276-1101)
e Small "hobby" motor, permanent-magnet type (Radio
Shack catalog # 273-223 or equivalent)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 3: "Diodes
and Rectifiers"
LEARNING OBJECTIVES
e Design of a bridge rectifier circuit
e Advantages and disadvantages of the bridge rectifier
circuit, compared to the center-tap circuit
SCHEMATIC DIAGRAM
AC
0 ft}
ILLUSTRATION
Low-voltage
AC power supply
Terminal
strip
INSTRUCTIONS
This circuit provides full-wave rectification without the
necessity of a center-tapped transformer. In applications
where a center-tapped, or sp/it-ohase, source is unavailable,
this is the only practical method of full-wave rectification.
In addition to requiring more diodes than the center-tap
circuit, the full-wave bridge suffers a slight performance
disadvantage as well: the additional voltage drop caused by
current having to go through two diodes in each half-cycle
rather than through only one. With a low-voltage source
such as the one you're using (6 volts RMS), this
disadvantage is easily measured. Compare the DC voltage
reading across the motor terminals with the reading
obtained from the last experiment, given the same AC power
supply and the same motor.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Netlist (make a text file containing the following text,
verbatim):
Fullwave bridge rectifier
v1 10 sin(0O 8.485 60 0 0)
rload 2 3 10k
d1 3 1 modl
d2 1 2 modl
d3 3 0 modl
d4 0 2 modl
.model modl d
.tran .5m 25m
.plot tran v(1,0) v(2,3)
end
Rectifier/filter circuit
PARTS AND MATERIALS
e Low-voltage AC power supply
e Bridge rectifier pack (Radio Shack catalog # 276-1185
or equivalent)
Electrolytic capacitor, 1000 uF, at least 25 WVDC (Radio
Shack catalog # 272-1047 or equivalent)
e Four "banana" jack style binding posts, or other terminal
hardware, for connection to potentiometer circuit (Radio
Shack catalog # 274-662 or equivalent)
Metal box
12-volt light bulb, 25 watt
e Lamp socket
A bridge rectifier "pack" is highly recommended over
constructing a bridge rectifier circuit from individual diodes,
because such "packs" are made to bolt onto a metal heat
sink. A metal box is recommended over a plastic box for its
ability to function as a heat sink for the rectifier.
A larger capacitor value is fine to use in this experiment, so
long as its working voltage is high enough. To be safe,
choose a capacitor with a working voltage rating at least
twice the RMS AC voltage output of the low-voltage AC
power supply.
High-wattage 12-volt lamps may be purchased from
recreational vehicle (RV) and boating supply stores.
Common sizes are 25 watt and 50 watt. This lamp will be
used as a "heavy" load for the power supply.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 2, chapter 8: "Filters"
LEARNING OBJECTIVES
e Capacitive filter function in an AC/DC power supply
e Importance of heat sinks for power semiconductors
SCHEMATIC DIAGRAM
. Rectifier
DC
out
ILLUSTRATION
=
AC
in
=
Rectifier
out
Joyoedey
S,
O
INSTRUCTIONS
This experiment involves constructing a rectifier and filter
circuit for attachment to the low-voltage AC power supply
constructed earlier. With this device, you will have a source
of low-voltage, DC power suitable as a replacement for a
battery in battery-powered experiments. If you would like to
make this device its own, self-contained 120VAC/DC power
supply, you may add all the componentry of the low-voltage
AC supply to the "AC in" side of this circuit: a transformer,
power cord, and plug. Even if you don't choose to do this, |
recommend using a metal box larger than necessary to
provide room for additional voltage regulation circuitry you
might choose to add to this project later.
The bridge rectifier unit should be rated for a current at least
as high as the transformer's secondary winding is rated for,
and for a voltage at least twice as high as the RMS voltage
of the transformer's output (this allows for peak voltage,
plus an additional safety margin). The Radio Shack rectifier
specified in the parts list is rated for 25 amps and 50 volts,
more than enough for the output of the low-voltage AC
power supply specified in the AC experiments chapter.
Rectifier units of this size are often equipped with "quick-
disconnect" terminals. Complementary "quick-disconnect"
lugs are sold that crimp onto the bare ends of wire. This is
the preferred method of terminal connection. You may solder
wires directly to the lugs of the rectifier, but | recommend
against direct soldering to any semiconductor component
for two reasons: possible heat damage during soldering, and
difficulty of replacing the component in the event of failure.
Semiconductor devices are more prone to failure than most
of the components covered in these experiments thus far,
and so if you have any intent of making a circuit permanent,
you should build it to be maintained. "Maintainable
construction" involves, among other things, making all
delicate components replaceable. It also means making "test
points" accessible to meter probes throughout the circuit, so
that troubleshooting may be executed with a minimum of
inconvenience. Terminal strips inherently provide test points
for taking voltage measurements, and they also allow for
easy disconnection of wires without sacrificing connection
durability.
Bolt the rectifier unit to the inside of the metal box. The
box's surface area will act as a radiator, keeping the rectifier
unit cool as it passes high currents. Any metal radiator
surface designed to lower the operating temperature of an
electronic component is called a heat sink. Semiconductor
devices in general are prone to damage from overheating, so
providing a path for heat transfer from the device(s) to the
ambient air is very important when the circuit in question
may handle large amounts of power.
A capacitor is included in the circuit to act as a fi/terto
reduce ripple voltage. Make sure that you connect the
Capacitor properly across the DC output terminals of the
rectifier, so that the polarities match. Being an electrolytic
capacitor, it is sensitive to damage by polarity reversal. In
this circuit especially, where the internal resistance of the
transformer and rectifier are low and the short-circuit current
consequently is high, the potential for damage is great.
Warning: a failed capacitor in this circuit will likely explode
with alarming force!
After the rectifier/filter circuit is built, connect it to the low-
voltage AC power supply like this:
Low-voltage
AC power supply
QO
oO
out
-
Joyoedey
Measure the AC voltage output by the low-voltage power
supply. Your meter should indicate approximately 6 volts if
the circuit is connected as shown. This voltage measurement
is the RMS voltage of the AC power supply.
Now, switch your multimeter to the DC voltage function and
measure the DC voltage output by the rectifier/filter circuit.
It should read substantially higher than the RMS voltage of
the AC input measured before. The filtering action of the
Capacitor provides a DC output voltage equal to the peak AC
voltage, hence the greater voltage indication:
Full-wave, rectified DC voltage
[VV V VN
Time —~
Full-wave, rectified DC voltage, with filtering
wscnctocn te
Time —~
Measure the AC ripple voltage magnitude with a digital
voltmeter set to AC volts (or AC millivolts). You should notice
a much smaller ripple voltage in this circuit than what was
measured in any of the unfiltered rectifier circuits previously
built. Feel free to use your audio detector to "listen" to the
AC ripple voltage output by the rectifier/filter unit. As usual,
connect a small "coupling" capacitor in series with the
detector so that it does not respond to the DC voltage, but
only the AC ripple. Very little sound should be heard.
After taking unloaded AC ripple voltage measurements,
connect the 25 watt light bulb to the output of the
rectifier/filter circuit like this:
Low-voltage
AC power supply
Joyioedey
Re-measure the ripple voltage present between the
rectifier/filter unit's "DC out" terminals. With a heavy load,
the filter capacitor becomes discharged between rectified
voltage peaks, resulting in greater ripple than before:
Full-wave, filtered DC voltage under heavy load
[SIN NT NS
Time —~
If less ripple is desired under heavy-load conditions, a larger
Capacitor may be used, or a more complex filter circuit may
be built using two capacitors and an inductor:
DC
out
If you choose to build such a filter circuit, be sure to use an
iron-core inductor for maximum inductance, and one with
thick enough wire to safely handle the full rated current of
power supply. Inductors used for the purpose of filtering are
sometimes referred to as chokes, because they "choke" AC
ripple voltage from getting to the load. If a suitable choke
cannot be obtained, the secondary winding of a step-down
power transformer like the type used to step 120 volts AC
down to 12 or 6 volts AC in the low-voltage power supply
may be used. Leave the primary (120 volt) winding open:
Leave these wires
disconnected!
pon
DC
out
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Netlist (make a text file containing the following text,
verbatim):
Fullwave bridge rectifier
v1 10 sin(0 8.485 60 0 0)
rload 2 3 10k
cl 2 3 1000u ic=0
d1 3 1 modl
d2 1 2 modl
d3 3 0 modl
d4 0 2 modl
.model modl d
.tran .5m 25m
.plot tran v(1,0) v(2,3)
.end
You may decrease the value of Rjgag in the simulation from
10 kQ to some lower value to explore the effects of loading
on ripple voltage. As it is with a 10 kQ load resistor, the
ripple is undetectable on the waveform plotted by SPICE.
Voltage regulator
PARTS AND MATERIALS
e Four 6 volt batteries
e Zener diode, 12 volt -- type 1N47 42 (Radio Shack
catalog # 276-563 or equivalent)
e One 10 kQ resistor
Any low-voltage zener diode is appropriate for this
experiment. The 1N4742 model listed here (zener voltage =
12 volts) is but one suggestion. Whatever diode model you
choose, | highly recommend one with a zener voltage rating
greater than the voltage of a single battery, for maximum
learning experience. It is important that you see how a zener
diode functions when exposed to a voltage /ess than its
breakdown rating.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 3: "Diodes
and Rectifiers"
LEARNING OBJECTIVES
e Zener diode function
SCHEMATIC DIAGRAM
10 kQ
Zener
— diode
ILLUSTRATION
INSTRUCTIONS
Build this simple circuit, being sure to connect the diode in
"reverse-bias" fashion (cathode positive and anode
negative), and measure the voltage across the diode with
one battery as a power source. Record this voltage drop for
future reference. Also, measure and record the voltage drop
across the 10 kQ resistor.
Modify the circuit by connecting two 6-volt batteries in
series, for 12 volts total power source voltage. Re-measure
the diode's voltage drop, as well as the resistor's voltage
drop, with a voltmeter:
Connect three, then four 6-volt batteries together in series,
forming an 18 volt and 24 volt power source, respectively.
Measure and record the diode's and resistor's voltage drops
for each new power supply voltage. What do you notice
about the diode's voltage drop for these four different source
voltages? Do you see how the diode voltage never exceeds a
level of 12 volts? What do you notice about the resistor's
voltage drop for these four different source voltage levels?
Zener diodes are frequently used as voltage regulating
devices, because they act to clamp the voltage drop across
themselves at a predetermined level. Whatever excess
voltage is supplied by the power source becomes dropped
across the series resistor. However, it is important to note
that a zener diode cannot make up for a deficiency in source
voltage. For instance, this 12-volt zener diode does not drop
12 volts when the power source is only 6 volts strong. It is
helpful to think of a zener diode as a voltage /imiter.
establishing a maximum voltage drop, but not a minimum
voltage drop.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
10 kQ
Zener
— diode
Netlist (make a text file containing the following text,
verbatim):
Zener diode
vl 10
rl 12 10k
d1 0 2 modi
.model modl d bv=12
.dc vl 18 18 1
print dc v(2,0)
.end
A zener diode may be simulated in SPICE with a normal
diode, the reverse breakdown parameter (bv=12) set to the
desired zener breakdown voltage.
Transistor as a switch
PARTS AND MATERIALS
Two 6-volt batteries
One NPN transistor -- models 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
One 100 kQ resistor
One 560 OQ resistor
One light-emitting diode (Radio Shack catalog # 276-
026 or equivalent)
Resistor values are not critical for this experiment. Neither is
the particular light emitting diode (LED) selected.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
LEARNING OBJECTIVES
e Current amplification of a bipolar junction transistor
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
The red wire shown in the diagram (the one terminating in
an arrowhead, connected to one end of the 100 kQ resistor)
is intended to remain loose, so that you may touch it
momentarily to other points in the circuit.
If you touch the end of the loose wire to any point in the
circuit more positive than it, such as the positive side of the
DC power source, the LED should light up. It takes 20 mA to
fully illuminate a standard LED, so this behavior should
strike you as interesting, because the 100 kQ resistor to
which the loose wire is attached restricts current through it
to a far lesser value than 20 mA. At most, a total voltage of
12 volts across a 100 kQ resistance yields a current of only
0.12 mA, or 120 UWA! The connection made by your touching
the wire to a positive point in the circuit conducts far less
current than 1 mA, yet through the amplifying action of the
transistor, is able to contro/ a much greater current through
the LED.
Try using an ammeter to connect the loose wire to the
positive side of the power source, like this:
You may have to select the most sensitive current range on
the meter to measure this small flow. After measuring this
controlling current, try measuring the LED's current (the
controlled current) and compare magnitudes. Don't be
surprised if you find a ratio in excess of 200 (the controlled
current 200 times as great as the controlling current)!
As you can see, the transistor is acting as a kind of
electrically-controlled switch, switching current on and off to
the LED at the command of a much smaller current signal
conducted through its base terminal.
To further illustrate just how miniscule the controlling
Current is, remove the loose wire from the circuit and try
"bridging" the unconnected end of the 100 kQ resistor to the
power source's positive pole with two fingers of one hand.
You may need to wet the ends of those fingers to maximize
conductivity:
Bridge the two points identified by arrows
with two fingers of one hand, to conduct a
small current to the transistor’s base.
Try varying the contact pressure of your fingers with these
two points in the circuit to vary the amount of resistance in
the controlling current's path. Can you vary the brightness
of the LED by doing so? What does this indicate about the
transistor's ability to act as more than just a switch; i.e. asa
variable
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Netlist (make a text file containing the following text,
verbatim):
Transistor as a switch
vl 10
rl 12 100k
r2 1 3 560
d1 3 4 mod2
ql 4 2 © modl
.model modl npn bf=200
.model mod2 d is=1le-28
.dc vl 12 12 1
.print dc v(2,0) v(4,0) v(1,2) v(1,3) v(3,4)
.end
In this simulation, the voltage drop across the 560 OQ resistor
v(1,3) turns out to be 10.26 volts, indicating a LED current
of 18.32 mA by Ohm's Law (I=E/R). Ry's voltage drop
(voltage between nodes 1 and 2) ends up being 11.15 volts,
which across 100 kQO gives a current of only 111.5 UA.
Obviously, a very small current is exerting control over a
much larger current in this circuit.
In case you were wondering, the is=1e-28 parameter in the
diode's .model line is there to make the diode act more like
an LED with a higher forward voltage drop.
Static electricity sensor
PARTS AND MATERIALS
e One N-channel junction field-effect transistor, models
2N3819 or J309 recommended (Radio Shack catalog #
276-2035 is the model 2N3819)
One 6 volt battery
One 100 kQ resistor
One light-emitting diode (Radio Shack catalog # 276-
026 or equivalent)
e Plastic comb
The particular junction field-effect transistor, or JFET, model
used in this experiment is not critical. P-channel JFETs are
also okay to use, but are not as popular as N-channel
transistors.
Beware that not all transistors share the same terminal
designations, or pinouts, even if they share the same
physical appearance. This will dictate how you connect the
transistors together and to other components, so be sure to
check the manufacturer's specifications (component
datasheet), easily obtained from the manufacturer's website.
Beware that it is possible for the transistor's package and
even the manufacturer's datasheet to show incorrect
terminal identification diagrams! Double-checking pin
identities with your multimeter's "diode check" function is
highly recommended. For details on how to identify junction
field-effect transistor terminals using a multimeter, consult
chapter 5 of the Semiconductor volume (volume III) of this
book series.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 5: "Junction
Field-Effect Transistors"
LEARNING OBJECTIVES
e How the JFET is used as an on/off switch
e How JFET current gain differs from a bipolar transistor
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
This experiment is very similar to the previous experiment
using a bipolar junction transistor (BJT) as a switching
device to control current through an LED. In this experiment,
a junction field-effect transistor is used instead, giving
dramatically improved sensitivity.
Build this circuit and touch the loose wire end (the wire
shown in red on the schematic diagram and in the
illustration, connected to the 100 kQ resistor) with your
hand. Simply touching this wire will likely have an effect on
the LED's status. This circuit makes a fine sensor of static
electricity! Try scuffing your feet on a carpet and then
touching the wire end if no effect on the light is seen yet.
For a more controlled test, touch the wire with one hand and
alternately touch the positive (+) and negative (-) terminals
of the battery with one finger of your other hand. Your body
acts as a conductor (albeit a poor one), connecting the gate
terminal of the JFET to either terminal of the battery as you
touch them. Make note which terminal makes the LED turn
on and which makes the LED turn off. Try to relate this
behavior with what you've read about JFETs in chapter 5 of
the Semiconductor volume.
The fact that a JFET is turned on and off so easily (requiring
so little control current), as evidenced by full on-and-off
control simply by conduction of a control current through
your body, demonstrates how great of a current gain it has.
With the BJT "switch" experiment, a much more "solid"
connection between the transistor's gate terminal and a
source of voltage was needed to turn it on. Not so with the
JFET. In fact, the mere presence of static electricity can turn
it on and off at a distance.
To further experiment with the effects of static electricity on
this circuit, brush your hair with the plastic comb and then
wave the comb near the transistor, watching the effect on
the LED. The action of combing your hair with a plastic
object creates a high static voltage between the comb and
your body. The strong electric field produced between these
two objects should be detectable by this circuit from a
significant distance!
In case you're wondering why there is no 560 Q "dropping"
resistor to limit current through the LED, many small-signal
JFETs tend to self-limit their controlled current to a level
acceptable by LEDs. The model 2N3819, for example, has a
typical saturated drain current (Ipss) of 10 mA and a
maximum of 20 mA. Since most LEDs are rated at a forward
current of 20 mA, there is no need for a dropping resistor to
limit circuit current: the JFET does it intrinsically.
Pulsed-light sensor
PARTS AND MATERIALS
Two 6-volt batteries
One NPN transistor -- models 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
e One light-emitting diode (Radio Shack catalog # 276-
026 or equivalent)
e Audio detector with headphones
If you don't have an audio detector already constructed, you
can use a nice set of audio headphones (closed-cup style,
that completely covers your ears) and a 120V/6V step-down
transformer to build a sensitive audio detector without
volume control or overvoltage protection, just for this
experiment.
Connect these portions of the headphone stereo plug to the
transformer's secondary (6 volt) winding:
To transformer To transformer
| if eee
common right left common right left
Speakers in series Speakers in parallel
Try both the series and the parallel connection schemes for
the loudest sound.
If you haven't made an audio detector as outlined in both
the DC and AC experiments chapters, you really should -- it
is a valuable piece of test equipment for your collection.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
LEARNING OBJECTIVES
e How to use a transistor as a crude common-emitter
amplifier
e How to use an LED as a light sensor
SCHEMATIC DIAGRAM
6V —
"4
ILLUSTRATION
headphones
fe) Sensitivity pl ug
INSTRUCTIONS
This circuit detects pulses of light striking the LED and
converts them into relatively strong audio signals to be
heard through the headphones. Forrest Mims teaches that
LEDs have the ability to produce current when exposed to
light, in a manner not unlike a semiconductor solar cell.
[MIM] By itself, the LED does not produce enough electrical
power to drive the audio detector circuit, so a transistor is
used to amplify the LED's signals. If the LED is exposed to a
pulsing source of light, a tone will be heard in the
headphones.
Sources of light suitable for this experiment include
fluorescent and neon lamps, which blink rapidly with the 60
Hz AC power energizing them. You may also try using bright
sunlight for a steady light source, then waving your fingers
in front of the LED. The rapidly passing shadows will cause
the LED to generate pulses of voltage, creating a brief
"buzzing" sound in the headphones.
LEDs serving as photo-detectors are narrow-band devices,
responding to a narrow band of wavelengths close, but not
identical, to that normally emitted. Infrared remote controls
are a good illumination source for near-infrared LEDs
employed as photo-sensors, producing a receiver sound.
[MIM3]
With a little imagination, it is not difficult to grasp the
concept of transmitting audio information -- such as music or
speech -- over a beam of pulsing light. Given a suitable
“transmitter” circuit to pulse an LED on and off with the
positive and negative crests of an audio waveform from a
microphone, the "receiver" circuit shown here would convert
those light pulses back into audio signals. [MIM2]
Voltage follower
PARTS AND MATERIALS
e One NPN transistor -- models 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
e Two 6-volt batteries
e Two 1 kOQ resistors
e One 10 kQ potentiometer, single-turn, linear taper
(Radio Shack catalog # 271-1715)
Beware that not all transistors share the same terminal
designations, or pinouts, even if they share the same
physical appearance. This will dictate how you connect the
transistors together and to other components, so be sure to
check the manufacturer's specifications (component
datasheet), easily obtained from the manufacturer's website.
Beware that it is possible for the transistor's package and
even the manufacturer's datasheet to show incorrect
terminal identification diagrams! Double-checking pin
identities with your multimeter's "diode check" function is
highly recommended. For details on how to identify bipolar
transistor terminals using a multimeter, consult chapter 4 of
the Semiconductor volume (volume III) of this book series.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
LEARNING OBJECTIVES
e Purpose of circuit "ground" when there is no actual
connection to earth ground
e Using a shunt resistor to measure current with a
voltmeter
e Measure amplifier voltage gain
e Measure amplifier current gain
e Amplifier impedance transformation
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
Again, beware that the transistor you select for this
experiment may not have the same terminal designations
shown here, and so the breadboard layout shown in the
illustration may not be correct for you. In my illustrations, |
show all TO-92 package transistors with terminals labeled
"CBE": Collector, Base, and Emitter, from left to right. This is
correct for the model 2N2222 transistor and some others,
but not for all; not even for all NPN-type transistors! As
usual, check with the manufacturer for details on the
particular component(s) you choose for a project. With
bipolar junction transistors, it is easy enough to verify
terminal assignments with a multimeter.
The voltage follower is the safest and easiest transistor
amplifier circuit to build. Its purpose is to provide
approximately the same voltage to a load as what is input to
the amplifier, but at a much greater current. In other words,
it has no voltage gain, but it does have current gain.
Note that the negative (-) side of the power supply is shown
in the schematic diagram to be connected to ground, as
indicated by the symbol in the lower-left corner of the
diagram. This does not necessarily represent a connection to
the actual earth. What it means is that this point in the
circuit -- and all points electrically common to it -- constitute
the default reference point for all voltage measurements in
the circuit. Since voltage is by necessity a quantity relative
between two points, a "common" point of reference
designated in a circuit gives us the ability to speak
meaningfully of voltage at particular, single points in that
Circuit.
These points
are all considered "ground"
For example, if | were to speak of voltage at the base of the
transistor (Vp), | would mean the voltage measured between
the transistor's base terminal and the negative side of the
power supply (ground), with the red probe touching the base
terminal and the black probe touching ground. Normally, it
is nonsense to speak of voltage ata single point, but having
an implicit reference point for voltage measurements makes
such statements meaningful:
Voltmeter measuring
base voltage (V,)
Build this circuit, and measure output voltage versus input
voltage for several different potentiometer settings. Input
voltage is the voltage at the potentiometer's wiper (voltage
between the wiper and circuit ground), while output voltage
is the load resistor voltage (voltage across the load resistor,
or emitter voltage: between emitter and circuit ground). You
should see a close correlation between these two voltages:
one is just a little bit greater than the other (about 0.6 volts
or so?), but a change in the input voltage gives almost equal
change in the output voltage. Because the relationship
between input change and output change is almost 1:1, we
say that the AC voltage gain of this amplifier is nearly 1.
Not very impressive, is it? Now measure current through the
base of the transistor (input current) versus current through
the load resistor (output current). Before you break the
circuit and insert your ammeter to take these
measurements, consider an alternative method: measure
voltage across the base and load resistors, whose resistance
values are known. Using Ohm's Law, current through each
resistor may be easily calculated: divide the measured
voltage by the known resistance (lI=E/R). This calculation is
particularly easy with resistors of 1 kQ value: there will be 1
milliamp of current for every volt of drop across them. For
best precision, you may measure the resistance of each
resistor rather than assume an exact value of 1 kQ, but it
really doesn't matter much for the purposes of this
experiment. When resistors are used to take current
measurements by "translating" a current into a
corresponding voltage, they are often referred to as shunt
resistors.
You should expect to find huge differences between input
and output currents for this amplifier circuit. In fact, it is not
uncommon to experience current gains well in excess of 200
for a small-signal transistor operating at low current levels.
This is the primary purpose of a voltage follower circuit: to
boost the current capacity of a "weak" signal without
altering its voltage.
Another way of thinking of this circuit's function is in terms
of impedance. The input side of this amplifier accepts a
voltage signal without drawing much current. The output
side of this amplifier delivers the same voltage, but at a
current limited only by load resistance and the current-
handling ability of the transistor. Cast in terms of
impedance, we could say that this amplifier has a high input
impedance (voltage dropped with very little current drawn)
and a low output impedance (voltage dropped with almost
unlimited current-sourcing capacity).
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Netlist (make a text file containing the following text,
verbatim):
Voltage follower
v1 10
rpotl 1 2 5k
rpot2 2 0 5k
rbase 2 3 1k
rload 4 0 1k
ql 1 3 4 modl
.model modl npn bf=200
.dc vl 12 12 1
print de v(2,0) v(4,0) v(2,3)
.end
When this simulation is run through the SPICE program, it
shows an input voltage of 5.937 volts and an output voltage
of 5.095 volts, with an input current of 25.35 YA (2.535E-02
volts dropped across the 1 kO Rpace resistor). Output Current
is, of course, 5.095 mA, inferred from the output voltage of
5.095 volts dropped across a load resistance of exactly 1 kQ.
You may change the "potentiometer" setting in this circuit
by adjusting the values of Ryot1 and Ryot2, always keeping
their sum at 10 kQ.
Common-emitter amplifier
PARTS AND MATERIALS
e One NPN transistor -- model 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
Two 6-volt batteries
e One 10 kQ potentiometer, single-turn, linear taper
(Radio Shack catalog # 271-1715)
One 1 MO resistor
One 100 kQ resistor
One 10 kQ resistor
One 1.5 kOQ resistor
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
LEARNING OBJECTIVES
e Design of a simple common-emitter amplifier circuit
e How to measure amplifier voltage gain
e The difference between an inverting and a noninverting
amplifier
e Ways to introduce negative feedback in an amplifier
Circuit
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
Build this circuit and measure output voltage (voltage
measured between the transistor's collector terminal and
ground) and input voltage (voltage measured between the
potentiometer's wiper terminal and ground) for several
position settings of the potentiometer. | recommend
determining the output voltage range as the potentiometer
is adjusted through its entire range of motion, then choosing
several voltages spanning that output range to take
measurements at. For example, if full rotation on the
potentiometer drives the amplifier circuit's output voltage
from 0.1 volts (low) to 11.7 volts (high), choose several
voltage levels between those limits (1 volt, 3 volts, 5 volts, 7
volts, 9 volts, and 11 volts). Measuring the output voltage
with a meter, adjust the potentiometer to obtain each of
these predetermined voltages at the output, noting the
exact figure for later reference. Then, measure the exact
input voltage producing that output voltage, and record that
voltage figure as well.
In the end, you should have a table of numbers representing
several different output voltages along with their
corresponding input voltages. Take any two pairs of voltage
figures and calculate voltage gain by dividing the difference
in output voltages by the difference in input voltages. For
example, if an input voltage of 1.5 volts gives me an output
voltage of 7.0 volts and an input voltage of 1.66 volts gives
me an output voltage of 1.0 volt, the amplifier's voltage gain
is (7.0 - 1.0)/(1.66 - 1.5), or 6 divided by 0.16: a gain ratio of
37.50.
You should immediately notice two characteristics while
taking these voltage measurements: first, that the input-to-
output effect is "reversed;" that is, an increasing input
voltage results in a decreasing output voltage. This effect is
Known as signal inversion, and this kind of amplifier as an
inverting amplifier. Secondly, this amplifier exhibits a very
strong voltage gain: a small change in input voltage results
in a large change in output voltage. This should stand in
stark contrast to the "voltage follower" amplifier circuit
discussed earlier, which had a voltage gain of about 1.
Common-emitter amplifiers are widely used due to their high
voltage gain, but they are rarely used in as crude a form as
this. Although this amplifier circuit works to demonstrate the
basic concept, it is very susceptible to changes in
temperature. Try leaving the potentiometer in one position
and heating the transistor by grasping it firmly with your
hand or heating it with some other source of heat such as an
electric hair dryer (WARNING: be careful not to get it so hot
that your plastic breadboard melts!). You may also explore
temperature effects by cooling the transistor: touch an ice
cube to its surface and note the change in output voltage.
When the transistor's temperature changes, its base-emitter
diode characteristics change, resulting in different amounts
of base current for the same input voltage. This in turn alters
the controlled current through the collector terminal, thus
affecting output voltage. Such changes may be minimized
through the use of signal feedback, whereby a portion of the
output voltage is "fed back" to the amplifier's input so as to
have a negative, or canceling, effect on voltage gain.
Stability is improved at the expense of voltage gain, a
compromise solution, but practical nonetheless.
Perhaps the simplest way to add negative feedback to a
common-emitter amplifier is to add some resistance
between the emitter terminal and ground, so that the input
voltage becomes divided between the base-emitter PN
junction and the voltage drop across the new resistance:
Repeat the same voltage measurement and recording
exercise with the 1.5 kQ resistor installed, calculating the
new (reduced) voltage gain. Try altering the transistor's
temperature again and noting the output voltage for a
steady input voltage. Does it change more or less than
without the 1.5 kQ resistor?
Another method of introducing negative feedback to this
amplifier circuit is to "couple" the output to the input
through a high-value resistor. Connecting a 1 MQ resistor
between the transistor's collector and base terminals works
well:
out
Although this different method of feedback accomplishes
the same goal of increased stability by diminishing gain, the
two feedback circuits will not behave identically. Note the
range of possible output voltages with each feedback
scheme (the low and high voltage values obtained with a
full sweep of the input voltage potentiometer), and how this
differs between the two circuits.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
V
supply
Netlist (make a text file containing the following text,
verbatim):
Common-emitter amplifier
vsupply 1 0 dc 12
vin 3 0
rc 1 2 10k
rb 3 4 100k
ql 2 4 0 modl
.model modl npn bf=200
.dc vin 0 2 0.05
.plot dc v(2,0) v(3,0)
.end
This SPICE simulation sets up a circuit with a variable DC
voltage source (vin) as the input signal, and measures the
corresponding output voltage between nodes 2 and O. The
input voltage is varied, or "swept," from 0 to 2 volts in 0.05
volt increments. Results are shown on a plot, with the input
voltage appearing as a straight line and the output voltage
as a "step" figure where the voltage begins and ends level,
with a steep change in the middle where the transistor is in
its active mode of operation.
Multi-stage amplifier
PARTS AND MATERIALS
e Three NPN transistors -- model 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
Two 6-volt batteries
One 10 kQ potentiometer, single-turn, linear taper
(Radio Shack catalog # 271-1715)
One 1 MOQ resistor
Three 100 kQ resistors
Three 10 kQ resistors
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
LEARNING OBJECTIVES
e Design of a multi-stage, direct-coupled common-emitter
amplifier circuit
e Effect of negative feedback in an amplifier circuit
SCHEMATIC DIAGRAM
ILLUSTRATION
Myo oo eo eco eo eo eo eo eee 8 8
ooso oooooo ooo oo o0oc8ce
oooooo oo oo oec0ecec0ec0e000
ooo ooo oocec0e00
eooooo oc oo cecesecece0ececee00
INSTRUCTIONS
By connecting three common-emitter amplifier circuit
together -- the collector terminal of the previous transistor to
the base (resistor) of the next transistor -- the voltage gains
of each stage compound to give a very high overall voltage
gain. | recommend building this circuit without the 1 MQ
feedback resistor to begin with, to see for yourself just how
high the unrestricted voltage gain is. You may find it
impossible to adjust the potentiometer for a stable output
voltage (that isn't saturated at full supply voltage or zero),
the gain being so high.
Even if you can't adjust the input voltage fine enough to
stabilize the output voltage in the active range of the last
transistor, you should be able to tell that the output-to-input
relationship is inverting; that is, the output tends to drive to
a high voltage when the input goes low, and vice versa.
Since any one of the common-emitter "stages" is inverting in
itself, an even number of staged common-emitter amplifiers
gives noninverting response, while an odd number of stages
gives inverting. You may experience these relationships by
measuring the collector-to-ground voltage at each transistor
while adjusting the input voltage potentiometer, noting
whether or not the output voltage increases or decreases
with an increase in input voltage.
Connect the 1 MQ feedback resistor into the circuit, coupling
the collector of the last transistor to the base of the first.
Since the overall response of this three-stage amplifier is
inverting, the feedback signal provided through the 1 MQ
resistor from the output of the last transistor to the input of
the first should be negative in nature. As such, it will act to
stabilize the amplifier's response and minimize the voltage
gain. You should notice the reduction in gain immediately by
the decreased sensitivity of the output signal on input signal
changes (changes in potentiometer position). Simply put,
the amplifier isn't nearly as "touchy" as it was without the
feedback resistor in place.
As with the simple common-emitter amplifier discussed in
an earlier experiment, it is a good idea here to make a table
of input versus output voltage figures with which you may
calculate voltage gain.
Experiment with different values of feedback resistance.
What effect do you think a decrease in feedback resistance
have on voltage gain? What about an increase in feedback
resistance? Try it and find out!
An advantage of using negative feedback to "tame" a high-
gain amplifier circuit is that the resulting voltage gain
becomes more dependent upon the resistor values and less
dependent upon the characteristics of the constituent
transistors. This is good, because it is far easier to
manufacture consistent resistors than consistent transistors.
Thus, it is easier to design an amplifier with predictable gain
by building a staged network of transistors with an
arbitrarily high voltage gain, then mitigate that gain
precisely through negative feedback. It is this same
principle that is used to make operational amplifier circuits
behave so predictably.
This amplifier circuit is a bit simplified from what you will
normally encounter in practical multi-stage circuits. Rarely is
a pure common-emitter configuration (i.e. with no emitter-
to-ground resistor) used, and if the amplifier's service is for
AC signals, the inter-stage coupling is often capacitive with
voltage divider networks connected to each transistor base
for proper biasing of each stage. Radio-frequency amplifier
circuits are often transformer-coupled, with capacitors
connected in parallel with the transformer windings for
resonant tuning.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
3 1MQ ¢
Netlist (make a text file containing the following text,
verbatim):
Multi-stage amplifier
vsupply 1 0 dc 12
vin 2 0
rl 2 3 100k
r2 1 4 10k
ql 4 3 O modl
r3 4 7 100k
r4 15 10k
q2 5 7 0 modl
r5 5 8 100k
r6 1 6 10k
q3 6 8 O modl
rf 3 6 1lmeg
.model modl npn bf=200
.dc vin 0 2.5 0.1
.plot dc v(6,0) v(2,0)
.end
This simulation plots output voltage against input voltage,
and allows comparison between those variables in numerical
form: a list of voltage figures printed to the left of the plot.
You may calculate voltage gain by taking any two analysis
points and dividing the difference in output voltages by the
difference in input voltages, just like you do for the real
circuit.
Experiment with different feedback resistance values (rf)
and see the impact on overall voltage gain. Do you notice a
pattern? Here's a hint: the overall voltage gain may be
closely approximated by using the resistance figures of r1
and rf, without reference to any other circuit component!
Current mirror
PARTS AND MATERIALS
e Two NPN transistors -- models 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
e Two 6-volt batteries
e One 10 kQ potentiometer, single-turn, linear taper
(Radio Shack catalog # 271-1715)
e Two 10 kQ resistors
e Four 1.5 kQ resistors
Small signal transistors are recommended so as to be able to
experience "thermal runaway" in the latter portion of the
experiment. Larger "power" transistors may not exhibit the
Same behavior at these low current levels. However, any pair
of identical NPN transistors may be used to build a current
mirror.
Beware that not all transistors share the same terminal
designations, or pinouts, even if they share the same
physical appearance. This will dictate how you connect the
transistors together and to other components, so be sure to
check the manufacturer's specifications (component
datasheet), easily obtained from the manufacturer's website.
Beware that it is possible for the transistor's package and
even the manufacturer's datasheet to show incorrect
terminal identification diagrams! Double-checking pin
identities with your multimeter's "diode check" function is
highly recommended. For details on how to identify bipolar
transistor terminals using a multimeter, consult chapter 4 of
the Semiconductor volume (volume III) of this book series.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
LEARNING OBJECTIVES
e How to build a current mirror circuit
e Current limitations of a current mirror circuit
e Temperature dependence of BJTs
e Experience a controlled "thermal runaway" situation
SCHEMATIC DIAGRAM
6V Rioad
TP3
Riad?
6V TP2
Rioadi
“ Riad through Rjoaas
are 1.5 kQ each
Rioad4s Riimits and Ragjust
are 10 kQ each
ILLUSTRATION
INSTRUCTIONS
A current mirror may be thought of as an adjustable current
regulator, the current limit being easily set by a single
resistance. It is a rather crude current regulator circuit, but
one that finds wide use due to its simplicity. In this
experiment, you will get the opportunity to build one of
these circuits, explore its current-regulating properties, and
also experience some of its practical limitations firsthand.
Build the circuit as shown in the schematic and illustration.
You will have one extra 1.5 kQ fixed-value resistor from the
parts specified in the parts list. You will be using it in the last
part of this experiment.
The potentiometer sets the amount of current through
transistor Q,. This transistor is connected to act as a simple
diode: just a PN junction. Why use a transistor instead of a
regular diode? Because it is important to match the junction
characteristics of these two transistors when using them ina
current mirror circuit. Voltage dropped across the base-
emitter junction of Q, is impressed across the base-emitter
junction of the other transistor, Q5, causing it to turn "on"
and likewise conduct current.
Since voltage across the two transistors' base-emitter
junctions is the same -- the two junction pairs being
connected in parallel with each other -- so should the
current be through their base terminals, assuming identical
junction characteristics and identical junction temperatures.
Matched transistors should have the same B ratios, as well,
SO equal base currents means equal collector currents. The
practical result of all this is Q,'s collector current mimicking
whatever current magnitude has been established through
the collector of Q, by the potentiometer. In other words,
current through Q, mirrors the current through Q).
Changes in load resistance (resistance connecting the
collector of Q, to the positive side of the battery) have no
effect on Q,'s current, and consequently have no effect
upon the base-emitter voltage or base current of Q5. With a
constant base current and a nearly constant B ratio, Q> will
drop as much or as little collector-emitter voltage as
necessary to hold its collector (load) current constant. Thus,
the current mirror circuit acts to regu/ate current at a value
set by the potentiometer, without regard to load resistance.
Well, that is how it is supposed to work, anyway. Reality isn't
quite so simple, as you are about to see. In the circuit
diagram shown, the load circuit of Q5 is completed to the
positive side of the battery through an ammeter, for easy
current measurement. Rather than solidly connect the
ammeter's black probe to a definite point in the circuit, I've
marked five test points, TP1 through TP5, for you to touch
the black test probe to while measuring current. This allows
you to quickly and effortlessly change load resistance:
touching the probe to TP1 results in practically no load
resistance, while touching it to TP5 results in approximately
14.5 kQ of load resistance.
To begin the experiment, touch the test probe to TP4 and
adjust the potentiometer through its range of travel. You
should see a small, changing current indicated by your
ammeter as you move the potentiometer mechanism: no
more than a few milliamps. Leave the potentiometer set to a
position giving a round number of milliamps and move the
meter's black test probe to TP3. The current indication
should be very nearly the same as before. Move the probe to
TP2, then TP1. Again, you should see a nearly unchanged
amount of current. Try adjusting the potentiometer to
another position, giving a different current indication, and
touch the meter's black probe to test points TP1 through
TP4, noting the stability of the current indications as you
change load resistance. This demonstrates the current
regulating behavior of this circuit.
You should note that the current regulation isn't perfect.
Despite regulating the current at nearly the value for load
resistances between 0 and 4.5 kQ, there is some variation
over this range. The regulation may be much worse if load
resistance is allowed to rise too high. Try adjusting the
potentiometer so that maximum current is obtained, as
indicated with the ammeter test probe connected to TP1.
Leaving the potentiometer at that position, move the meter
probe to TP2, then TP3, then TP4, and finally TP5, noting the
meter's indication at each connection point. The current
should be regulated at a nearly constant value until the
meter probe is moved to the last test point, TP5. There, the
Current indication will be substantially lower than at the
other test points. Why is this? Because too much load
resistance has been inserted into Q,'s circuit. Simply put, Q>
cannot "turn on" any more than it already has, to maintain
the same amount of current with this great a load resistance
as with lesser load resistances.
This phenomenon is common to all current-regulator
circuits: there is a limited amount of resistance a current
regulator can handle before it saturates. This stands to
reason, as any current regulator circuit capable of supplying
a constant amount of current through any load resistance
imaginable would require an unlimited source of voltage to
do it! Ohm's Law (E=IR) dictates the amount of voltage
needed to push a given amount of current through a given
amount of resistance, and with only 12 volts of power supply
voltage at our disposal, a finite limit of load current and load
resistance definitely exists for this circuit. For this reason, it
may be helpful to think of current regulator circuits as being
current /imiter circuits, for all they can really do is limit
current to some maximum value.
An important caveat for current mirror circuits in general is
that of equal temperature between the two transistors. The
Current "mirroring" taking place between the two transistors'
collector circuits depends on the base-emitter junctions of
those two transistors having the exact same properties. As
the "diode equation" describes, the voltage/current
relationship for a PN junction strongly depends on junction
temperature. The hotter a PN junction is, the more current it
will pass for a given amount of voltage drop. If one transistor
should become hotter than the other, it will pass more
collector current than the other, and the circuit will no
longer "mirror" current as expected. When building a real
current mirror circuit using discrete transistors, the two
transistors should be epoxy-glued together (back-to-back)
so that they remain at approximately the same temperature.
To illustrate this dependence on equal temperature, try
grasping one transistor between your fingers to heat it up.
What happens to the current through the load resistors as
the transistor's temperature increases? Now, let go of the
transistor and blow on it to cool it down to ambient
temperature. Grasp the othertransistor between your
fingers to heat it up. What does the load current do now?
In this next phase of the experiment, we will intentionally
allow one of the transistors to overheat and note the effects.
To avoid damaging a transistor, this procedure should be
conducted no longer than is necessary to observe load
current begin to "run away." To begin, adjust the
potentiometer for minimum current. Next, replace the 10 kQ
Riimit Fesistor with a 1.5 kQ resistor. This will allow a higher
current to pass through Q,, and consequently through Q, as
well.
Place the ammeter's black probe on TP1 and observe the
Current indication. Move the potentiometer in the direction
of increasing current until you read about 10 mA through
the ammeter. At that point, stop moving the potentiometer
and just observe the current. You will notice current begin to
increase all on its own, without further potentiometer
motion! Break the circuit by removing the meter probe from
TP1 when the current exceeds 30 mA, to avoid damaging
transistor Q>.
If you carefully touch both transistors with a finger, you
should notice Q> is warm, while Q, is cool. Warning: if Q,'s
current has been allowed to "run away" too far or for too
long atime, it may become very hot! You can receive a bad
burn on your fingertip by touching an overheated
semiconductor component, so be careful here!
What just happened to make Q, overheat and lose current
control? By connecting the ammeter to TP1, all load
resistance was removed, so Q, had to drop full battery
voltage between collector and emitter as it regulated
current. Transistor Q, at least had the 1.5 kQ resistance of
Riimit in place to drop most of the battery voltage, so its
power dissipation was far less than that of Q5. This gross
imbalance of power dissipation caused Q, to heat more than
Q,. As the temperature increased, Q, began to pass more
current for the same amount of base-emitter voltage drop.
This caused it to heat up even faster, as it was passing more
collector current while still dropping the full 12 volts
between collector and emitter. The effect is known as
thermal runaway, and it is possible in many bipolar junction
transistor circuits, not just current mirrors.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Vv
ammeter
Netlist (make a text file containing the following text,
verbatim):
Current mirror
vl 10
vammeter 1 3 dc 0
rlimit 1 2 10k
rload 3 4 3k
ql 2 2 0 modl
q2 4 2 0 modl
.model mod1l npn bf=100
.dc vl 12 12 1
.print dc i(vammeter)
.end
Vammeter |S Nothing more than a zero-volt DC battery
strategically placed to intercept load current. This is nothing
more than a trick to measure current in a SPICE simulation,
as no dedicated "ammeter" component exists in the SPICE
language.
It is important to remember that SPICE only recognizes the
first eight characters of a component's name. The name
“vammeter" is okay, but if we were to incorporate more than
one current-measuring voltage source in the circuit and
name them "vammeterl" and "vammeter2", respectively,
SPICE would see them as being two instances of the same
component "vammeter" (seeing only the first eight
characters) and halt with an error. Something to bear in
mind when altering the netlist or programming your own
SPICE simulation!
You will have to experiment with different resistance values
Of Rigag in this simulation to appreciate the current-
regulating nature of the circuit. With Rj,j¢ set to 10 kKQ anda
power supply voltage of 12 volts, the regulated current
through Rjgag Will be 1.1 MA. SPICE shows the regulation to
be perfect (isn't the virtual world of computer simulation so
nice?), the load current remaining at 1.1 mA for a wide
range of load resistances. However, if the load resistance is
increased beyond 10 kQ, even this simulation shows the
load current suffering a decrease as in real life.
JFET current regulator
PARTS AND MATERIALS
e One N-channel junction field-effect transistor, models
2N3819 or J309 recommended (Radio Shack catalog #
276-2035 is the model 2N3819)
e Two 6-volt batteries
e One 10 kQ potentiometer, single-turn, linear taper
(Radio Shack catalog # 271-1715)
e One 1 kO resistor
e One 10 kQ resistor
e Three 1.5 kQ resistors
For this experiment you will need an N-channel JFET, not a P-
channel!
Beware that not all transistors share the same terminal
designations, or pinouts, even if they share the same
physical appearance. This will dictate how you connect the
transistors together and to other components, so be sure to
check the manufacturer's specifications (Component
datasheet), easily obtained from the manufacturer's website.
Beware that it is possible for the transistor's package and
even the manufacturer's datasheet to show incorrect
terminal identification diagrams! Double-checking pin
identities with your multimeter's "diode check" function is
highly recommended. For details on how to identify junction
field-effect transistor terminals using a multimeter, consult
chapter 5 of the Semiconductor volume (volume III) of this
book series.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 5: "Junction
Field-Effect Transistors"
Lessons In Electric Circuits, Volume 3, chapter 3: "Diodes
and Rectifiers"
LEARNING OBJECTIVES
e How to use a JFET as a current regulator
e How the JFET is relatively immune to changes in
temperature
SCHEMATIC DIAGRAM
Rioaaa Rioaas
TP3
Rioaa2
TP2
Rioaat
ILLUSTRATION
Rioadi through Roads
are 1.5 kQ each
Rioaas and Ragiust
are 10 kQ each
Riimit is l kQ
INSTRUCTIONS
Previously in this chapter, you saw how a pair of bipolar
junction transistors (BJTs) could be used to form a current
mirror, whereby one transistor would try to maintain the
Same current through it as through the other, the other's
current level being established by a variable resistance. This
circuit performs the same task of regulating current, but
uses a Single junction field-effect transistor (JFET) instead of
two BJ Is.
The two series resistors Ragjust ANd Riimit Set the current
regulation point, while the load resistors and the test points
between them serve only to demonstrate constant current
despite changes in load resistance.
To begin the experiment, touch the test probe to TP4 and
adjust the potentiometer through its range of travel. You
should see a small, changing current indicated by your
ammeter as you move the potentiometer mechanism: no
more than a few milliamps. Leave the potentiometer set to a
position giving a round number of milliamps and move the
meter's black test probe to TP3. The current indication
should be very nearly the same as before. Move the probe to
TP2, then TP1. Again, you should see a nearly unchanged
amount of current. Try adjusting the potentiometer to
another position, giving a different current indication, and
touch the meter's black probe to test points TP1 through
TP4, noting the stability of the current indications as you
change load resistance. This demonstrates the current
regulating behavior of this circuit.
TP5, at the end of a 10 kO resistor, is provided for
introducing a large change in load resistance. Connecting
the black test probe of your ammeter to that test point gives
a combined load resistance of 14.5 kQ, which will be too
much resistance for the transistor to maintain maximum
regulated current through. To experience what I'm
describing here, touch the black test probe to TP1 and
adjust the potentiometer for maximum current. Now, move
the black test probe to TP2, then TP3, then TP4. For all these
test point positions, the current will remain approximately
constant. However, when you touch the black probe to TP5,
the current will fall dramatically. Why? Because at this level
of load resistance, there is insufficient voltage drop across
the transistor to maintain regulation. In other words, the
transistor will be saturated as it attempts to provide more
current than the circuit resistance will allow.
Move the black test probe back to TP1 and adjust the
potentiometer for minimum current. Now, touch the black
test probe to TP2, then TP3, then TP4, and finally TP5. What
do you notice about the current indication at all these
points? When the current regulation point is adjusted toa
lesser value, the transistor is able to maintain regulation
over a much larger range of load resistance.
An important caveat with the BJT current mirror circuit is
that both transistors must be at equal temperature for the
two currents to be equal. With this circuit, however,
transistor temperature is almost irrelevant. Try grasping the
transistor between your fingers to heat it up, noting the load
current with your ammeter. Try cooling it down afterward by
blowing on it. Not only is the requirement of transistor
matching eliminated (due to the use of just one transistor),
but the thermal effects are all but eliminated as well due to
the relative thermal immunity of the field-effect transistor.
This behavior also makes field-effect transistors immune to
thermal runaway; a decided advantage over bipolar junction
transistors.
An interesting application of this current-regulator circuit is
the so-called constant-current diode. Described in the
"Diodes and Rectifiers" chapter of volume III, this diode isn't
really a PN junction device at all. Instead, it is aJFET witha
fixed resistance connected between the gate and source
terminals:
Constant-current diode
Ancde
Symbol Actual
device
Anode
Cathode
Cathode
A normal PN-junction diode is included in series with the
JFET to protect the transistor against damage from reverse-
bias voltage, but otherwise the current-regulating facility of
this device is entirely provided by the field-effect transistor.
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Rica 4-5 kQ
2
Vecurce 0 |
3
1kQ
0 0 0
Netlist (make a text file containing the following text,
verbatim):
JFET current regulator
vsource 1 0
rload 1 2 4.5k
jl 2 0 3 modl
rlimit 3 0 lk
.model modl njf
.dc vsource 6 12 0.1
.plot dc i(vsource)
.end
SPICE does not allow for "sweeping" resistance values, so to
demonstrate the current regulation of this circuit over a wide
range of conditions, I've elected to sweep the source voltage
from 6 to 12 volts in 0.1 volt steps. If you wish, you can set
rload to different resistance values and verify that the circuit
current remains constant. With an rlimit value of 1 kQ, the
regulated current will be 291.8 WA. This current figure will
most likely not be the same as your actual circuit current,
due to differences in JFET parameters.
Many manufacturers give SPICE model parameters for their
transistors, which may be typed in the .model line of the
netlist for a more accurate circuit simulation.
Differential amplifier
PARTS AND MATERIALS
Two 6-volt batteries
Two NPN transistors -- models 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
Two 10 kQ potentiometers, single-turn, linear taper
(Radio Shack catalog # 271-1715)
Two 22 kOQ resistors
Two 10 kQ resistors
One 100 kQ resistor
One 1.5 kQ resistor
Resistor values are not especially critical in this experiment,
but have been chosen to provide high voltage gain fora
“comparator-like" differential amplifier behavior.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
Lessons In Electric Circuits, Volume 3, chapter 8:
“Operational Amplifiers"
LEARNING OBJECTIVES
e Basic design of a differential amplifier circuit.
e Working definitions of differential and common-mode
voltages
SCHEMATIC DIAGRAM
(noninv)
ILLUSTRATION
oninverting
Inverting
INSTRUCTIONS
This circuit forms the heart of most operational amplifier
circuits: the differential pair. In the form shown here, it is a
rather crude differential amplifier, quite nonlinear and
unsymmetrical with regard to output voltage versus input
voltage(s). With a high voltage gain created by a large
collector/emitter resistor ratio (100 kQ/1.5 kQ), though, it
acts primarily as a comparator: the output voltage rapidly
changing value as the two input voltage signals approach
equality.
Measure the output voltage (voltage at the collector of Q,
with respect to ground) as the input voltages are varied.
Note how the two potentiometers have different effects on
the output voltage: one input tends to drive the output
voltage in the same direction (noninverting), while the other
tends to drive the output voltage in the opposite direction
(inverting). This is the essential nature of a differential
amplifier. two complementary inputs, with contrary effects
on the output signal. Ideally, the output voltage of such an
amplifier is strictly a function of the difference between the
two input signals. This circuit falls considerably short of the
ideal, aS even a cursory test will reveal.
An ideal differential amplifier ignores all common-mode
voltage, which is whatever level of voltage common to both
inputs. For example, if the inverting input is at 3 volts and
the noninverting input at 2.5 volts, the differential voltage
will be 0.5 volts (3 - 2.5) but the common-mode voltage will
be 2.5 volts, since that is the lowest input signal level.
Ideally, this condition should produce the same output
signal voltage as if the inputs were set at 3.5 and 3 volts,
respectively (0.5 volts differential, with a 3 volt common-
mode voltage). However, this circuit does not give the same
result for the two different input signal scenarios. In other
words, its output voltage depends on both the differential
voltage and the common-mode voltage.
As imperfect as this differential amplifier is, its behavior
could be worse. Note how the input signal potentiometers
have been limited by 22 kQ resistors to an adjustable range
of approximately 0 to 4 volts, given a power supply voltage
of 12 volts. If you'd like to see how this circuit behaves
without any input signal limiting, just bypass the 22 kO
resistors with jumper wires, allowing full 0 to 12 volt
adjustment range from each potentiometer.
Do not worry about building up excessive heat while
adjusting potentiometers in this circuit! Unlike the current
mirror circuit, this circuit is protected from thermal runaway
by the emitter resistor (1.5 kQ), which doesn't allow enough
transistor current to cause any problem.
Si
ple op-amp
PARTS AND MATERIALS
Two 6-volt batteries
Four NPN transistors -- models 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
Two PNP transistors -- models 2N2907 or 2N3906
recommended (Radio Shack catalog # 276-1604 isa
package of fifteen PNP transistors ideal for this and other
experiments)
Two 10 kQ potentiometers, single-turn, linear taper
(Radio Shack catalog # 271-1715)
One 270 kQ resistor
Three 100 kO resistors
One 10 kO resistor
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
Lessons In Electric Circuits, Volume 3, chapter 8:
"Operational Amplifiers"
LEARNING OBJECTIVES
e Design of a differential amplifier circuit using current
mirrors.
e Effects of negative feedback on a high-gain differential
amplifier.
SCHEMATIC DIAGRAM
ILLUSTRATION
o o6°8 ooosd ooo SS aN
Inverting
INSTRUCTIONS
This circuit design improves on the differential amplifier
shown previously. Rather than use resistors to drop voltage
in the differential pair circuit, a set of current mirrors is used
instead, the result being higher voltage gain and more
predictable performance. With a higher voltage gain, this
circuit is able to function as a working operational amplifier,
or op-amp. Op-amps form the basis of a great many modern
analog semiconductor circuits, so understanding the internal
workings of an operational amplifier is important.
PNP transistors Q; and Q> form a current mirror which tries
to keep current split equally through the two differential pair
transistors Q3 and Q,. NPN transistors Qs and Q, form
another current mirror, setting the tota/ differential pair
current at a level predetermined by resistor Royg-
Measure the output voltage (voltage at the collector of Qy
with respect to ground) as the input voltages are varied.
Note how the two potentiometers have different effects on
the output voltage: one input tends to drive the output
voltage in the same direction (noninverting), while the other
tends to drive the output voltage in the opposite direction
(inverting). You will notice that the output voltage is most
responsive to changes in the input when the two input
Signals are nearly equal to each other.
Once the circuit's differential response has been proven (the
output voltage sharply transitioning from one extreme level
to another when one input is adjusted above and below the
other input's voltage level), you are ready to use this circuit
as areal op-amp. A simple op-amp circuit called a vo/tage
follower is a good configuration to try first. To make a
voltage follower circuit, directly connect the output of the
amplifier to its inverting input. This means connecting the
collector and base terminals of Q, together, and discarding
the "inverting" potentiometer:
Op-amp diagram
y 0 Noninverting
Ri ariree o 6
oo
a.
Note the triangular symbol of the op-amp shown in the lower
schematic diagram. The inverting and noninverting inputs
are designated with (-) and (+) symbols, respectively, with
the output terminal at the right apex. The feedback wire
connecting output to inverting input is shown in red in the
above diagrams.
As a voltage follower, the output voltage should "follow" the
input voltage very closely, deviating no more than a few
hundredths of a volt. This is a much more precise follower
circuit than that of a single common-collector transistor,
described in an earlier experiment!
A more complex op-amp circuit is called the noninverting
amplifier, and it uses a pair of resistors in the feedback loop
to "feed back" a fraction of the output voltage to the
inverting input, causing the amplifier to output a voltage
equal to some multiple of the voltage at the noninverting
input. If we use two equal-value resistors, the feedback
voltage will be 1/2 the output voltage, causing the output
voltage to become twice the voltage impressed at the
noninverting input. Thus, we have a voltage amplifier with a
precise gain of 2:
As you test this noninverting amplifier circuit, you may
notice slight discrepancies between the output and input
voltages. According to the feedback resistor values, the
voltage gain should be exactly 2. However, you may notice
deviations in the order of several hundredths of a volt
between what the output voltage is and what it should be.
These deviations are due to imperfections of the differential
amplifier circuit, and may be greatly diminished if we add
more amplification stages to increase the differential voltage
gain. However, one way we can maximize the precision of
the existing circuit is to change the resistance of Rorg. This
resistor sets the lower current mirror's control point, and in
so doing influences many performance parameters of the op-
amp. Try substituting difference resistance values, ranging
from 10 kQ to 1 MQ. Do not use a resistance less than 10 kQ,
or else the current mirror transistors may begin to overheat
and thermally "run away."
Some operational amplifiers available in prepackaged units
provide a way for the user to similarly "program" the
differential pair's current mirror, and are called
programmable op-amps. Most op-amps are not
programmable, and have their internal current mirror control
points fixed by an internal resistance, trimmed to precise
value at the factory.
Audio oscillator
PARTS AND MATERIALS
e Two 6-volt batteries
e Three NPN transistors -- models 2N2222 or 2N3403
recommended (Radio Shack catalog # 276-1617 isa
package of fifteen NPN transistors ideal for this and
other experiments)
e Two 0.1 UF capacitors (Radio Shack catalog # 272-135
or equivalent)
One 1 MQ resistor
Two 100 kQ resistors
One 1 kQ resistor
Assortment of resistor pairs, less than 100 kQO (ex: two
10 kQ, two 5 kQ, two 1 kQ)
One light-emitting diode (Radio Shack catalog # 276-
026 or equivalent)
Audio detector with headphones
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
Lessons In Electric Circuits, Volume 4, chapter 10:
"Multivibrators"
LEARNING OBJECTIVES
e How to build an astable multivibrator circuit using
discrete transistors
SCHEMATIC DIAGRAM
100 kQ
ILLUSTRATION
INSTRUCTIONS
The proper name for this circuit is "astab/e multivibrator'". |t
is a simple, free-running oscillator circuit timed by the sizes
of the resistors, capacitors, and power supply voltage.
Unfortunately, its output waveform is very distorted, neither
sine wave nor square. For the simple purpose of making an
audio tone, however, distortion doesn't matter much.
With a 12 volt supply, 100 kQ resistors, and 0.1 uF
Capacitors, the oscillation frequency will be in the low audio
range. You may listen to this signal with the audio detector
connected with one test probe to ground and the other to
one of the transistor's collector terminals. | recommend
placing a 1 MQ resistor in series with the audio detector to
minimize both circuit loading effects and headphone
loudness:
Use resistor lead
as test probe for
audio detector
The multivibrator itself is just two transistors, two resistors,
and two cross-connecting capacitors. The third transistor
shown in the schematic and illustration is there for driving
the LED, to be used as a visual indicator of oscillator action.
Use the probe wire connected to the base of this common-
emitter amplifier to detect voltage at different parts of the
circuit with respect to ground. Given the low oscillating
frequency of this multivibrator circuit, you should be able to
see the LED blink rapidly with the probe wire connected to
the collector terminal of either multivibrator transistor.
You may notice that the LED fails to blink with its probe wire
touching the base of either multivibrator transistor, yet the
audio detector tells you there is an oscillating voltage there.
Why is this? The LED's common-collector transistor amplifier
is a voltage follower, meaning that it doesn't amplify
voltage. Thus, if the voltage under test is less than the
minimum required by the LED to light up, it will not glow.
Since the forward-biased base-emitter junction of an active
transistor drops only about 0.7 volts, there is insufficient
voltage at either transistor base to energize the LED. The
audio detector, being extraordinarily sensitive, though,
detects this low voltage signal easily.
Feel free to substitute lower-value resistors in place of the
two 100 kQ units shown. What happens to the oscillation
frequency when you do so? | recommend using resistors at
least 1 kQ in size to prevent excessive transistor current.
One shortcoming of many oscillator circuits is its
dependence on a minimum amount of power supply voltage.
Too little voltage and the circuit ceases to oscillate. This
circuit is no exception. You might want to experiment with
lower supply voltages and determine the minimum voltage
necessary for oscillation, as well as experience the effect
supply voltage change has on oscillation frequency.
One shortcoming specific to this circuit is the dependence
on mismatched components for successful starting. In order
for the circuit to begin oscillating, one transistor must turn
on before the other one. Usually, there is enough mismatch
in the various component values to enable this to happen,
but it is possible for the circuit to "freeze" and fail to
oscillate at power-up. If this happens, try different
components (same values, but different units) in the circuit.
Vacuum tube audio amplifier
PARTS AND MATERIALS
e One 12AX7 dual triode vacuum tube
e Two power transformers, 120VAC step-down to 12VAC
(Radio Shack catalog # 273-1365, 273-1352, or 273-
1511).
e Bridge rectifier module (Radio Shack catalog # 276-
1173)
e Electrolytic capacitor, at least 47 uF, with a working
voltage of at least 200 volts DC.
Automotive ignition coil
Audio speaker, 8 O impedance
Two 100 kQ resistors
One 0.1 UF capacitor, 250 WVDC (Radio Shack catalog #
272-1053)
e "Low-voltage AC power supply" as shown in AC
Experiments chapter
One toggle switch, SPST ("Single-Pole, Single-Throw")
Radio, tape player, musical keyboard, or other source of
audio voltage signal
Where can you obtain a 12AX7 tube, you ask? These tubes
are very popular for use in the "preamplifier" stages of many
professional electric guitar amplifiers. Go to any good music
store and you will find them available for a modest price
($12 US or less). A Russian manufacturer named Sovtek
makes these tubes new, so you need not rely on "New-Old-
Stock" (NOS) components left over from defunct American
manufacturers. This model of tube was very popular in its
day, and may be found in old "tubed" electronic test
equipment (oscilloscopes, oscillators), if you happen to have
access to such equipment. However, | strongly suggest
buying a tube new rather than taking chances with tubes
Salvaged from antique equipment.
It is important to select an electrolytic capacitor with
sufficient working voltage (WVDC) to withstand the output
of this amplifier's power supply circuit (about 170 volts). |
strongly recommend choosing a capacitor with a voltage
rating well in excess of the expected operating voltage, so
as to handle unexpected voltage surges or any other event
that may tax the capacitor. | purchased the Radio Shack
electrolytic capacitor assortment (catalog # 272-802), and it
happened to contain two 47 uF, 250 WVDC capacitors. If you
are not as fortunate, you may build this circuit using five
Capacitors, each rated at 50 WVDC, to substitute for one 250
WVDC unit:
22 kQ
22 kQ equivalent to
(Each capacitor rated
for 50 volts DC) 2 ko
22 kQ
250 110kQ
WVDC
Bear in mind that the total capacitance for this five-
capacitor network will be 1/5, or 20%, of each capacitor's
value. Also, to ensure even charging of capacitors in the
network, be sure all capacitor values (in WF) and all resistor
values are identical.
An automotive ignition coil is a special-purpose high-voltage
transformer used in car engines to produce tens of
thousands of volts to "fire" the spark plugs. In this
experiment, it is used (very unconventionally, | might add!)
as an impedance-matching transformer between the
vacuum tube and an 8 Q audio speaker. The specific choice
of "coil" is not critical, so long as it is in good operating
condition. Here is a photograph of the coil | used for this
experiment:
The audio speaker need not be extravagant. I've used small
"bookshelf" speakers, automotive (6"x9") speakers, as well
as a large (100 watt) 3-way stereo speaker for this
experiment, and they all work fine. Do not use a set of
headphones under any circumstances, as the ignition coil
does not provide electrical isolation between the 170 volts
DC of the "plate" power supply and the speaker, thus
elevating the speaker connections to that voltage with
respect to ground. Since obviously placing wires on your
head with high voltage to ground would be very hazardous,
please do not use headphones!
You will need some source of audio-frequency AC as an input
signal to this amplifier circuit. | recommend a small battery-
powered radio or musical keyboard, with an appropriate
cable plugged into the "headphone" or "audio out" jack to
convey the signal to your amplifier.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 13: "Electron
Tubes"
Lessons In Electric Circuits, Volume 3, chapter 3: "Diodes
and Rectifiers"
Lessons In Electric Circuits, Volume 2, chapter 9:
"Transformers"
LEARNING OBJECTIVES
e Using a vacuum tube (triode) as an audio amplifier
e Using transformers in both step-down and step-up
operation
e How to build a high-voltage DC power supply
e Using a transformer to match impedances
SCHEMATIC DIAGRAM
B+ (100 to 300 volts DC)
Class-A single-ended
tube audio amplifier
8 Q speaker
Automotive
ignition "coil"
0.1 uF
12AX7 "hi-mu"
Audio i
input 100k dual triode tube
CaS
to filament power
(12 volts AC)
High-voltage "plate" DC power supply
Switch Fuse 1 ae 12
o—
12 volt AC
to filaments
Low-voltage AC
power supply
Transformer
Plate supply
switch
ANGER! High voltage!!
= 170 volts
B+
47 wh
250 WVDC
12/120
ratio
Transformer
ILLUSTRATION
Low-voltage
AC power supply
Filament
power
Power supply
INSTRUCTIONS
Welcome to the world of vacuum tube electronics! While not
exactly an application of semiconductor technology (power
supply rectifier excepted), this circuit is useful as an
introduction to vacuum tube technology, and an interesting
application for impedance-matching transformers. It should
be noted that building and operating this circuit
involves work with lethal voltages! You must exhibit the
utmost care while working with this circuit, as 170 volts DC
is capable of electrocuting you!! It is recommended that
beginners seek qualified assistance (experienced
electricians, electronics technicians, or engineers) if
attempting to build this amplifier.
WARNING: do not touch any wires or terminals while
the amplifier circuit is energized! If you must make
contact with the circuit at any point, turn off the "plate"
power supply switch and wait for the filter capacitor to
discharge below 30 volts before touching any part of the
circuit. If testing circuit voltages with the power on, use only
one hand if possible to avoid the possibility of an arm-to-arm
electric shock.
Building the high-voltage power supply
Vacuum tubes require fairly high DC voltage applied
between plate and cathode terminals in order to function
efficiently. Although it is possible to operate the amplifier
circuit described in this experiment on as low as 24 volts DC,
the power output will be miniscule and the sound quality
poor. The 12AX7 triode is rated at a maximum "plate
voltage" (voltage applied between plate and cathode
terminals) of 330 volts, so our power supply of 170 volts DC
specified here is well within that maximum limit. I've
operated this amplifier on as high as 235 volts DC, and
discovered that both sound quality and intensity improved
slightly, but not enough in my estimation to warrant the
additional hazard to experimenters.
The power supply actually has two different power outputs:
the "B+" DC output for plate power, and the "filament"
power, which is only 12 volts AC. Tubes require power
applied to a small filament (sometimes called a heater) in
order to function, as the cathode must be hot enough to
thermally emit electrons, and that doesn't happen at room
temperature! Using one power transformer to step
household 120 volt AC power down to 12 volts AC provides
low-voltage for the filaments, and another transformer
connected in step-up fashion brings the voltage back up to
120 volts. You might be wondering, "why step the voltage
back up to 120 volts with another transformer? Why not just
tap off the wall socket plug to obtain 120 volt AC power
directly, and then rectify that into 170 volts DC?" The
answer to this is twofold: first, running power through two
transformers inherently limits the amount of current that
may be sent into an accidental short-circuit on the plate-side
of the amplifier circuit. Second, it electrically isolates the
plate circuit from the wiring system of your house. If we were
to rectify wall-socket power with a diode bridge, it would
make both DC terminals (+ and -) elevated in voltage from
the safety ground connection of your house's electrical
system, thereby increasing the shock hazard.
Note the toggle switch connected between the 12-volt
windings of the two transformers, labeled "Plate supply
switch." This switch controls power to the step-up
transformer, thereby controlling plate voltage to the
amplifier circuit. Why not just use the main power switch
connected to the 120 volt plug? Why have a second switch
to shut off the DC high voltage, when shutting off one main
switch would accomplish the same thing? The answer lies in
proper vacuum tube operation: like incandescent light
bulbs, vacuum tubes "wear" when their filaments are
powered up and down repeatedly, so having this additional
switch in the circuit allows you to shut off the DC high
voltage (for safety when modifying or adjusting the circuit)
without having to shut off the filament. Also, it is a good
habit to wait for the tube to reach full operating temperature
before applying plate voltage, and this second switch allows
you to delay the application of plate voltage until the tube
has had time to reach operating temperature.
During operation, you should have a voltmeter connected to
the "B+" output of the power supply (between the B+
terminal and ground), continuously providing indication of
the power supply voltage. This meter will show you when
the filter capacitor has discharged below the shock-hazard
limit (30 volts) when you turn off the "Plate supply switch"
to service the amplifier circuit.
The "ground" terminal shown on the DC output of the power
supply circuit need not connect to earth ground. Rather, it is
merely a symbol showing a common connection with a
corresponding ground terminal symbol in the amplifier
circuit. In the circuit you build, there will be a piece of wire
connecting these two "ground" points together. As always,
the designation of certain common points in a circuit by
means of a shared symbol is standard practice in electronic
schematics.
You will note that the schematic diagram shows a 100 kQ
resistor in parallel with the filter capacitor. This resistor is
quite necessary, as it provides the capacitor a path for
discharge when the AC power is turned off. Without this
“bleeder" resistor in the circuit, the capacitor would likely
retain a dangerous charge for a long time after "power-
down," posing an additional shock hazard to you. In the
circuit | built -- with a 47 UF capacitor and a 100 kQ bleeder
resistor -- the time constant of this RC circuit was a brief 4.7
seconds. If you happen to find a larger filter capacitor value
(good for minimizing unwanted power supply "hum" in the
speaker), you will need to use a correspondingly smaller
value of bleeder resistor, or wait longer for the voltage to
bleed off each time you turn the "Plate supply" switch off.
Be sure you have the power supply safely constructed and
working reliably before attempting to power the amplifier
circuit with it. This is good circuit-building practice in
general: build and troubleshoot the power supply first, then
build the circuit you intend to power with it. If the power
supply does not function as it should, then neither will the
powered circuit, no matter how well it may be designed and
built.
Building the amplifier
One of the problems with building vacuum tube circuits in
the 21st century is that sockets for these components can
be difficult to find. Given the limited lifetime of most
"receiver" tubes (a few years), most "tubed" electronic
devices used sockets for mounting the tubes, so that they
could be easily removed and replaced. Though tubes may
still be obtained (from music supply stores) with relative
ease, the sockets they plug into are considerably scarcer --
your local Radio Shack will not have them in stock! How,
then, do we build circuits with tubes, if we might not be able
to obtain sockets for them to plug in to?
For small tubes, this problem may be circumvented by
directly soldering short lengths of 22-gauge solid copper
wire to the pins of the tube, thus enabling you to "plug" the
tube into a solderless breadboard. Here is a photograph of
my tube amplifier, showing the 12AX7 in an inverted
position (pin-side-up). Please disregard the 10-segment LED
bargraph to the left and the 8-position DIP switch assembly
to the right in the photograph, as these are leftover
components from a digital circuit experiment assembled
previously on my breadboard.
AVEAAAAL
One benefit of mounting the tube in this position is ease of
pin identification, since most "pin connection diagrams" for
tubes are shown from a bottom view:
12AX7 dual triode tube
Filament 1
Filament 2 ©)
(a)
Cathode 2 © \ Grid 1
Grid 2 v (Ls ; ; Cathode 1
G)
©) Filament
Plate 2 tap
(view from bottom)
You will notice on the amplifier schematic that both triode
elements inside the 12AX7's glass envelope are being used,
in parallel: plate connected to plate, grid connected to grid,
and cathode connected to cathode. This is done to maximize
power output from the tube, but it is not necessary for
demonstrating basic operation. You may use just one of the
triodes, for simplicity, if you wish.
The 0.1 UF capacitor shown on the schematic "couples" the
audio signal source (radio, musical keyboard, etc.) to the
tube's grid(s), allowing AC to pass but blocking DC. The 100
kQ resistor ensures that the average DC voltage between
grid and cathode is zero, and cannot "float" to some high
level. Typically, bias circuits are used to keep the grid
slightly negative with respect to ground, but for this purpose
a bias circuit would introduce more complexity than its
worth.
When | tested my amplifier circuit, | used the output of a
radio receiver, and later the output of a compact disk (CD)
player, as the audio signal source. Using a "mono'"-to-
“ohono" connector extension cord plugged into the
headphone jack of the receiver/CD player, and alligator clip
jumper wires connecting the "mono" tip of the cord to the
input terminals of the tube amplifier, | was able to easily
send the amplifier audio signals of varying amplitude to test
its performance over a wide range of conditions:
Amplifier circuit
| "Mono" headphone
plug
A transformer is essential at the output of the amplifier
circuit for "matching" the impedances of vacuum tube and
speaker. Since the vacuum tube is a high-voltage, low-
Current device, and most speakers are low-voltage, high-
current devices, the mismatch between them would result in
very audio low power output if they were directly connected.
To successfully match the high-voltage, low-current source
to the low-voltage, high current load, we must use a step-
down transformer.
Since the vacuum tube circuit's Thevenin resistance ranges
in the tens of thousands of ohms, and the speaker only has
about 8 ohms impedance, we will need a transformer with an
impedance ratio of about 10,000:1. Since the impedance
ratio of a transformer is the square of its turns ratio (or
voltage ratio), we're looking for a transformer with a turns
ratio of about 100:1. A typical automotive ignition coil has
approximately this turns ratio, and it is also rated for
extremely high voltage on the high-voltage winding, making
it well suited for this application.
The only bad aspect of using an ignition coil is that it
provides no electrical isolation between primary and
secondary windings, since the device is actually an
autotransformer, with each winding sharing a common
terminal at one end. This means that the speaker wires will
be at a high DC voltage with respect to circuit ground. So
long as we know this, and avoid touching those wires during
operation, there will be no problem. Ideally, though, the
transformer would provide complete isolation as well as
impedance matching, and the speaker wires would be
perfectly safe to touch during use.
Remember, make all connections in the circuit with the
power turned off! After checking connections visually and
with an ohmmeter to ensure that the circuit is built as per
the schematic diagram, apply power to the filaments of the
tube and wait about 30 seconds for it to reach operating
temperature. The both filaments should emit a soft, orange
glow, visible from both the top and bottom views of the
tube.
Turn the volume control of your radio/CD player/musical
keyboard signal source to minimum, then turn on the plate
supply switch. The voltmeter you have connected between
the power supply's B+ output terminal and "ground" should
register full voltage (about 170 volts). Now, increase the
volume control on the signal source and listen to the
speaker. If all is well, you should hear the correct sounds
clearly through the speaker.
Troubleshooting this circuit is best done with the sensitive
audio detector described in the DC and AC chapters of this
Experiments volume. Connect a 0.1 UF capacitor in series
with each test lead to block DC from the detector, then
connect one of the test leads to ground, while using the
other test lead to check for audio signal at various points in
the circuit. Use capacitors with a high voltage rating, like
the one used on the input of the amplifier circuit:
Using the sensitive audio headphones
detector as a troubleshooting
B+ instrument for the amplifier
0.1 WF
Amplifier circuit
bo) Sensitivity plug
Using two coupling capacitors instead of just one adds an
additional degree of safety, in helping to isolate the unit
from any (high) DC voltage. Even without the extra
Capacitor, though, the detector's internal transformer should
provide sufficient electrical isolation for your safety in using
it to test for signals in a high-voltage circuit like this,
especially if you built your detector using a 120 volt power
transformer (rather than an "audio output" transformer) as
suggested. Use it to test for a good signal at the input, then
at the grid pin(s) of the tube, then at the plate of the tube,
etc. until the problem is found. Being capacitively coupled,
the detector is also able to test for excessive power supply
“hum:" touch the free test lead to the supply's B+ terminal
and listen for a loud 60 Hz humming noise. The noise should
be very soft, not loud. If it is loud, the power supply is not
filtered adequately enough, and may need additional filter
Capacitance.
After testing a point in the amplifier circuit with large DC
voltage to ground, the coupling capacitors on the detector
may build up substantial voltage. To discharge this voltage,
briefly touch the free test lead to the grounded test lead. A
"pop" sound should be heard in the headphones as the
coupling capacitors discharge.
If you would rather use a voltmeter to test for the presence
of audio signal, you may do so, setting it to a sensitive AC
voltage range. The indication you get from a voltmeter,
though, doesn't tell you anything about the quality of the
signal, just its mere presence. Bear in mind that most AC
voltmeters will register a transient voltage when initially
connected across a source of DC voltage, so don't be
surprised to see a "Spike" (a strong, momentary voltage
indication) at the very moment contact is made with the
meter's probes to the circuit, rapidly decreasing to the true
AC signal value.
You may be pleasantly surprised at the quality and depth of
tone from this little amplifier circuit, especially given its low
power output: less than 1 watt of audio power. Of course, the
circuit is quite crude and sacrifices quality for simplicity and
parts availability, but it serves to demonstrate the basic
principle of vacuum tube amplification. Advanced hobbyists
and students may wish to experiment with biasing networks,
negative feedback, different output transformers, different
power supply voltages, and even different tubes, to obtain
more power and/or better sound quality.
Here is a photo of a very similar amplifier circuit, built by the
husband-and-wife team of Terry and Chery! Goetz,
illustrating what can be done when care and craftsmanship
are applied to a project like this.
Bibliography
1. [MIM]Forrest M. Mims III, “Sun Photometer with Light-
Emitting Diodes as Spectrally Selective Detectors”,
Applied Optics, 31, 33, 6965-6967, 1992.
2. [MIM2]Forrest M. Mims Ill,“Light Emitting Diodes”
Howard W. Sams & Co., 1973, pp. 118-119.
3. [MIM3]Forrest M. Mims Ill, Private communications,
February 29, 2008.
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
|| 4] l_—
—/ | 4]
Lessons In Electric Circuits
-- Volume VI
Chapter 6
ANALOG INTEGRATED
CIRCUITS
Introduction
Voltage comparator
Precision voltage follower
Noninverting amplifier
High-impedance voltmeter
Integrator
555 audio oscillator
555 ramp generator
PWM _power controller
Class B audio amplifier
Introduction
Analog circuits are circuits dealing with signals free to vary
from zero to full power supply voltage. This stands in
contrast to digita/ circuits, which almost exclusively employ
“all or nothing" signals: voltages restricted to values of zero
and full supply voltage, with no valid state in between those
extreme limits. Analog circuits are often referred to as /inear
circuits to emphasize the valid continuity of signal range
forbidden in digital circuits, but this label is unfortunately
misleading. Just because a voltage or current signal is
allowed to vary smoothly between the extremes of zero and
full power supply limits does not necessarily mean that all
mathematical relationships between these signals are linear
in the "straight-line" or "proportional" sense of the word. As
you will see in this chapter, many so-called "linear" circuits
are quite nonlinear in their behavior, either by necessity of
physics or by design.
The circuits in this chapter make use of /C, or integrated
circuit, components. Such components are actually networks
of interconnected components manufactured on a single
wafer of semiconducting material. Integrated circuits
providing a multitude of pre-engineered functions are
available at very low cost, benefitting students, hobbyists
and professional circuit designers alike. Most integrated
circuits provide the same functionality as "discrete"
semiconductor circuits at higher levels of reliability and ata
fraction of the cost. Usually, discrete-component circuit
construction is favored only when power dissipation levels
are too high for integrated circuits to handle.
Perhaps the most versatile and important analog integrated
circuit for the student to master is the operational amplifier,
or op-amp. Essentially nothing more than a differential
amplifier with very high voltage gain, op-amps are the
workhorse of the analog design world. By cleverly applying
feedback from the output of an op-amp to one or more of its
inputs, a wide variety of behaviors may be obtained from
this single device. Many different models of op-amp are
available at low cost, but circuits described in this chapter
will incorporate only commonly available op-amp models.
Voltage comparator
PARTS AND MATERIALS
e Operational amplifier, model 1458 or 353 recommended
(Radio Shack catalog # 276-038 and 900-6298,
respectively)
Three 6 volt batteries
Two 10 kQ potentiometers, linear taper (Radio Shack
catalog # 271-1715)
e One light-emitting diode (Radio Shack catalog # 276-
026 or equivalent)
e One 330 OQ resistor
e One 470 O resistor
This experiment only requires a single operational amplifier.
The model 1458 and 353 are both "dual" op-amp units, with
two complete amplifier circuits housed in the same 8-pin DIP
package. | recommend that you purchase and use "dual" op-
amps over "single" op-amps even if a project only requires
one, because they are more versatile (the same op-amp unit
can function in projects requiring only one amplifier as well
as in projects requiring two). In the interest of purchasing
and stocking the least number of components for your home
laboratory, this makes sense.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 8:
"Operational Amplifiers"
LEARNING OBJECTIVES
¢ How to use an op-amp as a comparator
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
A comparator circuit compares two voltage signals and
determines which one is greater. The result of this
comparison is indicated by the output voltage: if the op-
amp's output is saturated in the positive direction, the
noninverting input (+) is a greater, or more positive, voltage
than the inverting input (-), all voltages measured with
respect to ground. If the op-amp's voltage is near the
negative supply voltage (in this case, O volts, or ground
potential), it means the inverting input (-) has a greater
voltage applied to it than the noninverting input (+).
This behavior is much easier understood by experimenting
with a comparator circuit than it is by reading someone's
verbal description of it. In this experiment, two
potentiometers supply variable voltages to be compared by
the op-amp. The output status of the op-amp is indicated
visually by the LED. By adjusting the two potentiometers
and observing the LED, one can easily comprehend the
function of a comparator circuit.
For greater insight into this circuit's operation, you might
want to connect a pair of voltmeters to the op-amp input
terminals (both voltmeters referenced to ground) so that
both input voltages may be numerically compared with each
other, these meter indications compared to the LED status:
Comparator circuits are widely used to compare physical
measurements, provided those physical variables can be
translated into voltage signals. For instance, if a small
generator were attached to an anemometer wheel to
produce a voltage proportional to wind speed, that wind
speed signal could be compared with a "set-point" voltage
and compared by an op-amp to drive a high wind speed
alarm:
LED lights up when wind speed exceeds
"set-point" limit established by the
potentiometer position.
Precision voltage follower
PARTS AND MATERIALS
e Operational amplifier, model 1458 or 353 recommended
(Radio Shack catalog # 276-038 and 900-6298,
respectively)
e Three 6 volt batteries
e One 10 kQ potentiometer, linear taper (Radio Shack
catalog # 271-1715)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 8:
"Operational Amplifiers"
LEARNING OBJECTIVES
e How to use an op-amp as a voltage follower
e Purpose of negative feedback
e Troubleshooting strategy
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
In the previous op-amp experiment, the amplifier was used
in "open-loop" mode; that is, without any feedback from
output to input. As such, the full voltage gain of the
operational amplifier was available, resulting in the output
voltage saturating for virtually any amount of differential
voltage applied between the two input terminals. This is
good if we desire comparator operation, but if we want the
op-amp to behave as a true amplifier, we need it to exhibit a
manageable voltage gain.
Since we do not have the luxury of disassembling the
integrated circuitry of the op-amp and changing resistor
values to give a lesser voltage gain, we are limited to
external connections and componentry. Actually, this is not
a disadvantage as one might think, because the
combination of extremely high open-loop voltage gain
coupled with feedback allows us to use the op-amp for a
much wider variety of purposes, much easier than if we were
to exercise the option of modifying its internal circuitry.
If we connect the output of an op-amp to its inverting (-)
input, the output voltage will seek whatever level is
necessary to balance the inverting input's voltage with that
applied to the noninverting (+) input. If this feedback
connection is direct, as in a straight piece of wire, the output
voltage will precisely "follow" the noninverting input's
voltage. Unlike the voltage follower circuit made from a
single transistor (see chapter 5: Discrete Semiconductor
Circuits), which approximated the input voltage to within
several tenths of a volt, this voltage follower circuit will
output a voltage accurate to within mere microvolts of the
input voltage!
Measure the input voltage of this circuit with a voltmeter
connected between the op-amp's noninverting (+) input
terminal and circuit ground (the negative side of the power
supply), and the output voltage between the op-amp's
output terminal and circuit ground. Watch the op-amp's
output voltage follow the input voltage as you adjust the
potentiometer through its range.
You may directly measure the difference, or error, between
output and input voltages by connecting the voltmeter
between the op-amp's two input terminals. Throughout most
of the potentiometer's range, this error voltage should be
almost zero.
Try moving the potentiometer to one of its extreme
positions, far clockwise or far counterclockwise. Measure
error voltage, or compare output voltage against input
voltage. Do you notice anything unusual? If you are using
the model 1458 or model 353 op-amp for this experiment,
you should measure a substantial error voltage, or difference
between output and input. Many op-amps, the specified
models included, cannot "swing" their output voltage
exactly to full power supply ("rail") voltage levels. In this
case, the "rail" voltages are +18 volts and 0 volts,
respectively. Due to limitations in the 1458's internal
circuitry, its output voltage is unable to exactly reach these
high and low limits. You may find that it can only go within a
volt or two of the power supply "rails." This is a very
important limitation to understand when designing circuits
using operational amplifiers. If full "rail-to-rail" output
voltage swing is required in a circuit design, other op-amp
models may be selected which offer this capability. The
model 3130 is one such op-amp.
Precision voltage follower circuits are useful if the voltage
signal to be amplified cannot tolerate "loading;" that is, if it
has a high source impedance. Since a voltage follower by
definition has a voltage gain of 1, its purpose has nothing to
do with amplifying voltage, but rather with amplifying a
signal's capacity to deliver current to a load.
Voltage follower circuits have another important use for
circuit builders: they allow for simple linear testing of an op-
amp. One of the troubleshooting techniques | recommend is
to simplify and rebuild. Suppose that you are building a
circuit using One or more op-amps to perform some
advanced function. If one of those op-amps seems to be
causing a problem and you suspect it may be faulty, try re-
connecting it as a simple voltage follower and see if it
functions in that capacity. An op-amp that fails to work as a
voltage follower certainly won't work as anything more
complex!
COMPUTER SIMULATION
Schematic with SPICE node numbers:
.
_ 2
1
to
V —_ Rjoad
input
Netlist (make a text file containing the following text,
verbatim):
Voltage follower
vinput 1 0
rbogus 1 0 1meg
el 2 0 1 2 999meg
rload 2 0 10k
.dc vinput 55 1
.print dc v(1,0) v(2,0) v(1,2)
.end
An ideal operational amplifier may be simulated in SPICE
using a dependent voltage source (e1 in the netlist). The
output nodes are specified first (2 0), then the two input
nodes, non-inverting input first (1 2). Open-loop gain is
specified last (999meg) in the dependent voltage source line.
Because SPICE views the input impedance of a dependent
source as infinite, some finite amount of resistance must be
included to avoid an analysis error. This is the purpose of
Rpogus: to provide DC path to ground for the Vinpur voltage
source. Such "bogus" resistances should be arbitrarily large.
In this simulation | chose 1 MQ for an Rpogus value.
A load resistor is included in the circuit for much the same
reason: to provide a DC path for current at the output of the
dependent voltage source. As you can see, SPICE doesn't
like open circuits!
Noninverting amplifier
PARTS AND MATERIALS
e Operational amplifier, model 1458 or 353 recommended
(Radio Shack catalog # 276-038 and 900-6298,
respectively)
e Three 6 volt batteries
e Two 10 kQ potentiometers, linear taper (Radio Shack
catalog # 271-1715)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 8:
"Operational Amplifiers"
LEARNING OBJECTIVES
e How to use an op-amp as a single-ended amplifier
e Using divided, negative feedback
SCHEMATIC DIAGRAM
6V
'/, 1458 :
6V
6V
ILLUSTRATION
INSTRUCTIONS
This circuit differs from the voltage follower in only one
respect: output voltage is "fed back" to the inverting (-)
input through a voltage-dividing potentiometer rather than
being directly connected. With only a fraction of the output
voltage fed back to the inverting input, the op-amp will
output a corresponding multiple of the voltage sensed at the
noninverting (+) input in keeping the input differential
voltage near zero. In other words, the op-amp will now
function as an amplifier with a controllable voltage gain,
that gain being established by the position of the feedback
potentiometer (R>).
Set Rz to approximately mid-position. This should give a
voltage gain of about 2. Measure both input and output
voltage for several positions of the input potentiometer Rj.
Move R> to a different position and re-take voltage
measurements for several positions of R;. For any given R>
position, the ratio between output and input voltage should
be the same.
You will also notice that the input and output voltages are
always positive with respect to ground. Because the output
voltage increases in a positive direction for a positive
increase of the input voltage, this amplifier is referred to as
noninverting. If the output and input voltages were related
to one another in an inverse fashion (i.e. positive increasing
input voltage results in positive decreasing or negative
increasing output), then the amplifier would be known as an
inverting type.
The ability to leverage an op-amp in this fashion to create an
amplifier with controllable voltage gain makes this circuit an
extremely useful one. It would take quite a bit more design
and troubleshooting effort to produce a similar circuit using
discrete transistors.
Try adjusting R»z for maximum and minimum voltage gain.
What is the /owest voltage gain attainable with this
amplifier configuration? Why do you think this is?
COMPUTER SIMULATION
Schematic with SPICE node numbers:
Netlist (make a text file containing the following text,
verbatim):
Noninverting amplifier
vinput 1 0
r2 3 2 5k
rl 2 0 5k
rbogus 1 0 1meg
el 3 0 1 2 999meg
rload 3 0 10k
.dc vinput 55 1
.print dc v(1,0) v(3,0)
.end
With R, and R> set equally to 5 kQ in the simulation, it
mimics the feedback potentiometer of the real circuit at mid-
position (50%). To simulate the potentiometer at the 75%
position, set Rz to 7.5 kO and R, to 2.5 kQ.
High-impedance voltmeter
PARTS AND MATERIALS
e Operational amplifier, model TLO82 recommended
(Radio Shack catalog # 276-1715)
e Operational amplifier, model LM1458 recommended
(Radio Shack catalog # 276-038)
Four 6 volt batteries
One meter movement, 1 mA full-scale deflection (Radio
Shack catalog #22-410)
e 15 kQ precision resistor
e Four 1 MO resistors
The 1 mA meter movement sold by Radio Shack is
advertised as a 0-15 VDC meter, but is actually a 1 mA
movement sold with a 15 kQ +/- 1% tolerance multiplier
resistor. If you get this Radio Shack meter movement, you
can use the included 15 kQ resistor for the resistor specified
in the parts list.
This meter experiment is based on a JFET-input op-amp such
as the TLO82. The other op-amp (model 1458) is used in this
experiment to demonstrate the absence of latch-up: a
problem inherent to the TLO82.
You don't need 1 MQ resistors, exactly. Any very high
resistance resistors will suffice.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 8:
"Operational Amplifiers"
LEARNING OBJECTIVES
e Voltmeter loading: its causes and its solution
e How to make a high-impedance voltmeter using an op-
amp
e What op-amp "latch-up" is and how to avoid it
SCHEMATIC DIAGRAM
ILLUSTRATION
Oto1 mA
meter
movement
INSTRUCTIONS
An ideal voltmeter has infinite input impedance, meaning
that it draws zero current from the circuit under test. This
way, there will be no "impact" on the circuit as the voltage is
being measured. The more current a voltmeter draws from
the circuit under test, the more the measured voltage will
"sag" under the loading effect of the meter, like a tire-
pressure gauge releasing air out of the tire being measured:
the more air released from the tire, the more the tire's
pressure will be impacted in the act of measurement. This
loading is more pronounced on circuits of high resistance,
like the voltage divider made of 1 MQ resistors, shown in the
schematic diagram.
If you were to build a simple 0-15 volt range voltmeter by
connecting the 1 mA meter movement in series with the 15
kQ precision resistor, and try to use this voltmeter to
measure the voltages at TP1, TP2, or TP3 (with respect to
ground), you'd encounter severe measurement errors
induced by meter "impact:"
TP3 should be 9 volts
6vV — TP2 should be 6 volts 1 MQ
| TP1 should be 3 volts om
al = 1 MQ
However, the meter will fail to TPI
measure these voltages correctly 1 MQ
due to the meter’s "loading" effect!
Try using the meter movement and 15 kQ resistor as shown
to measure these three voltages. Does the meter read falsely
high or falsely low? Why do you think this is?
If we were to increase the meter's input impedance, we
would diminish its current draw or "load" on the circuit
under test and consequently improve its measurement
accuracy. An op-amp with high-impedance inputs (using a
JFET transistor input stage rather than a BJT input stage)
works well for this application.
Note that the meter movement is part of the op-amp's
feedback loop from output to inverting input. This circuit
drives the meter movement with a current proportional to
the voltage impressed at the noninverting (+) input, the
requisite current supplied directly from the batteries through
the op-amp's power supply pins, not from the circuit under
test through the test probe. The meter's range is set by the
resistor connecting the inverting (-) input to ground.
Build the op-amp meter circuit as shown and re-take voltage
measurements at TP1, TP2, and TP3. You should enjoy far
better success this time, with the meter movement
accurately measuring these voltages (approximately 3, 6,
and 9 volts, respectively).
You may witness the extreme sensitivity of this voltmeter by
touching the test probe with one hand and the most positive
battery terminal with the other. Notice how you can drive
the needle upward on the scale simply by measuring battery
voltage through your body resistance: an impossible feat
with the original, unamplified voltmeter circuit. If you touch
the test probe to ground, the meter should read exactly 0
volts.
After you've proven this circuit to work, modify it by
changing the power supply from dual to split. This entails
removing the center-tap ground connection between the
2nd and 3rd batteries, and grounding the far negative
battery terminal instead:
This alteration in the power supply increases the voltages at
TP1, TP2, and TP3 to 6, 12, and 18 volts, respectively. With a
15 kQ range resistor and a 1 mA meter movement,
measuring 18 volts will gently "peg" the meter, but you
should be able to measure the 6 and 12 volt test points just
fine.
Try touching the meter's test probe to ground. This should
drive the meter needle to exactly O volts as before, but it will
not! What is happening here is an op-amp phenomenon
called /atch-up: where the op-amp output drives to a
positive voltage when the input common-mode voltage
exceeds the allowable limit. In this case, as with many JFET-
input op-amps, neither input should be allowed to come
close to either power supply rail voltage. With a single
supply, the op-amp's negative power rail is at ground
potential (0 volts), so grounding the test probe brings the
noninverting (+) input exactly to that rail voltage. This is
bad for aJFET op-amp, and drives the output strongly
positive, even though it doesn't seem like it should, based
on how op-amps are supposed to function.
When the op-amp ran on a "dual" supply (+12/-12 volts,
rather than a "single" +24 volt supply), the negative power
supply rail was 12 volts away from ground (0 volts), so
grounding the test probe didn't violate the op-amp's
common-mode voltage limit. However, with the "single" +24
volt supply, we have a problem. Note that some op-amps do
not "latch-up" the way the model TLO82 does. You may
replace the TLO82 with an LM1458 op-amp, which is pin-for-
pin compatible (no breadboard wiring changes needed). The
model 1458 will not "latch-up" when the test probe is
grounded, although you may still get incorrect meter
readings with the measured voltage exactly equal to the
negative power supply rail. As a general rule, you should
always be sure the op-amp's power supply rail voltages
exceed the expected input voltages.
Integrator
PARTS AND MATERIALS
Four 6 volt batteries
Operational amplifier, model 1458 recommended (Radio
Shack catalog # 276-038)
e One 10 kQ potentiometer, linear taper (Radio Shack
catalog # 271-1715)
e Two capacitors, 0.1 uF each, non-polarized (Radio Shack
catalog # 272-135)
e Two 100 kO resistors
e Three 1 MOQ resistors
Just about any operational amplifier model will work fine for
this integrator experiment, but I'm specifying the model
1458 over the 353 because the 1458 has much higher input
bias currents. Normally, high input bias current is a bad
characteristic for an op-amp to have in a precision DC
amplifier circuit (and especially an integrator circuit!).
However, | want the bias current to be high in order that its
bad effects may be exaggerated, and so that you will learn
one method of counteracting its effects.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 8:
"Operational Amplifiers"
LEARNING OBJECTIVES
e Method for limiting the span of a potentiometer
e Purpose of an integrator circuit
e How to compensate for op-amp bias current
SCHEMATIC DIAGRAM
ILLUSTRATION
oo oc 09 89 8808 8
ooo oc 0 8080800
INSTRUCTIONS
As you can see from the schematic diagram, the
potentiometer is connected to the "rails" of the power source
through 100 kQ resistors, one on each end. This is to limit
the span of the potentiometer, so that full movement
produces a fairly small range of input voltages for the op-
amp to operate on. At one extreme of the potentiometer's
motion, a voltage of about 0.5 volt (with respect the the
ground point in the middle of the series battery string) will
be produced at the potentiometer wiper. At the other
extreme of motion, a voltage of about -0.5 volt will be
produced. When the potentiometer is positioned dead-
center, the wiper voltage should measure zero volts.
Connect a voltmeter between the op-amp's output terminal
and the circuit ground point. Slowly move the potentiometer
control while monitoring the output voltage. The output
voltage should be changing at a rate established by the
potentiometer's deviation from zero (center) position. To use
calculus terms, we would say that the output voltage
represents the /ntegra/ (with respect to time) of the input
voltage function. That is, the input voltage level establishes
the output voltage rate of change over time. This is
precisely the opposite of differentiation, where the derivative
of a signal or function is its instantaneous rate of change.
If you have two voltmeters, you may readily see this
relationship between input voltage and output vo/tage rate
of change by measuring the wiper voltage (between the
potentiometer wiper and ground) with one meter and the
output voltage (between the op-amp output terminal and
ground) with the other. Adjusting the potentiometer to give
zero volts should result in the slowest output voltage rate-of-
change. Conversely, the more voltage input to this circuit,
the faster its output voltage will change, or "ramp."
Try connecting the second 0.1 uF capacitor in parallel with
the first. This will double the amount of capacitance in the
op-amp's feedback loop. What affect does this have on the
circuit's integration rate for any given potentiometer
position?
Try connecting another 1 MQ resistor in parallel with the
input resistor (the resistor connecting the potentiometer
wiper to the inverting terminal of the op-amp). This will
halve the integrator's input resistance. What affect does this
have on the circuit's integration rate?
Integrator circuits are one of the fundamental "building-
block" functions of an analog computer. By connecting
integrator circuits with amplifiers, summers, and
potentiometers (dividers), almost any differential equation
could be modeled, and solutions obtained by measuring
voltages produced at various points in the network of
circuits. Because differential equations describe so many
physical processes, analog computers are useful as
simulators. Before the advent of modern digital computers,
engineers used analog computers to simulate such
processes as machinery vibration, rocket trajectory, and
control system response. Even though analog computers are
considered obsolete by modern standards, their constituent
components still work well as learning tools for calculus
concepts.
Move the potentiometer until the op-amp's output voltage is
as close to zero as you can get it, and moving as slowly as
you can make it. Disconnect the integrator input from the
potentiometer wiper terminal and connect it instead to
ground, like this:
Connect integrator input directly to ground ==
Connect integrator input directly to ground
Applying exactly zero voltage to the input of an integrator
circuit should, ideally, cause the output voltage rate-of-
change to be zero. When you make this change to the
circuit, you should notice the output voltage remaining at a
constant level or changing very slowly.
With the integrator input still shorted to ground, short past
the 1 MQ resistor connecting the op-amp's noninverting (+)
input to ground. There should be no need for this resistor in
an ideal op-amp circuit, so by shorting past it we will see
what function it provides in this very rea/op-amp circuit:
=
Connect integrator input directly to ground ==
Short past the "grounding" resistor
Connect integrator input directly to ground
Short past the "grounding" resistor
As soon as the "grounding" resistor is shorted with a jumper
wire, the op-amp's output voltage will start to change, or
drift. Ideally, this should not happen, because the integrator
circuit still has an input signal of zero volts. However, real
operational amplifiers have a very small amount of current
entering each input terminal called the bias current. These
bias currents will drop voltage across any resistance in their
path. Since the 1 MQ input resistor conducts some amount
of bias current regardless of input signal magnitude, it will
drop voltage across its terminals due to bias current, thus
"offsetting" the amount of signal voltage seen at the
inverting terminal of the op-amp. If the other (noninverting)
input is connected directly to ground as we have done here,
this "offset" voltage incurred by voltage drop generated by
bias current will cause the integrator circuit to slowly
"integrate" as though it were receiving a very small input
signal.
The "grounding" resistor is better known as a compensating
resistor, because it acts to compensate for voltage errors
created by bias current. Since the bias currents through
each op-amp input terminal are approximately equal to each
other, an equal amount of resistance placed in the path of
each bias current will produce approximately the same
voltage drop. Equal voltage drops seen at the
complementary inputs of an op-amp cancel each other out,
thus nulling the error otherwise induced by bias current.
Remove the jumper wire shorting past the compensating
resistor and notice how the op-amp output returns to a
relatively stable state. It may still drift some, most likely due
to bias voltage error in the op-amp itself, but that is another
subject altogether!
COMPUTER SIMULATION
Schematic with SPICE node numbers:
IMQ 2. OBE
Netlist (make a text file containing the following text,
verbatim):
DC integrator
vinput 1 0 dc 0.05
rl 12 1meg
cl 2 3 0.1lu ic=0
el 3 0 0 2 999k
.tran 1 30 uic
.plot tran v(1,0) v(3,0)
.end
555 audio oscillator
PARTS AND MATERIALS
Two 6 volt batteries
One capacitor, 0.1 UF, non-polarized (Radio Shack
catalog # 272-135)
One 555 timer IC (Radio Shack catalog # 276-1723)
e Two light-emitting diodes (Radio Shack catalog # 276-
026 or equivalent)
One 1 MOQ resistor
One 100 kQ resistor
Two 510 OQ resistors
Audio detector with headphones
Oscilloscope (recommended, but not necessary)
A oscilloscope would be useful in analyzing the waveforms
produced by this circuit, but it is not essential. An audio
detector is a very useful piece of test equipment for this
experiment, especially if you don't have an oscilloscope.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 10:
"Multivibrators"
LEARNING OBJECTIVES
e How to use the 555 timer as an astable multivibrator
e Working knowledge of duty cycle
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
The "555" integrated circuit is a general-purpose timer
useful for a variety of functions. In this experiment, we
explore its use as an astable multivibrator, or oscillator.
Connected to a capacitor and two resistors as shown, it will
oscillate freely, driving the LEDs on and off with a square-
wave output voltage.
This circuit works on the principle of alternately charging
and discharging a capacitor. The 555 begins to discharge
the capacitor by grounding the Disch terminal when the
voltage detected by the Thresh terminal exceeds 2/3 the
power supply voltage (V,,). It stops discharging the
Capacitor when the voltage detected by the Trig terminal
falls below 1/3 the power supply voltage. Thus, when both
Thresh and Trig terminals are connected to the capacitor's
positive terminal, the capacitor voltage will cycle between
1/3 and 2/3 power supply voltage in a "sawtooth" pattern.
During the charging cycle, the capacitor receives charging
current through the series combination of the 1 MO and 100
kQ resistors. As soon as the Disch terminal on the 555 timer
goes to ground potential (a transistor inside the 555
connected between that terminal and ground turns on), the
Capacitor's discharging current only has to go through the
100 kQ resistor. The result is an RC time constant that is
much longer for charging than for discharging, resulting ina
charging time greatly exceeding the discharging time.
The 555's out terminal produces a square-wave voltage
signal that is "high" (nearly V..) when the capacitor is
charging, and "low" (nearly 0 volts) when the capacitor is
discharging. This alternating high/low voltage signal drives
the two LEDs in opposite modes: when one is on, the other
will be off. Because the capacitor's charging and discharging
times are unequal, the "high" and "low" times of the output's
square-wave waveform will be unequal as well. This can be
seen in the relative brightness of the two LEDs: one will be
much brighter than the other, because it is on for a longer
period of time during each cycle.
The equality or inequality between "high" and "low" times of
a square wave Is expressed as that wave's duty cycle. A
square wave with a 50% duty cycle is perfectly symmetrical:
its "high" time is precisely equal to its "low" time. A square
wave that is "high" 10% of the time and "low" 90% of the
time is said to have a 10% duty cycle. In this circuit, the
output waveform has a "high" time exceeding the "low"
time, resulting in a duty cycle greater than 50%.
Use the audio detector (or an oscilloscope) to investigate
the different voltage waveforms produced by this circuit. Try
different resistor values and/or capacitor values to see what
effects they have on output frequency or charge/discharge
times.
555 ramp generator
PARTS AND MATERIALS
Two 6 volt batteries
One capacitor, 470 uF electrolytic, 35 WVDC (Radio
Shack catalog # 272-1030 or equivalent)
e One capacitor, 0.1 UF, non-polarized (Radio Shack
catalog # 272-135)
One 555 timer IC (Radio Shack catalog # 276-1723)
Two PNP transistors -- models 2N2907 or 2N3906
recommended (Radio Shack catalog # 276-1604 isa
package of fifteen PNP transistors ideal for this and other
experiments)
e Two light-emitting diodes (Radio Shack catalog # 276-
026 or equivalent)
One 100 kQ resistor
One 47 kQ resistor
Two 510 Q resistors
Audio detector with headphones
The voltage rating on the 470 uF capacitor is not critical, so
long as it generously exceeds the maximum power supply
voltage. In this particular circuit, that maximum voltage is
12 volts. Be sure you connect this capacitor in the circuit
properly, respecting polarity!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 13:
"Capacitors"
Lessons In Electric Circuits, Volume 4, chapter 10:
"Multivibrators"
LEARNING OBJECTIVES
e How to use the 555 timer as an astable multivibrator
e A practical use for a current mirror circuit
e Understanding the relationship between capacitor
current and capacitor voltage rate-of-change
SCHEMATIC DIAGRAM
100 kQ
ILLUSTRATION
INSTRUCTIONS
Again, we are using a 555 timer IC as an astable
multivibrator, or oscillator. This time, however, we will
compare its operation in two different capacitor-charging
modes: traditional RC and constant-current.
Connecting test point #1 (TP1) to test point #3 (TP3) using
a jumper wire. This allows the capacitor to charge through a
47 kQ resistor. When the capacitor has reached 2/3 supply
voltage, the 555 timer switches to "discharge" mode and
discharges the capacitor to a level of 1/3 supply voltage
almost immediately. The charging cycle begins again at this
point. Measure voltage directly across the capacitor with a
voltmeter (a digital voltmeter is preferred), and note the rate
of capacitor charging over time. It should rise quickly at first,
then taper off as it builds up to 2/3 supply voltage, just as
you would expect from an RC charging circuit.
Remove the jumper wire from TP3, and re-connect it to TP2.
This allows the capacitor to be charged through the
controlled-current leg of a current mirror circuit formed by
the two PNP transistors. Measure voltage directly across the
Capacitor again, noting the difference in charging rate over
time as compared to the last circuit configuration.
By connecting TP1 to TP2, the capacitor receives a nearly
constant charging current. Constant capacitor charging
current yields a voltage curve that is linear, as described by
the equation | = C(de/dt). If the capacitor's current is
constant, so will be its rate-of-change of voltage over time.
The result is a "ramp" waveform rather than a "sawtooth"
waveform:
fVvwvw\
Sawtooth waveform (RC circuit)
ZVAV (|
Ramp waveform (constant current)
The capacitor's charging current may be directly measured
by substituting an ammeter in place of the jumper wire. The
ammeter will need to be set to measure a current in the
range of hundreds of microamps (tenths of a milliamp).
Connected between TP1 and TP3, you should see a current
that starts at a relatively high value at the beginning of the
charging cycle, and tapers off toward the end. Connected
between TP1 and TP2, however, the current will be much
more stable.
It is an interesting experiment at this point to change the
temperature of either current mirror transistor by touching it
with your finger. As the transistor warms, it will conduct
more collector current for the same base-emitter voltage. If
the controlling transistor (the one connected to the 100 kO
resistor) is touched, the current decreases. If the controlled
transistor is touched, the current increases. For the most
stable current mirror operation, the two transistors should be
cemented together so that their temperatures never differ
by any substantial amount.
This circuit works just as well at high frequencies as it does
at low frequencies. Replace the 470 uF capacitor with a 0.1
UF capacitor, and use an audio detector to sense the voltage
waveform at the 555's output terminal. The detector should
produce an audio tone that is easy to hear. The capacitor's
voltage will now be changing much too fast to view with a
voltmeter in the DC mode, but we can still measure
Capacitor current with an ammeter.
With the ammeter connected between TP1 and TP3 (RC
mode), measure both DC microamps and AC microamps.
Record these current figures on paper. Now, connect the
ammeter between TP1 and TP2 (constant-current mode).
Measure both DC microamps and AC microamps, noting any
differences in current readings between this circuit
configuration and the last one. Measuring AC current in
addition to DC current is an easy way to determine which
circuit configuration gives the most stable charging current.
If the current mirror circuit were perfect -- the capacitor
charging current absolutely constant -- there would be zero
AC current measured by the meter.
PWM power controller
PARTS AND MATERIALS
Four 6 volt batteries
One capacitor, 100 uF electrolytic, 35 WVDC (Radio
Shack catalog # 272-1028 or equivalent)
One capacitor, 0.1 UF, non-polarized (Radio Shack
catalog # 272-135)
One 555 timer IC (Radio Shack catalog # 276-1723)
Dual operational amplifier, model 1458 recommended
(Radio Shack catalog # 276-038)
One NPN power transistor -- (Radio Shack catalog # 276-
2041 or equivalent)
Three 1N4001 rectifying diodes (Radio Shack catalog #
276-1101)
One 10 kQ potentiometer, linear taper (Radio Shack
catalog # 271-1715)
One 33 kQ resistor
12 volt automotive tail-light lamp
Audio detector with headphones
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 8:
“Operational Amplifiers"
Lessons In Electric Circuits, Volume 2, chapter 7: "Mixed-
Frequency AC Signals"
LEARNING OBJECTIVES
e How to use the 555 timer as an astable multivibrator
e How to use an op-amp as a comparator
e How to use diodes to drop unwanted DC voltage
e How to control power to a load by pulse-width
modulation
SCHEMATIC DIAGRAM
Power
transistor
O.1 LP
ILLUSTRATION
INSTRUCTIONS
This circuit uses a 555 timer to generate a sawtooth voltage
waveform across a capacitor, then compares that signal
against a steady voltage provided by a potentiometer, using
an op-amp as a comparator. The comparison of these two
voltage signals produces a square-wave output from the op-
amp, varying in duty cycle according to the potentiometer's
position. This variable duty cycle signal then drives the base
of a power transistor, switching current on and off through
the load. The 555's oscillation frequency is much higher
than the lamp filament's ability to thermally cycle (heat and
cool), so any variation in duty cycle, or pulse width, has the
effect of controlling the total power dissipated by the load
over time.
Output (low power to load)
AYVIVWU
E
Output (high power to load)
Controlling electrical power through a load by means of
quickly switching it on and off, and varying the "on" time, is
known as pulse-width modulation, or PWM. \t is a very
efficient means of controlling electrical power because the
controlling element (the power transistor) dissipates
comparatively little power in switching on and off, especially
if compared to the wasted power dissipated of a rheostat in
a similar situation. When the transistor is in cutoff, its power
dissipation is zero because there is no current through it.
When the transistor is saturated, its dissipation is very low
because there is little voltage dropped between collector
and emitter while it is conducting current.
PWM is a concept easier understood through
experimentation than reading. It would be nice to view the
Capacitor voltage, potentiometer voltage, and op-amp
output waveforms all on one (triple-trace) oscilloscope to see
how they relate to one another, and to the load power.
However, most of us have no access to a triple-trace
oscilloscope, much less any oscilloscope at all, so an
alternative method is to slow the 555 oscillator down
enough that the three voltages may be compared with a
simple DC voltmeter. Replace the 0.1 UF capacitor with one
that is 100 uF or larger. This will slow the oscillation
frequency down by a factor of at least a thousand, enabling
you to measure the capacitor voltage s/owly rise over time,
and the op-amp output transition from "high" to "low" when
the capacitor voltage becomes greater than the
potentiometer voltage. With such a slow oscillation
frequency, the load power will not be proportioned as before.
Rather, the lamp will turn on and off at regular intervals.
Feel free to experiment with other capacitor or resistor
values to speed up the oscillations enough so the lamp
never fully turns on or off, but is "throttled" by quick on-and-
off pulsing of the transistor.
When you examine the schematic, you will notice two
operational amplifiers connected in parallel. This is done to
provide maximum current output to the base terminal of the
power transistor. A single op-amp (one-half of a 1458 IC)
may not be able to provide sufficient output current to drive
the transistor into saturation, so two op-amps are used in
tandem. This should only be done if the op-amps in question
are overload-protected, which the 1458 series of op-amps
are. Otherwise, it is possible (though unlikely) that one op-
amp could turn on before the other, and damage result from
the two outputs short-circuiting each other (one driving
"high" and the other driving "low" simultaneously). The
inherent short-circuit protection offered by the 1458 allows
for direct driving of the power transistor base without any
need for a current-limiting resistor.
The three diodes in series connecting the op-amps' outputs
to the transistor's base are there to drop voltage and ensure
the transistor falls into cutoff when the op-amp outputs go
"low." Because the 1458 op-amp cannot swing its output
voltage all the way down to ground potential, but only to
within about 2 volts of ground, a direct connection from the
op-amp to the transistor would mean the transistor would
never fully turn off. Adding three silicon diodes in series
drops approximately 2.1 volts (0.7 volts times 3) to ensure
there is minimal voltage at the transistor's base when the
Op-amp outputs go "low."
It is interesting to listen to the op-amp output signal through
an audio detector as the potentiometer is adjusted through
its full range of motion. Adjusting the potentiometer has no
effect on signal frequency, but it greatly affects duty cycle.
Note the difference in tone quality, or timbre, as the
potentiometer varies the duty cycle from 0% to 50% to
100%. Varying the duty cycle has the effect of changing the
harmonic content of the waveform, which makes the tone
sound different.
You might notice a particular uniqueness to the sound heard
through the detector headphones when the potentiometer is
in center position (50% duty cycle -- 50% load power),
versus a kind of similarity in sound just above or below 50%
duty cycle. This is due to the absence or presence of even-
numbered harmonics. Any waveform that is symmetrical
above and below its centerline, such as a square wave with a
50% duty cycle, contains no even-numbered harmonics,
only odd-numbered. If the duty cycle is below or above 50%,
the waveform will not exhibit this symmetry, and there will
be even-numbered harmonics. The presence of these even-
numbered harmonic frequencies can be detected by the
human ear, as some of them correspond to octaves of the
fundamental frequency and thus "fit" more naturally into the
tone scheme.
Class B audio amplifier
PARTS AND MATERIALS
Four 6 volt batteries
Dual operational amplifier, model TLO82 recommended
(Radio Shack catalog # 276-1715)
e One NPN power transistor in a TO-220 package -- (Radio
Shack catalog # 276-2020 or equivalent)
e One PNP power transistor in a TO-220 package -- (Radio
Shack catalog # 276-2027 or equivalent)
e One 1N914 switching diode (Radio Shack catalog # 276-
1620)
e One capacitor, 47 uF electrolytic, 35 WVDC (Radio
Shack catalog # 272-1015 or equivalent)
e Two capacitors, 0.22 uF, non-polarized (Radio Shack
catalog # 272-1070)
e One 10 kQ potentiometer, linear taper (Radio Shack
catalog # 271-1715)
Be sure to use an op-amp that has a high s/ew rate. Avoid
the LM741 or LM1458 for this reason.
The closer matched the two transistors are, the better. If
possible, try to obtain TIP41 and TIP42 transistors, which are
closely matched NPN and PNP power transistors with
dissipation ratings of 65 watts each. If you cannot get a
TIP41 NPN transistor, the TIP3055 (available from Radio
Shack) is a good substitute. Do not use very large (i.e. TO-3
case) power transistors, as the op-amp may have trouble
driving enough current to their bases for good operation.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 4: "Bipolar
Junction Transistors"
Lessons In Electric Circuits, Volume 3, chapter 8:
"Operational Amplifiers"
LEARNING OBJECTIVES
e How to build a "push-pull" class B amplifier using
complementary bipolar transistors
e The effects of "crossover distortion" in a push-pull
amplifier circuit
e Using negative feedback via an op-amp to correct circuit
nonlinearities
SCHEMATIC DIAGRAM
22 WF TIP41
or
0
slanal —
TIP3055
+ sO
ait speaker
TIP42 —
Power
supply
ILLUSTRATION
Volume
control
Speaker
INSTRUCTIONS
This project is an audio amplifier suitable for amplifying the
output signal from a small radio, tape player, CD player, or
any other source of audio signals. For stereo operation, two
identical amplifiers must be built, one for the left channel
and other for the right channel. To obtain an input signal for
this amplifier to amplify, just connect it to the output of a
radio or other audio device like this:
Amplifier circuit
1 "Phono" plug
| "Mono" headphone
plug
This amplifier circuit also works well in amplifying "line-
level" audio signals from high-quality, modular stereo
components. It provides a surprising amount of sound power
when played through a large speaker, and may be run
without heat sinks on the transistors (though you should
experiment with it a bit before deciding to forego heat sinks,
as the power dissipation varies according to the type of
speaker used).
The goal of any amplifier circuit is to reproduce the input
waveshape as accurately as possible. Perfect reproduction is
impossible, of course, and any differences between the
output and input waveshapes is known as distortion. In an
audio amplifier, distortion may cause unpleasant tones to be
superimposed on the true sound. There are many different
configurations of audio amplifier circuitry, each with its own
advantages and disadvantages. This particular circuit is
called a "class B," push-pull circuit.
Most audio "power" amplifiers use a class B configuration,
where one transistor provides power to the load during one-
half of the waveform cycle (it pushes) and a second
transistor provides power to the load for the other half of the
cycle (it pulls). In this scheme, neither transistor remains
"on" for the entire cycle, giving each one a time to "rest" and
cool during the waveform cycle. This makes for a power-
efficient amplifier circuit, but leads to a distinct type of
nonlinearity Known as "crossover distortion."
Shown here is a sine-wave shape, equivalent to a constant
audio tone of constant volume:
ONS NS NS
In a push-pull amplifier circuit, the two transistors take turns
amplifying the alternate half-cycles of the waveform like
this:
Transistor #1 Transistor #1 Transistor #1
Transistor #2 Transistor #2 Transistor #2
If the "hand-off" between the two transistors is not precisely
synchronized, though, the amplifier's output waveform may
look something like this instead of a pure sine wave:
Transistor #1 Transistor #1 Transistor #1
ae Ae Ae,
Transistor #2 Transistor #2 Transistor #2
Here, distortion results from the fact that there is a delay
between the time one transistor turns off and the other
transistor turns on. This type of distortion, where the
waveform "flattens" at the crossover point between positive
and negative half-cycles, is called crossover distortion. One
common method of mitigating crossover distortion is to bias
the transistors so that their turn-on/turn-off points actually
overlap, so that both transistors are in a state of conduction
for a brief moment during the crossover period:
Transistor #1 Transistor #1 Transistor #1
both oth both th both
Transistor #2 Transistor #2 Transistor #2
This form of amplification is technically known as class AB
rather than class B, because each transistor is "on" for more
than 50% of the time during a complete waveform cycle.
The disadvantage to doing this, though, is increased power
consumption of the amplifier circuit, because during the
moments of time where both transistors are conducting,
there is current conducted through the transistors that is not
going through the load, but is merely being "shorted" from
one power supply rail to the other (from -V to +V). Not only
is this a waste of energy, but it dissipates more heat energy
in the transistors. When transistors increase in temperature,
their characteristics change (V,. forward voltage drop, B,
junction resistances, etc.), making proper biasing difficult.
In this experiment, the transistors operate in pure class B
mode. That is, they are never conducting at the same time.
This saves energy and decreases heat dissipation, but lends
itself to crossover distortion. The solution taken in this
circuit is to use an op-amp with negative feedback to quickly
drive the transistors through the "dead" zone producing
crossover distortion and reduce the amount of "flattening" of
the waveform during crossover.
The first (leftmost) op-amp shown in the schematic diagram
is nothing more than a buffer. A buffer helps to reduce the
loading of the input capacitor/resistor network, which has
been placed in the circuit to filter out any DC bias voltage
out of the input signal, preventing any DC voltage from
becoming amplified by the circuit and sent to the speaker
where it might cause damage. Without the buffer op-amp,
the capacitor/resistor filtering circuit reduces the low-
frequency ("bass") response of the amplifier, and
accentuates the high-frequency ("treble").
The second op-amp functions as an inverting amplifier
whose gain is controlled by the 10 kQ potentiometer. This
does nothing more than provide a volume control for the
amplifier. Usually, inverting op-amp circuits have their
feedback resistor(s) connected directly from the op-amp
output terminal to the inverting input terminal like this:
Input
Output
-V
If we were to use the resulting output signal to drive the
base terminals of the push-pull transistor pair, though, we
would experience significant crossover distortion, because
there would be a "dead" zone in the transistors' operation as
the base voltage went from + 0.7 volts to - 0.7 volts:
Top transistor doesn’t turn
on until V,, exceeds +0.7 volts
Bottom transistor doesn’t turn
on until V,,. drops below -0.7 volts
If you have already constructed the amplifier circuit in its
final form, you may simplify it to this form and listen to the
difference in sound quality. If you have not yet begun
construction of the circuit, the schematic diagram shown
above would be a good starting point. It will amplify an
audio signal, but it will sound horrible!
The reason for the crossover distortion is that when the op-
amp output signal is between + 0.7 volts and - 0.7 volts,
neither transistor will be conducting, and the output voltage
to the speaker will be O volts for the entire 1.4 volts span of
base voltage swing. Thus, there is a "zone" in the input
signal range where no change in speaker output voltage will
occur. Here is where intricate biasing techniques are usually
introduced to the circuit to reduce this 1.4 volt "gap" in
transistor input signal response. Usually, something like this
is done:
+V
60
: speaker
Input —
signal os
-V
The two series-connected diodes will drop approximately 1.4
volts, equivalent to the combined V,,.. forward voltage drops
of the two transistors, resulting in a scenario where each
transistor is just on the verge of turning on when the input
signal is zero volts, eliminating the 1.4 volt "dead" signal
zone that existed before.
Unfortunately, though, this solution is not perfect: as the
transistors heat up from conducting power to the load, their
Vpbe forward voltage drops will decrease from 0.7 volts to
something less, such as 0.6 volts or 0.5 volts. The diodes,
which are not subject to the same heating effect because
they do not conduct any substantial current, will not
experience the same change in forward voltage drop. Thus,
the diodes will continue to provide the same 1.4 volt bias
voltage even though the transistors require less bias voltage
due to heating. The result will be that the circuit drifts into
class AB operation, where both transistors will be in a state
of conduction part of the time. This, of course, will result in
more heat dissipation through the transistors, exacerbating
the problem of forward voltage drop change.
A common solution to this problem is the insertion of
temperature-compensation "feedback" resistors in the
emitter legs of the push-pull transistor circuit:
+V
8 QO
speaker
Input
signal
-V
This solution doesn't prevent simultaneous turn-on of the
two transistors, but merely reduces the severity of the
problem and prevents thermal runaway. It also has the
unfortunate effect of inserting resistance in the load current
path, limiting the output current of the amplifier. The
solution | opted for in this experiment is one that capitalizes
on the principle of op-amp negative feedback to overcome
the inherent limitations of the push-pull transistor output
circuit. | use one diode to provide a 0.7 volt bias voltage for
the push-pull pair. This is not enough to eliminate the
"dead" signal zone, but it reduces it by at least 50%:
+V
Since the voltage drop of a single diode will always be less
than the combined voltage drops of the two transistors’
base-emitter junctions, the transistors can never turn on
simultaneously, thereby preventing class AB operation.
Next, to help get rid of the remaining crossover distortion,
the feedback signal of the op-amp is taken from the output
terminal of the amplifier (the transistors’ emitter terminals)
like this:
Audio
signal
The op-amp's function is to output whatever voltage signal
it has to in order to keep its two input terminals at the same
voltage (0 volts differential). By connecting the feedback
wire to the emitter terminals of the push-pull transistors, the
op-amp has the ability to sense any "dead" zone where
neither transistor is conducting, and output an appropriate
voltage signal to the bases of the transistors to quickly drive
them into conduction again to "keep up" with the input
signal waveform. This requires an op-amp with a high s/ew
rate (the ability to produce a fast-rising or fast-falling output
voltage), which is why the TLO82 op-amp was specified for
this circuit. Slower op-amps such as the LM741 or LM1458
may not be able to keep up with the high dv/dt (voltage
rate-of-change over time, also Known as de/at) necessary for
low-distortion operation.
Only a couple of capacitors are added to this circuit to bring
it into its final form: a 47 UF capacitor connected in parallel
with the diode helps to keep the 0.7 volt bias voltage
constant despite large voltage swings in the op-amp's
output, while a 0.22 UF capacitor connected between the
base and emitter of the NPN transistor helps reduce
crossover distortion at low volume settings:
TIP41
or
TIP3055
80
speaker
Power
supply
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
— 4 —>
—/ | 4]
Lessons In Electric Circuits
-- Volume VI
Chapter 7
DIGITAL INTEGRATED
CIRCUITS
Introduction
Basic gate function
NOR gate S-R latch
NAND gate S-R enabled latch
NAND gate S-R flip-flop
LED sequencer
Simple combination lock
3-bit binary counter
7-segment display
Introduction
Digital circuits are circuits dealing with signals restricted to
the extreme limits of zero and some full amount. This stands
in contrast to analog circuits, in which signals are free to
vary continuously between the limits imposed by power
supply voltage and circuit resistances. These circuits find
use in "true/false" logical operations and digital
computation.
The circuits in this chapter make use of /C, or integrated
circuit, components. Such components are actually networks
of interconnected components manufactured on a single
wafer of semiconducting material. Integrated circuits
providing a multitude of pre-engineered functions are
available at very low cost, benefitting students, hobbyists
and professional circuit designers alike. Most integrated
circuits provide the same functionality as "discrete"
semiconductor circuits at higher levels of reliability and ata
fraction of the cost.
Circuits in this chapter will primarily use CMOS technology,
as this form of IC design allows for a broad range of power
supply voltage while maintaining generally low power
consumption levels. Though CMOS circuitry is susceptible to
damage from static electricity (high voltages will puncture
the insulating barriers in the MOSFET transistors), modern
CMOS ICs are far more tolerant of electrostatic discharge
than the CMOS ICs of the past, reducing the risk of chip
failure by mishandling. Proper handling of CMOS involves
the use of anti-static foam for storage and transport of IC's,
and measures to prevent static charge from building up on
your body (use of a grounding wrist strap, or frequently
touching a grounded object).
Circuits using 77L technology require a regulated power
supply voltage of 5 volts, and will not tolerate any
substantial deviation from this voltage level. Any TTL
circuits in this chapter will be adequately labeled as such,
and it will be expected that you realize its unique power
supply requirements.
When building digital circuits using integrated circuit
"chips," it is highly recommended that you use a breadboard
with power supply "rail" connections along the length. These
are sets of holes in the breadboard that are electrically
common along the entire length of the board. Connect one
to the positive terminal of a battery, and the other to the
negative terminal, and DC power will be available to any
area of the breadboard via connection through short jumper
wires:
These points electrically common
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With so many of these integrated circuits having "reset,"
"enable," and "disable" terminals needing to be maintained
ina "high" or "low" state, not to mention the Vpp (or V¢c)
and ground power terminals which require connection to the
power supply, having both terminals of the power supply
readily available for connection at any point along the
board's length is very useful.
Most breadboards that | have seen have these power supply
"rail" holes, but some do not. Up until this point, I've been
illustrating circuits using a breadboard lacking this feature,
just to show how it isn't absolutely necessary. However,
digital circuits seem to require more connections to the
power supply than other types of breadboard circuits,
making this feature more than just a convenience.
Basic gate function
PARTS AND MATERIALS
e 4011 quad NAND gate (Radio Shack catalog # 276-
2411)
e Eight-position DIP switch (Radio Shack catalog # 275-
1301)
Ten-segment bargraph LED (Radio Shack catalog # 276-
081)
One 6 volt battery
Two 10 kQ resistors
Three 470 Q resistors
Caution! The 4011 IC is CMOS, and therefore sensitive to
static electricity!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 3: "Logic
Gates"
LEARNING OBJECTIVES
e Purpose of a "pulldown" resistor
e How to experimentally determine the truth table of a
gate
e How to connect logic gates together
How to create different logical functions by using NAND
gates
SCHEMATIC DIAGRAM
ILLUSTRATION
INSTRUCTIONS
To begin, connect a single NAND gate to two input switches
and one LED, as shown. At first, the use of an 8-position
switch and a 10-segment LED bargraph may seem
excessive, since only two switches and one LED are needed
to show the operation of a single NAND gate. However, the
presence of those extra switches and LEDs make it very
convenient to expand the circuit, and help make the circuit
layout both clean and compact.
It is highly recommended that you have a datasheet for the
4011 chip available when you build your circuit. Don't just
follow the illustration shown above! It is important that you
develop the skill of reading datasheets, especially "pinout"
diagrams, when connecting IC terminals to other circuit
elements. The datasheet's connection diagram is an
essential piece of information to have. Shown here is my
own rendition of what any 4011 datasheet shows:
"Pinout," or "connection" diagram for
the 4011 quad NAND gate
In the breadboard illustration, I've shown the circuit built
using the lower-left NAND gate: pin #'s 1 and 2 are the
inputs, and pin #3 is the output. Pin #'s 14 and 7 conduct
DC power to all four gate circuits inside the IC chip, "Vpp"
representing the positive side of the power supply (+V), and
"Gnd" representing the negative side of the power supply (-
V), or ground. Sometimes the negative power supply
terminal will be labeled "Vo." instead of "Gnd" ona
datasheet, but it means the same thing.
Digital logic circuitry does not make use of split power
supplies as op-amps do. Like op-amp circuits, though,
ground is still the implicit point of reference for all voltage
measurements. If | were to speak of a "high" signal being
present on a certain pin of the chip, | would mean that there
was full voltage between that pin and the negative side of
the power supply (ground).
Note how all inputs of the unused gates inside the 4011 chip
are connected either to Vpp or ground. This is not a mistake,
but an act of intentional design. Since the 4011 is a CMOS
integrated circuit, and CMOS circuit inputs left unconnected
(floating) can assume any voltage level merely from
intercepting a static electric charge from a nearby object,
leaving inputs floating means that those unused gates may
receive any random combinations of "high" and "low"
Signals.
Why is this undesirable, if we aren't using those gates? Who
cares what signals they receive, if we are not doing anything
with their outputs? The problem is, if static voltage signals
appear at the gate inputs that are not fully "high" or fully
"low," the gates' internal transistors may begin to turn on in
such a way as to draw excessive current. At worst, this could
lead to damage of the chip. At best it means excessive
power consumption. It matters little if we choose to connect
these unused gate inputs "high" (Vpp) or "low" (ground), so
long as we connect them to one of those two places. In the
breadboard illustration, | show all the top inputs connected
to Vpp, and all the bottom inputs (of the unused gates)
connected to ground. This was done merely because those
power supply rail holes were closer and did not require long
jumper wires!
Please note that none of the unused gate outputs have been
connected to Vpp or ground, and for good reason! If | were to
do that, | may be forcing a gate to assume the opposite
output state that its trying to achieve, which isa
complicated way of saying that | would have created a short-
circuit. Imagine a gate that is supposed to output a "high"
logic level (fora NAND gate, this would be true if any of its
inputs were "low"). If such a gate were to have its output
terminal directly connected to ground, it could never reach a
"high" state (being made electrically common to ground
through the jumper wire connection). Instead, its upper (P-
channel) output transistor would be turned on in vain,
sourcing maximum current to a nonexistent load. This would
very likely damage the gate! Gate output terminals, by their
very nature, generate their own logic levels and never
"float" in the same way that CMOS gate inputs do.
The two 10 kO resistors are placed in the circuit to avoid
floating input conditions on the used gate. With a switch
closed, the respective input will be directly connected to
Vpp and therefore be "high." With a switch open, the 10 kQ
"pulldown" resistor provides a resistive connection to
ground, ensuring a secure "low" state at the gate's input
terminal. This way, the input will not be susceptible to stray
static voltages.
With the NAND gate connected to the two switches and one
LED as shown, you are ready to develop a "truth table" for
the NAND gate. Even if you already know what a NAND gate
truth table looks like, this is a good exercise in
experimentation: discovering a circuit's behavioral
principles by induction. Draw a truth table on a piece of
paper like this:
TAT [ Ouiput |
ofol
ef
The "A" and "B" columns represent the two input switches,
respectively. When the switch is on, its state is "high" or 1.
When the switch is off, its state is "low," or 0, as ensured by
its pulldown resistor. The gate's output, of course, is
represented by the LED: whether it is lit (1) or unlit (0). After
placing the switches in every possible combination of states
and recording the LED's status, compare the resulting truth
table with what a NAND gate's truth table should be.
As you can imagine, this breadboard circuit is not limited to
testing NAND gates. Any gate type may be tested with two
switches, two pulldown resistors, and an LED to indicate
output status. Just be sure to double-check the chip's
"pinout" diagram before substituting it pin-for-pin in place of
the 4011. Not all "quad" gate chips have the same pin
assignments!
An improvement you might want to make to this circuit is to
assign a couple of LEDs to indicate input status, in addition
to the one LED assigned to indicate the output. This makes
operation a little more interesting to observe, and has the
further benefit of indicating if a switch fails to close (or
open) by showing the true input signal to the gate, rather
than forcing you to infer input status from switch position:
NOR gate S-R latch
PARTS AND MATERIALS
4001 quad NOR gate (Radio Shack catalog # 276-2401)
Eight-position DIP switch (Radio Shack catalog # 275-
1301)
Ten-segment bargraph LED (Radio Shack catalog # 276-
081)
One 6 volt battery
Two 10 kQ resistors
Two 470 Q resistors
Two 100 Q resistors
Caution! The 4001 IC is CMOS, and therefore sensitive to
static electricity!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 3: "Logic
Gates"
Lessons In Electric Circuits, Volume 4, chapter 10:
"Multivibrators"
LEARNING OBJECTIVES
e The effects of positive feedback in a digital circuit
e What is meant by the "invalid" state of a latch circuit
e What a race condition is in a digital circuit
e The importance of valid "high" CMOS signal voltage
levels
SCHEMATIC DIAGRAM
ILLUSTRATION
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INSTRUCTIONS
The 4001 integrated circuit is a CMOS quad NOR gate,
identical in input, output, and power supply pin assignments
to the 4011 quad NAND gate. Its "pinout," or "connection,"
diagram is as such:
"Pinout," or "connection" diagram for
the 4001 quad NOR gate
Vpp
When two NOR gates are cross-connected as shown in the
schematic diagram, there will be positive feedback from
output to input. That is, the output signal tends to maintain
the gate in its last output state. Just as in op-amp circuits,
positive feedback creates hysteresis. This tendency for the
circuit to remain in its last output state gives it a sort of
"memory." In fact, there are solid-state computer memory
technologies based on circuitry like this!
If we designate the left switch as the "Set" input and the
right switch as the "Reset," the left LED will be the "Q"
output and the right LED the "Q-not" output. With the Set
input "high" (switch on) and the Reset input "low," Q will go
"high" and Q-not will go "low." This is known as the set state
of the circuit. Making the Reset input "high" and the Set
input "low" reverses the latch circuit's output state: Q "low"
and Q-not "high." This is Known as the reset state of the
circuit. If both inputs are placed into the "low" state, the
circuit's Q and Q-not outputs will remain in their last states,
"remembering" their prior settings. This is known as the
latched state of the circuit.
Because the outputs have been designated "Q" and "Q-not,"
it is implied that their states will always be complementary
(opposite). Thus, if something were to happen that forced
both outputs to the same state, we would be inclined to call
that mode of the circuit "invalid." This is exactly what will
happen if we make both Set and Reset inputs "high:" both Q
and Q-not outputs will be forced to the same "low" logic
state. This is Known as the invalid or i/lega/ state of the
circuit, not because something has gone wrong, but because
the outputs have failed to meet the expectations established
by their labels.
Since the "latched" state is a hysteretic condition whereby
the last output states are "remembered," one might wonder
what will happen if the circuit powers up this way, with no
previous state to hold. To experiment, place both switches in
their off positions, making both Set and Reset inputs low,
then disconnect one of the battery wires from the
breadboard. Then, quickly make and break contact between
that battery wire and its proper connection point on the
breadboard, noting the status of the two LEDs as the circuit
iS powered up again and again:
make and break contact!
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When a latch circuit such as this is powered up into its
"latched" state, the gates race against each other for
control. Given the "low" inputs, both gates try to output
"high" signals. If one of the gates reaches its "high" output
state before the other, that "high" state will be fed back to
the other gate's input to force its output "low," and the race
is won by the faster gate.
Invariably, one gate wins the race, due to internal variations
between gates in the chip, and/or external resistances and
capacitances that act to delay one gate more than the other.
What this usually means is that the circuit tends to power up
in the same mode, over and over again. However, if you are
persistent in your powering/unpowering cycles, you should
see at least a few times where the latch circuit powers up
latched in the opposite state from normal.
Race conditions are generally undesirable in any kind of
system, as they lead to unpredictable operation. They can
be particularly troublesome to locate, as this experiment
shows, because of the unpredictability they create. Imagine
a scenario, for instance, where one of the two NOR gates was
exceptionally slow-acting, due to a defect in the chip. This
handicap would cause the other gate to win the power-up
race every time. In other words, the circuit will be very
predictable on power-up with both inputs "low." However,
suppose that the unusual chip were to be replaced by one
with more evenly matched gates, or by a chip where the
other NOR gate were consistently slower. Normal circuit
behavior is not supposed to change when a component is
replaced, but if race conditions are present, a change of
components may very well do just that.
Due to the inherent race tendency of an S-R latch, one
should not design a circuit with the expectation of a
consistent power-up state, but rather use external means to
"force" the race so that the desired gate always "wins."
An interesting modification to try in this circuit is to replace
one of the 470 Q LED "dropping" resistors with a lower-value
unit, such as 100 Q. The obvious effect of this alteration will
be increased LED brightness, as more current is allowed
through. A not-so-obvious effect will also result, and it is this
effect which holds great learning value. Try replacing one of
the 470 O resistors with a 100 O resistor, and operate the
input signal switches through all four possible setting
combinations, noting the behavior of the circuit.
You should note that the circuit refuses to latch in one of its
states (either Set or Reset), but only in the other state, when
the input switches are both set "low" (the "latch" mode).
Why is this? Take a voltmeter and measure the output
voltage of the gate whose output is "high" when both inputs
are "low." Note this voltage indication, then set the input
switches in such a way that the other state (either Reset or
Set) is forced, and measure the output voltage of the other
gate when its output is "high." Note the difference between
the two gate output voltage levels, one gate loaded by an
LED with a 470 Q resistor, and the other loaded by an LED
with a 100 O resistor. The one loaded down by the "heavier"
load (100 Q resistor) will be much less: so much less that
this voltage will not be interpreted by the other NOR gate's
input as a "high" signal at all as it is fed back! All logic gates
have permissible "high" and "low" input signal voltage
ranges, and if the voltage of a digital signal falls outside this
permissible range, it might not be properly interpreted by
the receiving gate. In a latch circuit such as this, which
depends on a solid "high" signal fed back from the output of
one gate to the input of the other, a "weak" signal will not
be able to maintain the positive feedback necessary to keep
the circuit latched in one of its states.
This is one reason | favor the use of a voltmeter as a logic
"probe" for determining digital signal levels, rather than an
actual logic probe with "high" and "low" lights. A logic probe
may not indicate the presence of a "weak" signal, whereas a
voltmeter definitely will by means of its quantitative
indication. This type of problem, common in circuits where
different "families" of integrated circuits are mixed (TTL and
CMOS, for example), can only be found with test equipment
providing quantitative measurements of signal level.
NAND gate S-R enabled latch
PARTS AND MATERIALS
4011 quad NAND gate (Radio Shack catalog # 276-
2411)
Eight-position DIP switch (Radio Shack catalog # 275-
1301)
Ten-segment bargraph LED (Radio Shack catalog # 276-
081)
One 6 volt battery
Three 10 kQ resistors
Two 470 Q resistors
Caution! The 4011 IC is CMOS, and therefore sensitive to
static electricity!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 3: "Logic
Gates"
Lessons In Electric Circuits, Volume 4, chapter 10:
"Multivibrators"
LEARNING OBJECTIVES
e Principle and function of an enabled latch circuit
SCHEMATIC DIAGRAM
(power connections to gates not
Set Enable Reset shown for simplicity)
ILLUSTRATION
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INSTRUCTIONS
Although this circuit uses NAND gates instead of NOR gates,
its behavior is identical to that of the NOR gate S-R latch (a
"high" Set input drives Q "high," and a "high" Reset input
drives Q-not "high"), except for the presence of a third
input: the Enable. The purpose of the Enable input is to
enable or disable the Set and Reset inputs from having
effect over the circuit's output status. When the Enable
input is "high," the circuit acts just like the NOR gate S-R
latch. When the Enable input is "low," the Set and Reset
inputs are disabled and have no effect whatsoever on the
outputs, leaving the circuit in its latched state.
This kind of latch circuit (also called a gated S-R latch), may
be constructed from two NOR gates and two AND gates, but
the NAND gate design is easier to build since it makes use of
all four gates in a single integrated circuit.
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NAND gate S-R flip-flop
PARTS AND MATERIALS
e 4011 quad NAND gate (Radio Shack catalog # 276-
2411)
4001 quad NOR gate (Radio Shack catalog # 276-2401)
Eight-position DIP switch (Radio Shack catalog # 275-
1301)
e Ten-segment bargraph LED (Radio Shack catalog # 276-
081)
e One 6 volt battery
e Three 10 kQ resistors
e Two 470 OQ resistors
Caution! The 4011 IC is CMOS, and therefore sensitive to
static electricity!
Although the parts list calls for a ten-segment LED unit, the
illustration shows two individual LEDs being used instead.
This is due to lack of room on my breadboard to mount the
switch assembly, two integrated circuits, and the bargraph.
If you have room on your breadboard, feel free to use the
bargraph as called for in the parts list, and as shown in prior
latch circuits.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 3: "Logic
Gates"
Lessons In Electric Circuits, Volume 4, chapter 10:
"Multivibrators"
LEARNING OBJECTIVES
e The difference between a gated latch and a flip-flop
e How to build a "pulse detector" circuit
e Learn the effects of switch contact "bounce" on digital
circuits
SCHEMATIC DIAGRAM
(power connections to gates not
Set Clock Reset shown for simplicity)
ILLUSTRATION
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INSTRUCTIONS
The only difference between a gated (or enabled) latch and
a flip-flop is that a flip-flop is enabled only on the rising or
falling edge of a "clock" signal, rather than for the entire
duration of a "high" enable signal. Converting an enabled
latch into a flip-flop simply requires that a "pulse detector"
circuit be added to the Enable input, so that the edge of a
clock pulse generates a brief "high" Enable pulse:
Delayed input
Input
Delayedinput™ __|___ == s=f
Output
pe! fag —
Brief period of time when
both inputs of the NOR gate
are low
The single NOR gate and three inverter gates create this
effect by exploiting the propagation delay time of multiple,
cascaded gates. In this experiment, | use three NOR gates
with paralleled inputs to create three inverters, thus using
all four NOR gates of a 4001 integrated circuit:
Pulse detector circuit
Output
Normally, when using a NOR gate as an inverter, one input
would be grounded while the other acts as the inverter
input, to minimize input capacitance and increase speed.
Here, however, slow response is desired, and so | parallel the
NOR inputs to make inverters rather than use the more
conventional method.
Please note that this particular pulse detector circuit
produces a "high" output pulse at every falling edge of the
clock (input) signal. This means that the flip-flop circuit
should be responsive to the Set and Reset input states only
when the middle switch is moved from "on" to "off," not from
"off" to "on."
When you build this circuit, though, you may discover that
the outputs respond to Set and Reset input signals during
both transitions of the Clock input, not just when it is
switched from a "high" state to a "low" state. The reason for
this is contact bounce: the effect of a mechanical switch
rapidly making-and-breaking when its contacts are first
closed, due to the elastic collision of the metal contact pads.
Instead of the Clock switch producing a single, clean low-to-
high signal transition when closed, there will most likely be
several low-high-low "cycles" as the contact pads "bounce"
upon off-to-on actuation. The first high-to-low transition
caused by bouncing will trigger the pulse detector circuit,
enabling the S-R latch for that moment in time, making it
responsive to the Set and Reset inputs.
Ideally, of course, switches are perfect and bounce-free. In
the real world, though, contact bounce is a very common
problem for digital gate circuits operated by switch inputs,
and must be understood well if it is to be overcome.
LED sequencer
PARTS AND MATERIALS
4017 decade counter/divider (Radio Shack catalog #
276-2417)
555 timer IC (Radio Shack catalog # 276-1723)
Ten-segment bargraph LED (Radio Shack catalog # 276-
081)
One SPST switch
One 6 volt battery
10 kQ resistor
1 MQ resistor
0.1 uF capacitor (Radio Shack catalog # 272-135 or
equivalent)
Coupling capacitor, 0.047 to 0.001 UF
Ten 470 Q resistors
Audio detector with headphones
Caution! The 4017 IC is CMOS, and therefore sensitive to
static electricity!
Any single-pole, single-throw switch is adequate. A
household light switch will work fine, and is readily available
at any hardware store.
The audio detector will be used to assess signal frequency. If
you have access to an oscilloscope, the audio detector is
unnecessary.
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 3: "Logic
Gates"
Lessons In Electric Circuits, Volume 4, chapter 4: "Switches"
Lessons In Electric Circuits, Volume 4, chapter 11:
"Counters"
LEARNING OBJECTIVES
e Use of a 555 timer circuit to produce "clock" pulses
(astab/e multivibrator)
e Use of a 4017 decade counter/divider circuit to produce
a sequence of pulses
e Use of a 4017 decade counter/divider circuit for
frequency division
e Using a frequency divider and timepiece (watch) to
measure frequency
Purpose of a "pulldown" resistor
Learn the effects of switch contact "bounce" on digital
circuits
e Use of a 555 timer circuit to "debounce" a mechanical
switch (monostable multivibrator)
SCHEMATIC DIAGRAM
" Clk ClkEn Rst Carry
alll Von -4017 Gnd
Out 4
Ten-segment
LED bargraph
470 Q each
ILLUSTRATION
INSTRUCTIONS
The model 4017 integrated circuit is a CMOS counter with
ten output terminals. One of these ten terminals will be ina
"high" state at any given time, with all others being "low,"
giving a "one-of-ten" output sequence. If low-to-high voltage
pulses are applied to the "clock" (Clk) terminal of the 4017,
it will increment its count, forcing the next output into a
"high" state.
With a 555 timer connected as an astable multivibrator
(oscillator) of low frequency, the 4017 will cycle through its
ten-count sequence, lighting up each LED, one at a time,
and "recycling" back to the first LED. The result is a visually
pleasing sequence of flashing lights. Feel free to experiment
with resistor and capacitor values on the 555 timer to create
different flash rates.
Try disconnecting the jumper wire leading from the 4017's
"Clock" terminal (pin #14) to the 555's "Output" terminal
(pin #3) where it connects to the 555 timer chip, and hold
its end in your hand. If there is sufficient 60 Hz power-line
“noise” around you, the 4017 will detect it as a fast clock
signal, causing the LEDs to blink very rapidly.
Two terminals on the 4017 chip, "Reset" and "Clock Enable,"
are maintained in a "low" state by means of a connection to
the negative side of the battery (ground). This is necessary if
the chip is to count freely. If the "Reset" terminal is made
"high," the 4017's output will be reset back to 0 (pin #3
"high," all other output pins "low"). If the "Clock Enable" is
made "high," the chip will stop responding to the clock
signal and pause in its counting sequence.
If the 4017's "Reset" terminal is connected to one of its ten
output terminals, its counting sequence will be cut short, or
truncated. You may experiment with this by disconnecting
the "Reset" terminal from ground, then connecting a long
jumper wire to the "Reset" terminal for easy connection to
the outputs at the ten-segment LED bargraph. Notice how
many (or how few) LEDs light up with the "Reset" connected
to any one of the outputs:
touch end of long jumper wire
to an LED terminal
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ocoocooco coo eoododcdsdcs0 06
Counters such as the 4017 may be used as digital frequency
dividers, to take a clock signal and produce a pulse
occurring at some integer factor of the clock frequency. For
example, if the clock signal from the 555 timer is 200 Hz,
and the 4017 is configured for a full-count sequence (the
"Reset" terminal connected to ground, giving a full, ten-step
count), a signal with a period ten times as long (20 Hz) will
be present at any of the 4017's output terminals. In other
words, each output terminal will cycle once for every ten
cycles of the clock signal: a frequency ten times as slow.
To experiment with this principle, connect your audio
detector between output O (pin #3) of the 4017 and ground,
through a very small capacitor (0.047 UF to 0.001 UF). The
capacitor is used for "coupling" AC signals only, to that you
may audibly detect pulses without placing a DC (resistive)
load on the counter chip output. With the 4017 "Reset"
terminal grounded, you will have a full-count sequence, and
you will hear a "click" in the headphones every time the "0"
LED lights up, corresponding to 1/10 of the 555's actual
output frequency:
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headphones
In fact, knowing this mathematical relationship between
clicks heard in the headphone and the clock frequency
allows us to measure the clock frequency to a fair degree of
precision. Using a stopwatch or other timepiece, count the
number of clicks heard in one full minute while connected to
the 4017's "0" output. Using a 1 MQ resistor and 0.1 UF
Capacitor in the 555 timing circuit, and a power supply
voltage of 13 volts (instead of 6), | counted 79 clicks in one
minute from my circuit. Your circuit may produce slightly
different results. Multiply the number of pulses counted at
the "0" output by 10 to obtain the number of cycles
produced by the 555 timer during that same time (my
circuit: 79 x 10 = 790 cycles). Divide this number by 60 to
obtain the number of timer cycles elapsed in each second
(my circuit: 790/60 = 13.17). This final figure is the clock
frequency in Hz.
Now, leaving one test probe of the audio detector connected
to ground, take the other test probe (the one with the
coupling capacitor connected in series) and connect it to pin
#3 of the 555 timer. The buzzing you hear is the undivided
clock frequency:
headphones
By connecting the 4017's "Reset" terminal to one of the
output terminals, a truncated sequence will result. If we are
using the 4017 as a frequency divider, this means the
output frequency will be a different factor of the clock
frequency: 1/9, 1/8, 1/7, 1/6, 1/5, 1/4, 1/3, or 1/2, depending
on which output terminal we connect the "Reset" jumper
wire to. Re-connect the audio detector test probe to output
"0" of the 4017 (pin #3), and connect the "Reset" terminal
jumper to the sixth LED from the left on the bargraph. This
should produce a 1/5 frequency division ratio:
—s«ofeecene
4017 output frequency is
1/5 of input (clock) frequency
Counting the number of clicks heard in one minute again,
you should obtain a number approximately twice as large as
what was counted with the 4017 configured for a 1/10 ratio,
because 1/5 is twice as large a ratio as 1/10. If you do not
obtain a count that is exactly twice what you obtained
before, it is because of error inherent to the method of
counting cycles: coordinating your sense of hearing with the
display of a stopwatch or other time-keeping device.
Try replacing the 1 MQ timing resistor in the 555 circuit with
one of greatly lesser value, such as 10 kQ. This will increase
the clock frequency driving the 4017 chip. Use the audio
detector to listen to the divided frequency at pin #3 of the
4017, noting the different tones produced as you move the
"Reset" jumper wire to different outputs, creating different
frequency division ratios. See if you can produce octaves by
dividing the original frequency by 2, then by 4, and then by
8 (each descending octave represents one-half the previous
frequency). Octaves are readily distinguished from other
divided frequencies by their similar pitches to the original
tone.
A final lesson that may be learned from this circuit is that of
switch contact "bounce." For this, you will need a switch to
provide clock signals to the 4017 chip, instead of the 555
timer. Re-connect the "Reset" jumper wire to ground to
enable a full ten-step count sequence, and disconnect the
555's output from the 4017's "Clock" input terminal.
Connect a switch in series with a 10 kQ pul/down resistor,
and connect this assembly to the 4017 "Clock" input as
shown:
] Clk ClkEn Rst 7
V 4017 Gnd
DD
Ten-segment q
LED bargraph
470 9 each
The purpose of a "pulldown" resistor is to provide a definite
"low" logic state when the switch contact opens. Without
this resistor in place, the 4017's "Clock" input wire would be
floating whenever the switch contact was opened, leaving it
susceptible to interference from stray static voltages or
electrical "noise," either one capable of making the 4017
count randomly. With the pulldown resistor in place, the
4017's "Clock" input will have a definite, albeit resistive,
connection to ground, providing a stable "low" logic state
that precludes any interference from static electricity or
"noise" coupled from nearby AC circuit wiring.
Actuate the switch on and off, noting the action of the LEDs.
With each off-to-on switch transition, the 4017 should
increment once in its count. However, you may notice some
strange behavior: sometimes, the LED sequence will "skip"
one or even several steps with a single switch closure. Why
is this? It is due to very rapid, mechanical "bouncing" of the
switch contacts. When two metallic contacts are brought
together rapidly as does happen inside most switches, there
will be an elastic collision. This collision results in the
contacts making and breaking very rapidly as they "bounce"
off one another. Normally, this "bouncing" is much to rapid
for you to see its effects, but in a digital circuit such as this
where the counter chip is able to respond to very quick clock
pulses, these "bounces" are interpreted as distinct clock
signals, and the count incremented accordingly.
One way to combat this problem is to use a timing circuit to
produce a single pulse for any number of input pulse signals
received within a short amount of time. The circuit is called
a monostable multivibrator, and any technique eliminating
the false pulses caused by switch contact "bounce" is called
debouncing.
The 555 timer circuit is capable of functioning as a
debouncer, if the "Trigger" input is connected to the switch
as such:
Using the 555 timer to “"debounce” the switch
.[ |,
Clk ClkEn Rst Carry
Vop 4017 Gnd
Disch
Thresh
Trig
0.1 pF == Gnd
Please note that since we are using the 555 once again to
provide a clock signal to the 4017, we must re-connect pin
#3 of the 555 chip to pin #14 of the 4017 chip! Also, if you
have altered the values of the resistor or capacitor in the
555 timer circuit, you should return to the original 1 MQ and
0.1 WF components.
Actuate the switch again and note the counting behavior of
the 4017. There should be no more "skipped" counts as
there were before, because the 555 timer outputs a single,
crisp pulse for every on-to-off actuation (notice the inversion
of operation here!) of the switch. It is important that the
timing of the 555 circuit be appropriate: the time to charge
the capacitor should be longer than the "settling" period of
the switch (the time required for the contacts to stop
bouncing), but not so long that the timer would "miss" a
rapid sequence of switch actuations, if they were to occur.
Simple combination lock
PARTS AND MATERIALS
4001 quad NOR gate (Radio Shack catalog # 276-2401)
4070 quad XOR gate (Radio Shack catalog # 900-6906)
Two, eight-position DIP switches (Radio Shack catalog #
275-1301)
e Two light-emitting diodes (Radio Shack catalog # 276-
026 or equivalent)
Four 1N914 "switching" diodes (Radio Shack catalog #
276-1122)
Ten 10 kQ resistors
Two 470 Q resistors
Pushbutton switch, normally open (Radio Shack catalog
# 275-1556)
e Two 6 volt batteries
Caution! Both the 4001 and 4070 ICs are CMOS, and
therefore sensitive to static electricity!
This experiment may be built using only one 8-position DIP
switch, but the concept is easier to understand if two switch
assemblies are used. The idea is, one switch acts to hold the
correct code for unlocking the lock, while the other switch
serves as a data entry point for the person trying to open
the lock. In real life, of course, the switch assembly with the
"key" code set on it must be hidden from the sight of the
person opening the lock, which means it must be physically
located e/sewhere from where the data entry switch
assembly is. This requires two switch assemblies. However, if
you understand this concept clearly, you may build a
working circuit with only one 8-position switch, using the left
four switches for data entry and the right four switches to
hold the "key" code.
For extra effect, choose different colors of LED: green for
"Go" and red for "No go."
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 3: "Logic
Gates"
LEARNING OBJECTIVES
e Using XOR gates as bit comparators
e How to build simple gate functions with diodes and a
pullup/down resistor
e Using NOR gates as controlled inverters
SCHEMATIC DIAGRAM
tdq "Keycode"
Vid "salon
10 kQ
Vdd Data entry
switch
10 kQ
ILLUSTRATION
ace pelelZi be Maecenas
INSTRUCTIONS
This circuit illustrates the use of XOR (Exclusive-OR) gates
as bit comparators. Four of these XOR gates compare the
respective bits of two 4-bit binary numbers, each number
"entered" into the circuit via a set of switches. If the two
numbers match, bit for bit, the green "Go" LED will light up
when the "Enter" pushbutton switch is pressed. If the two
numbers do not exactly match, the red "No go" LED will light
up when the "Enter" pushbutton is pressed.
Because four bits provides a mere sixteen possible
combinations, this lock circuit is not very sophisticated. If it
were used in a real application such as a home security
system, the "No go" output would have to be connected to
some kind of siren or other alarming device, so that the
entry of an incorrect code would deter an unauthorized
person from attempting another code entry. Otherwise, it
would not take much time to try all combinations (0000
through 1111) until the correct one was found! In this
experiment, | do not describe how to work this circuit into a
real security system or lock mechanism, but only how to
make it recognize a pre-entered code.
The "key" code that must be matched at the data entry
switch array should be hidden from view, of course. If this
were part of a real security system, the data entry switch
assembly would be located outside the door, and the key
code switch assembly behind the door with the rest of the
circuitry. In this experiment, you will likely locate the two
switch assemblies on two different breadboards, but it is
entirely possible to build the circuit using just a single (8-
position) DIP switch assembly. Again, the purpose of the
experiment is not to make a real security system, but merely
to introduce you to the principle of XOR gate code
comparison.
It is the nature of an XOR gate to output a "high" (1) signal if
the input signals are not the same logic state. The four XOR
gates' output terminals are connected through a diode
network which functions as a four-input OR gate: if any of
the four XOR gates outputs a "high" signal -- indicating that
the entered code and the key code are not identical -- then a
"high" signal will be passed on to the NOR gate logic. If the
two 4-bit codes are identical, then none of the XOR gate
outputs will be "high," and the pull-down resistor connected
to the common sides of the diodes will provide a "low" signal
state to the NOR logic.
The NOR gate logic performs a simple task: prevent either of
the LEDs from turning on if the "Enter" pushbutton is not
pressed. Only when this pushbutton is pressed can either of
the LEDs energize. If the Enter switch is pressed and the
XOR outputs are all "low," the "Go" LED will light up,
indicating that the correct code has been entered. If the
Enter switch is pressed and any of the XOR outputs are
"high," the "No go" LED will light up, indicating that an
incorrect code has been entered. Again, if this were a real
security system, it would be wise to have the "No go" output
do something that deters an unauthorized person from
discovering the correct code by trial-and-error. In other
words, there should be some sort of pena/ty for entering an
incorrect code. Let your imagination guide your design of
this detail!
3-bit binary counter
PARTS AND MATERIALS
555 timer IC (Radio Shack catalog # 276-1723)
e One 1N914 "switching" diode (Radio Shack catalog #
276-1122)
Two 10 kO resistors
One 100 uF capacitor (Radio Shack catalog # 272-1028)
4027 dual J-K flip-flop (Radio Shack catalog # 900-4394)
Ten-segment bargraph LED (Radio Shack catalog # 276-
081)
e Three 470 O resistors
One 6 volt battery
Caution! The 4027 IC is CMOS, and therefore sensitive to
static electricity!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 10:
"Multivibrators"
Lessons In Electric Circuits, Volume 4, chapter 11:
"Counters"
LEARNING OBJECTIVES
e Using the 555 timer as a square-wave oscillator
e How to make an asynchronous counter using J-K flip-
flops
SCHEMATIC DIAGRAM
ILLUSTRATION
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INSTRUCTIONS
In a sense, this circuit "cheats" by using only two J-K flip-
flops to make a three-bit binary counter. Ordinarily, three
flip-flops would be used -- one for each binary bit -- but in
this case we can use the clock pulse (555 timer output) as a
bit of its own. When you build this circuit, you will find that
it is a "down" counter. That is, its count sequence goes from
111 to 110 to 101 to 100 to 011 to 010 to 001 to 000 and
then back to 111. While it is possible to construct an "up"
counter using J-K flip-flops, this would require additional
components and introduce more complexity into the circuit.
The 555 timer operates as a slow, square-wave oscillator
with a duty cycle of approximately 50 percent. This duty
cycle is made possible by the use of a diode to "bypass" the
lower resistor during the capacitor's charging cycle, so that
the charging time constant is only RC and not 2RC as it
would be without the diode in place.
It is highly recommended, in this experiment as in all
experiments, to build the circuit in stages: identify portions
of the circuit with specific functions, and build those
portions one at a time, testing each one and verifying its
performance before building the next. A very common
mistake of new electronics students is to build an entire
circuit at once without testing sections of it during the
construction process, and then be faced with the possibility
of several problems simultaneously when it comes time to
finally apply power to it. Remember that a small amount of
extra attention paid to detail near the beginning of a project
is worth an enormous amount of troubleshooting work near
the end! Students who make the mistake of not testing
circuit portions before attempting to operate the entire
circuit often (falsely) think that the time it would take to test
those sections is not worth it, and then spend days trying to
figure out what the problem(s) might be with their
experiment.
Following this philosophy, build the 555 timer circuit first,
before even plugging the 4027 IC into the breadboard.
Connect the 555's output (pin #3) to the "Least Significant
Bit" (LSB) LED, so that you have visual indication of its
status. Make sure that the output oscillates in a slow,
square-wave pattern (LED is "lit" for about as long as it is
"off" in a cycle), and that it is a reliable signal (no erratic
behavior, no unexplained pauses). If the 555 timer is not
working properly, neither will the rest of the counter circuit!
Once the timer circuit has been proven good, proceed to
plug the 4027 IC into the breadboard and complete the rest
of the necessary connections between it, the 555 timer
circuit, and the LED assembly.
7-segment display
PARTS AND MATERIALS
e 4511 BCD-to-7 seg latch/decoder/driver (Radio Shack
catalog # 900-4437)
e Common-cathode 7-segment LED display (Radio Shack
catalog # 276-075)
e Eight-position DIP switch (Radio Shack catalog # 275-
1301)
Four 10 kQ resistors
Seven 470 QO resistors
One 6 volt battery
Caution! The 4511 IC is CMOS, and therefore sensitive to
static electricity!
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 4, chapter 9:
“Combinational Logic Functions"
LEARNING OBJECTIVES
e How to use the 4511 7-segment decoder/display driver
IC
Gain familiarity with the BCD code
How to use 7-Ssegment LED assemblies to create decimal
digit displays
e How to identify and use both "active-low" and "active-
high" logic inputs
SCHEMATIC DIAGRAM
ILLUSTRATION
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INSTRUCTIONS
This experiment is more of an introduction to the 4511
decoder/display driver IC than it is a lesson in how to "build
up" a digital function from lower-level components. Since 7 -
segment displays are very common components of digital
devices, it is good to be familiar with the "driving" circuits
behind them, and the 4511 is a good example of a typical
driver IC.
Its operating principle is to input a four-bit BCD (Binary-
Coded Decimal) value, and energize the proper output lines
to form the corresponding decimal digit on the 7-segment
LED display. The BCD inputs are designated A, B, C, and Din
order from least-significant to most-significant. Outputs are
labeled a, b, c, d, e, f, and g, each letter corresponding toa
standardized segment designation for 7-segment displays.
Of course, since each LED segment requires its own
dropping resistor, we must use seven 470 OQ resistors placed
in series between the 4511's output terminals and the
corresponding terminals of the display unit.
Most 7-segment displays also provide for a decimal point
(sometimes two!), a separate LED and terminal designated
for its operation. All LEDs inside the display unit are made
common to each other on one side, either cathode or anode.
The 4511 display driver IC requires a common-cathode 7-
segment display unit, and so that is what is used here.
After building the circuit and applying power, operate the
four switches in a binary counting sequence (0000 to 1111),
noting the 7-segment display. A 0000 input should result in
a decimal "0" display, a 0001 input should result ina
decimal "1" display, and so on through 1001 (decimal "9").
What happens for the binary numbers 1010 (10) through
1111 (15)? Read the datasheet on the 4511 IC and see what
the manufacturer specifies for operation above an input
value of 9. In the BCD code, there is no real meaning for
1010, 1011, 1100, 1101, 1110, or 1111. These are binary
values beyond the range of a single decimal digit, and so
have no function in a BCD system. The 4511 IC is built to
recognize this, and output (or not output!) accordingly.
Three inputs on the 4511 chip have been permanently
connected to either Vyg or ground: the "Lamp Test,"
"Blanking Input," and "Latch Enable." To learn what these
inputs do, remove the short jumpers connecting them to
either power supply rail (one at a time!), and replace the
short jumper with a longer one that can reach the other
power supply rail. For example, remove the short jumper
connecting the "Latch Enable" input (pin #5) to ground, and
replace it with a long jumper wire that can reach all the way
to the Vyg power supply rail. Experiment with making this
input "high" and "low," observing the results on the 7 -
segment display as you alter the BCD code with the four
input switches. After you've learned what the input's
function is, connect it to the power supply rail enabling
normal operation, and proceed to experiment with the next
input (either "Lamp Test" or "Blanking Input").
Once again, the manufacturer's datasheet will be
informative as to the purpose of each of these three inputs.
Note that the "Lamp Test" (LT) and "Blanking Input" (BI)
input labels are written with boolean complementation bars
over the abbreviations. Bar symbols designate these inputs
as active-low, meaning that you must make each one "low"
in order to invoke its particular function. Making an active-
low input "high" places that particular input into a "passive"
state where its function will not be invoked. Conversely, the
"Latch Enable" (LE) input has no complementation bar
written over its abbreviation, and correspondingly it is
shown connected to ground ("low") in the schematic so as to
not invoke that function. The "Latch Enable" input is an
active-high input, which means it must be made "high"
(connected to Vggq) in order to invoke its function.
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
|| 4]\l\—
—| | +4/l—
Lessons In Electric Circuits
-- Volume VI
Chapter 8
555 TIMER CIRCUITS
The 555 IC
555 Schmitt Trigger
555 HYSTERETIC OSCILLATOR
555 MONOSTABLE MULTIVIBRATOR
CMOS 555 LONG DURATION MINIMUM PARTS RED LED
FLASHER
CMOS 555 LONG DURATION BLUE LED FLASHER
CMOS 555 LONG DURATION FLYBACK LED FLASHER
HOW TO MAKE AN INDUCTOR
CMOS 555 LONG DURATION RED LED FLASHER
Original author: Bill Marsden
The 555 IC
The 555 integrated circuit is the most popular chip ever
manufactured. Independently manufactured by more than 10
manufacturers, still in current production, and almost 40
years old, this little circuit has withstood the test of time. It
has been redesigned, improved, and reconfigured in many
ways, yet the original design can be bought from many
vendors. The design of this chip was right the first time.
Originally conceived in 1970 and created by Hans R.
Camenzind in 1971, over 1 billion of these ICs were made in
2003 with no apparent reduction in demand. It has been
used in everything from toys to spacecraft. Due to its
versatility, availability, and low cost it remains a hobbyist
favorite.
One of the secrets to its success Is it is a true black box, its
symbolized schematic is simple and accurate enough that
designs using this simplification as a reference tend to work
first time. You don't need to understand every transistor in
the base schematic to make it work.
It has been used to derive the 556, a dual 555, each
independent of the other in one 14 pin package, and is the
inspiration of the 558, a quad timer in a 16 pin package.
What few weak points the original design has have been
addressed by redesigns into CMOS technology, with its
dramatically reduced current and expanded voltage
requirements, and yet the original version remains.
Originally conceived as a simple timer, the 555 has been
used for oscillators, waveform generators, VCO's, FM
discrimination, and a lot more. It really is an all purpose
circuit.
SOURCES
e The 555 Timer IC - An Interview with Hans Camenzind (
http://semiconductormuseum.com/Transistors/LectureHal
l/Camenzind/Camenzind Index.htm )
555 Tutorial (
http://www.sentex.ca/~mec1995/gadgets/555/555.html )
e 555 Timer IC Encyclopedia Article (
http://www.nationmaster.com/encyclopedia/555-timer-IC
)
555 Schmitt Trigger
PARTS AND MATERIALS
One 9V Battery
Battery Clip (Radio Shack catalog # 270-325)
Mini Hook Clips (soldered to Battery Clip, Radio Shack
catalog # 270-372)
One Potentiometer, 10 KQ, 15-Turn (Radio Shack catalog
# 271-343)
One 555 timer IC (Radio Shack catalog # 276-1723)
One red light-emitting diode (Radio Shack catalog #
276-041 or equivalent)
e One green light-emitting diode (Radio Shack catalog #
276-022 or equivalent)
Two 1 KQ Resistors
One DVM (Digital Volt Meter) or VOM (Volt Ohm Meter)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 3, chapter 8: “Positive
Feedback”
Lessons In Electric Circuits, Volume 4, chapter 3: “Logic
Signal Voltage Levels”
LEARNING OBJECTIVES
e Learn how a Schmitt Trigger works
e How to use the 555 timer as an Schmitt Trigger
SCHEMATIC DIAGRAM
Schmitt Triggers have a convention to show a gate that is
also a Schmitt Trigger, shown below.
The same schematic redrawn to reflect this convention looks
something like this:
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oO
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ILLUSTRATION
|
INSTRUCTIONS
The 555 timer is probably one of the more versatile "black
box" chips. Its 3 resistor voltage divider, 2 comparators, and
built in set reset flip flop are wired to form a Schmitt Trigger
in this design. It is interesting to note that the configuration
isn't even close to the op amp configuration shown
elsewhere, but the end result is identical.
Try adjusting the potentiometer until the lights flip states,
then measure the voltage. Compare this voltage to the power
supply voltage. Adjust the potentiometer the other way until
the LED's flip states again, and measure the voltage. How
close to the 1/3 and 2/3 marks did you get?
Try substituting the 9V battery with a 6 volt battery, or two 6
volt batteries, and see how close the thresholds are to the
1/3 and 2/3 marks.
Schmitt Triggers are a fundamental circuit with several uses.
One is signal processing, they can pull digital data out of
some extremely noisy environments. Other big uses will be
shown in following projects, such as an extremely simple RC
oscillator.
THEORY OF OPERATION
The defining characteristic of any Schmitt Trigger is its
hysteresis. In this case it is 1/3 and 2/3 of the power supply
voltage, defined by the built in resistor voltage divider on the
555. The built in comparators C1 and C2 compare the input
voltage to the references provided by the voltage divider and
use the comparison to trip the built in flip flop, which drives
the output driver, another nice feature of the 555. The 555
can drive up to 200ma off either side of the power supply
rail, the output driver creates a very low conduction path to
either side of the power supply connections. The circuit
"shorts" each side of the LED circuit, leaving the other side to
light up.
The 5KQ resistors are not very accurate. It is interesting to
note that IC fabrication doesn't generally allow precision
resistors, but the resistors compared to each other are
extremely close in value, which is critical to the circuit's
operation.
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555 Functional Schematic £Gnd
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Control Voltage—
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555 HYSTERETIC OSCILLATOR
PARTS AND MATERIALS
e One 9V Battery
Battery Clip (Radio Shack catalog # 270-325)
Mini Hook Clips (soldered to Battery Clip, Radio Shack
catalog # 270-372)
Ul - 555 timer IC (Radio Shack catalog # 276-1723)
D1 - Red light-emitting diode (Radio Shack catalog #
276-041 or equivalent)
e D2 - Green light-emitting diode (Radio Shack catalog #
276-022 or equivalent)
R1,R2 - 1 KQ 1/4W Resistors
R3 - 10 © 1/4W Resistor
R4 - 10 KQ, 15-Turn Potentiometer (Radio Shack catalog
# 271-343)
e Cl - 1 uF Capacitor (Radio Shack catalog 272-1434 or
equivalent)
e Cl - 100 uF Capacitor (Radio Shack catalog 272-1028 or
equivalent)
CROSS-REFERENCES
Lessons In Electric CircuitsVolume 1, chapter 16: Voltage and
current calculations
Lessons In Electric Circuits, Volume 1, chapter 16: Solving for
unknown time
Lessons In Electric Circuits, Volume 4, chapter 10:
Multivibrators
Lessons in Electric Circuits, Volume 3, chapter 8: Positive
Feedback
LEARNING OBJECTIVES
e Learn how to use a Schmitt Trigger for a simple RC
Oscillator
e Learn a practical application for a RC time constant
e Learn one of several 555 timer Astable Multivibrator
Configurations
SCHEMATIC DIAGRAM
Here is one way of drawing the schematic:
As mentioned in the previous experiment, there is also
another convention, shown below:
ILLUSTRATION
R3 R4
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INSTRUCTIONS
This is one of the most basic RC oscillators. It is simple and
very predictable. Any inverting Schmitt Trigger will work in
this design, although the frequency will shift somewhat
depending on the hysteresis of the gate.
This circuit has a lower end frequency of 0.7 Hertz, which
means each LED will alternate and be lit for just under a
second each. As you turn the potentiometer
counterclockwise the frequency will increase, going well into
the high end audio range. You can verify this with the Audio
Detector (Vol. VI, Chapter 3, Section 12) or a piezoelectric
Speaker, as you continue to turn the potentiometer the pitch
of the sound will rise. You can increase the frequency 100
times by replacing the capacitor with the 1uF capacitor,
which will also raise the maximum frequency well into the
ultrasonic range, around 7 OKhz.
The 555 does not go rail to rail (it doesn't quite reach the
upper supply voltage) because of its output Darlington
transistors, and this causes the oscillators square wave to be
not quite symmetrical. Can you see this looking at the LEDs?
The higher the power supply voltage, the less pronounced
this asymmetry is, while it gets worse with lower power
supply voltages. If the output were true rail to rail it would be
a 50% square wave, which can be attained if one uses the
CMOS version of the 555, such as the TLC555 (Radio Shack
P/N 276-1718).
R3 was added to prevent shorting the IC output through Cl,
as the capacitor shorts the AC portion of the 555 output to
ground. On a discharged battery it is not noticeable, but with
a fresh 9V the 555 IC will get very hot. If you eliminate the
resistor and adjust R4 for maximum frequency you can test
this, it is not good for the battery or the 555, but they will
survive a short test.
THEORY OF OPERATION
This is a hysteretic oscillator, which is a type of relaxation
oscillator. It is also an astable multivibrator. It is a logical
offshoot of the 555 Schmitt Trigger experiment shown earlier.
The formula to calculate the frequency with this
configuration using a 555 is:
o
f= :
x
‘o
The 555 hysteresis is dependent on the supply voltage, so
the frequency of the oscillator would be relatively
independent of the supply voltage if it weren't for the lack of
rail to rail output.
The output of a 555 either goes to ground, or relatively close
to the plus voltage. This allows the resistor and capacitor to
charge and discharge through the output pin. Since this isa
digital type signal, the LEDs interact very little in its
operation. The first pulse generated by the oscillator is a bit
longer than the rest. This and the charge/discharge curves
are shown in the following illustration, which also shows why
the asymmetrical square wave is created.
Voc
y, =
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Gnd =
Oscillator Input Oscillator Output
(Capacitor Charge Curves) (Based on Input)
555 Oscillator Waveforms
555 MONOSTABLE MULTIVIBRATOR
PARTS AND MATERIALS
e One 9V Battery
e Battery Clip (Radio Shack catalog # 270-325)
Mini Hook Clips (soldered to Battery Clip, Radio Shack
catalog # 270-372)
e A Watch with a second hand/display or a Stop Watch
e Awire, 11/2" to 2" (3.8 mm to 5 mm) long, folded in half
(shown as red wire in illustration)
Ul - 555 timer IC (Radio Shack catalog # 276-1723)
D1 - Red light-emitting diode (Radio Shack catalog #
276-041 or equivalent)
e D2 - Green light-emitting diode (Radio Shack catalog #
276-022 or equivalent)
R1,R2 - 1 KO 1/4W Resistors
Rt - 27 KQ 1/4W Resistor
Rt - 270 KQ 1/4W Resistor
C1,C2 - 0.1 uF Capacitor (Radio Shack catalog 272-1069
or equivalent)
e Ct - 10 uF Capacitor (Radio Shack catalog 272-1025 or
equivalent)
e Ct - 100 uF Capacitor (Radio Shack catalog 272-1028 or
equivalent)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 13: “Electric
fields and capacitance”
Lessons In Electric Circuits, Volume 1, chapter 13:
“Capacitors and calculus”
Lessons In Electric Circuits, Volume 1, chapter 16: “Voltage
and current calculations”
Lessons In Electric Circuits, Volume 1, chapter 16: “Solving
for unknown time”
Lessons In Electric Circuits, Volume 4, chapter 10:
“Monostable multivibrators”
LEARNING OBJECTIVES
e Learn how a Monostable Multivibrator works
e Learn a practical application for a RC time constant
e How to use the 555 timer as a Monostable Multivibrator
SCHEMATIC DIAGRAM
T=1.1 Re Cy * 3 sec
ILLUSTRATION
FOUTS
FGOUT4S
ABCODE
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= ny 7 w =
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—— mo VY
Note Cy polarity! : QV Battery wy
INSTRUCTIONS
This is one of the most basic 555 circuits. This circuit is part
of this chips datasheet, complete with the math needed to
design to specification, and is one of the reasons a 555 is
referred to as a timer. The green LED shown on the
illustration lights when the 555 output is high (i.e., switched
to Vcc), and the red LED lights when the 555 output is low
(switched to ground).
This particular monostable multivibrator (also known as a
monostable or timer) is not a retriggerable type. This means
once triggered it will ignore further inputs during a timing
cycle, with one exception, which will be discussed in the next
paragraph. The timer starts when the input goes low, or
switched to the ground level, and the output goes high. You
can prove this by connecting the red wire shown on the
illustration between ground and point B, disconnecting it,
and reconnecting it.
It is an illegal condition for the input to stay low for this
design past timeout. For this reason R3 and Cl were added to
create a Signal conditioner, which will allow edge only
triggering and prevent the illegal input. You can prove this by
connecting the red wire between ground and point A. The
timer will start when the wire is inserted into the protoboard
between these two points, and ignore further contacts. If you
force the timer input to stay low past timeout the output will
stay high, even though the timer has finished. As soon as this
ground is removed the timer will go low.
Rt and Ct were selected for 3 seconds timing duration. You
can verify this with a watch, 3 seconds is long enough that
we slow humans can actually measure it. Try swapping Rt
and Ct with the 27 KQ resistor and the 100 uF capacitor.
Since the answer to the formula is the same there should be
no difference in how it operates. Next try swapping Rt with
the 270 KO resistor, since the RC time constant is now 10
times greater you should get close to 30 seconds. The
resistor and capacitor are probably 5% and 20% tolerance
respectively, so the calculated times you measure can vary
as much as 25%, though it will usually be much closer.
Another nice feature of the 555 is its immunity from the
power supply voltage. If you were to swap the 9V battery
with a 6V or 12 battery you should get identical results,
though the LED light intensity will change.
C2 isn't actually necessary. The 555 IC has this option in case
the timer is being used in an environment where the power
supply line is noisy. You can remove it and not notice a
difference. The 555 itself is a source of noise, since there is a
very brief period of time that the transistors on both sides of
the output are both conducting, creating a power surge
(measured in nanoseconds) from the power supply.
THEORY OF OPERATION
Looking at the functional schematic shown (Figure below),
you can see that pin 7 is a transistor going to ground.
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This transistor is simply a switch that normally conducts until
pin 2 (which is connected through the comparator C1, which
feeds the internal flip flop) is brought low, allowing the
capacitor Ct to start charging. Pin 7 stays off until the voltage
on Ct charges to 2/3 of the power supply voltage, where the
timer times out and pin 7 transistor turns on again, its normal
state in this circuit.
The following (Figure below) will show the sequence of
switching, with red being the higher voltages and green
being ground (0 volts), with the spectrum in between since
this is fundamentally an analog circuit.
Timing
Trigger Time
Out
555 Timing Cycle
This graph shows the charge curve across the Ct.
555 Functional Schematic £/'Gnd
| 9V¥cc
FUP-PIOB iiies
Reset
Reset
Figure 1
Figure 1 is the starting and ending point for this circuit,
where it is waiting for a trigger to start a timing cycle. At this
point the pin 7 transistor is on, keeping the capacitor Ct
discharged.
. as:
555 Functional Schematic ©; 6nd
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Set Out
lip-Flo
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Reset Reset
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Figure 2
Figure 2 shows what happens when the 555 receives a
trigger, starting the sequence. Ct hasn't had time to
accumulate voltage, but the charging has started.
. ae:
555 Functional Schematic ©; 6"
?¥oc
: Pad 7
Pee inhibit /
Reset Reset
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Figure 3
Figure 3 shows the capacitor charging, during this time the
circuit is in a stable configuration and the output is high.
555 Functional Schematic Gnd
| Vcc
Reset
Reset
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Figure 4
Figure 4 shows the circuit in the middle of switching off when
it hits timeout. The capacitor has charged to 67%, the upper
limit of the 555 circuit, causing its internal flip flop to switch
states. As shown, the transistor hasn't switched yet, which
will discharge Ct when it does.
555 Functional Schematic £/'Gnd
| 9¥cc
Reset Reset
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Figure 5
Figure 5 shows the circuit after it has settled down, which is
basically the same as shown in Figure L.
CMOS 555 LONG DURATION MINIMUM
PARTS RED LED FLASHER
PARTS AND MATERIALS
Two AAA Batteries
Battery Clip (Radio Shack catalog # 270-398B)
One DVM or VOM
U1 - T One CMOS TLC555 timer IC (Radio Shack catalog
# 276-1718 or equivalent)
e D1 - Red light-emitting diode (Radio Shack catalog #
276-041 or equivalent)
R1- 1.5 MQ 1/4W 5% Resistor
R2 - 47 KO 1/4W 5% Resistor
e Cl - 1 uF Tantalum Capacitor (Radio Shack catalog 27 2-
1025 or equivalent)
e C2 - 100 uF Electrolytic Capacitor (Radio Shack catalog
272-1028 or equivalent)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 16: “Voltage
and current calculations”
Lessons In Electric Circuits, Volume 1, chapter 16: “Solving
for unknown time”
Lessons In Electric Circuits, Volume 3, chapter 9 :
“ElectroStatic Discharge”
Lessons In Electric Circuits, Volume 4, chapter 10:
“Multivibrators”
LEARNING OBJECTIVES
e Learn a practical application for a RC time constant
e Learn one of several 555 timer Astable Multivibrator
Configurations
e Working knowledge of duty cycle
e Learn how to handle ESD sensitive parts
SCHEMATIC DIAGRAM
Vop RST
TLCS55
Disch Out
Trig
Thresh
Ctrl
Gnd
ILLUSTRATION
Remove red jumper for normal operation. , i
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INSTRUCTIONS
NOTE! This project uses a static sensitive part, the CMOS
555. If you do not use protection as described in Volume 3,
Chapter 9, ElectroStatic Discharge, you run the risk of
destroying it.
The 555 is not a power hog, but it is a child of the 1970's,
created in 1971. It will suck a battery dry in days, if not
hours. Fortunately, the design has been reinvented using
CMOS technology. The new implementation isn't perfect, as it
lacks the fantastic current drive of the original, but fora
CMOS device the output current is still very good. The main
advantages include wider supply voltage range (power
supply specifications are 2V to 18V, and it will work using a
11/2V battery) and low power. This project uses the TLC555,
a Texas Instruments design. There are other CMOS 555's out
there, very similar but with some differences. These chips are
designed to be drop in replacements, and do very well as
long as the output is not substantially loaded.
This design turns a deficit into an advantage as the current
drive only gets worse at lower power supply voltages, its
specifications are not more than 3ma for 2VDC. This design
tries to make the batteries last as absolutely long as possible
using several different approaches. The CMOS IC is extremely
low current, and sends the LED a pulse of 30ms (which is a
very short time but within persistence of human vision) as
well as using a slow flash rate (1 second) using really large
resistors to minimize current. With a duty cycle of 3%, this
circuit soends most of its time off, and (assuming 20ma for
the LED) the average current is 0.6ma. The big problem is
using the built in current limitation of this IC, as is it is not
rated for a specific current, and the LED current can vary a
lot between different CMOS ICs.
It is possible to run into problems with electrolytic capacitors
when dealing with very low currents (2ua in this case) in that
the leakage can be excessive, a borderline failure condition.
If your experiment seems to do this it might be fixed by
charging across the battery, then discharging the capacitor
Cl across any conductor several times.
When you complete this circuit the LED should start flashing,
and would continue to do so for several months. If you use
larger batteries, such as D cells, this duration will increase
dramatically.
To measure the current draw feeding the LED, connect C1+
to Vcc with a jumper (shown in red on the Illustration), which
will turn the TLC555 on. Measure the amperage flowing from
the battery to the circuit. The target current is 20ma, |
measured 9ma to 24ma using different CMOS 555s. This isn't
critical, though it will affect the battery life.
THEORY OF OPERATION
An observant reader will note that this is fundamentally the
same circuit that was used in the 555 AUDIO OSCILLATOR
experiment. Many designs use the same basic designs and
concepts several different ways, this is such a case. A
conventional 555 IC would work in this design if the power
supply weren't so low and a LED current limiting resistor is
used. Other than the type of transistors used the block
diagram shown in Figure 1 is basically the same as a
conventional 555.
TLC555 (CMOS 555) |
Functional Schematic end
T Yop
Figure 2
This particular oscillator depends on the pin 7 transistor,
much like the 555 Monostable Multivibrator shown in an
earlier experiment. The startup condition is with the
Capacitor discharged, the output high, and pin 7 transistor
off. The capacitor starts charging as shown in Figure 2.
_— TLC555 (CMOS 555) |
Functional Schematic
Ms Gnd
Reset Q
Flip-Fl
eee P inhibit /
1 Set Reset
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Figure 2
Ypp° 8 Yop
R2
When the voltage across pins 2 and 6 reaches 2/3 of the
power supply the flip flop is reset via internal comparator Cl,
which turns on the Pin 7 transistor, and starts the capacitor
Cl discharging through R2 as shown in Figure 3. The current
shown through RI1 is incidental, and not important other than
it drains the battery. This is why this resistor value is so large.
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TLC555 (CMOS 555)
Functional Schematic
° Ven c+ Gnd
Figure 3
When the voltage across pins 2 and 6 reaches 1/3 of the
power supply the flip flop is set via internal comparator C2,
when turns off the pin 7 transistor, allowing the capacitor to
start charging again through R1 and R2, as shown in Figure
2. This cycle repeats.
Capacitor C2 extends the life of the batteries, since it will
store the voltage during the 97% of time the circuit is off,
and provide the current during the 3% it is on. This simple
addition will take the batteries beyond their useful life by a
large margin.
In running this experiment there was a feedback mechanism
| hadn't anticipated. The output current of the TLC555 is not
proportional, as the power supply voltage goes down the
output current reduces a lot more. My flasher lasted for 6
months before | terminated the experiment. It was still
flashing, it was just very dim.
CMOS 555 LONG DURATION BLUE LED
FLASHER
PARTS AND MATERIALS
Two AAA Batteries
Battery Clip (Radio Shack catalog # 270-398B)
Ul - 1CMOS TLC555 timer IC (Radio Shack catalog #
276-1718 or equivalent)
Q1 - 2N3906 PNP Transistor (Radio Shack catalog #276-
1604 (15 pack) or equivalent)
Q2 - 2N2222 NPN Transistor (Radio Shack catalog #276-
1617 (15 pack) or equivalent)
CR1 - 1N914 Diode (Radio Shack catalog #276-1122 (10
pack) or equivalent, see Instructions)
D1 - Blue light-emitting diode (Radio Shack catalog #
276-311 or equivalent)
R1-1.5 MO 1/4W 5% Resistor
R2 - 47 KQ 1/4W 5% Resistor
R3 - 2.2 KO 1/4W 5% Resistor
R4 - 620 QO 1/4W 5% Resistor
R5 - 82 0 1/4W 5% Resistor
Cl - 1 uF Tantalum Capacitor (Radio Shack catalog 27 2-
1025 or equivalent)
C2 - 100 uF Electrolytic Capacitor (Radio Shack catalog
272-1028 or equivalent)
C3 - 470 uF Electrolytic Capacitor (Radio Shack catalog
272-1030 or equivalent)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 16: “Voltage
and current calculations“
Lessons In Electric Circuits, Volume 1, chapter 16: “Solving
for unknown time”
Lessons In Electric Circuits, Volume 3, chapter 4 : “Bipolar
Junction Transistors”
Lessons In Electric Circuits, Volume 3, chapter 9 :
“ElectroStatic Discharge”
Lessons In Electric Circuits, Volume 4, chapter 10:
“Multivibrators”
LEARNING OBJECTIVES
Learn a practical application for a RC time constant
e Learn one of several 555 timer Astable Multivibrator
Configurations
Working knowledge of duty cycle
How to handle ESD sensitive parts
How to use transistors to improve current gain
How to use a Capacitor to double voltage with a switch
SCHEMATIC DIAGRAM
ILLUSTRATION
ESD Sensitive
Asajieg vve
INSTRUCTIONS
NOTE! This project uses a static sensitive part, the CMOS
555. If you do not use protection as described in Volume 3,
Chapter 9, ElectroStatic Discharge, you run the risk of
destroying it.
This circuit builds on the previous two experiments, using
their features and adding to them. Blue and white LEDs have
a higher Vf (forward dropping voltage) than most, around
3.6V. 3V batteries can't drive them without help, so extra
circuitry is required.
As in the previous circuits, the LED is given a 0.03 second
(30ms) pulse. C3 is used to double the voltage of this pulse,
but it can only do this for a short time. Measuring the current
though the LED is impractical with this circuit because of this
short duration, but blue LEDs are generally more predictable
because they were invented later.
This particular design can also be used with a single 1 1/2V
battery. The base concept was created with a now obsolete
IC, the LM3909, which used a red LED, the IC, and a
capacitor. As with this circuit, it could flash a red LED for over
a year with a single D cell. When newer red LEDs increased
their Vf from 1.5V to 2.5V this old chip was no longer
practical, and is still missed by many hobbyists. If you want
to try a 11/2V battery change R5 to 100 and use a red LED
with a better CR1 (see next paragraph) .
CR1 is not the best choice for this component, it was selected
because it is a common part and it works. Almost any diode
will work in this application. Schottky and germanium diodes
drop much less voltage, a silicon diode drops 0.6-0.7V, while
a Schottky diode drops 0.1-0.2V, and a germanium diode
drops 0.2V-0.3V. If these components are used the reduced
voltage drop would translate into brighter LED intensity, as
the circuits efficiency is increased.
THEORY OF OPERATION
Q2 is a switch, which this circuit uses. When Q2 is off C3 is
charged to the battery voltage, minus the diode drop, as
shown in Figure 1. Since the blue LED Vf is 3.4V to 3.6V it is
effectively out of the circuit.
Figure 2
Figure 2 shows what happens when Q2 turns on. The
Capacitor C3 + side is grounded, which moves the - side to
-2.4V. The diode CR1 is now back biased, and is out of the
circuit. The -2.4V is discharged through R5 and D1 to the
+3.0V of the batteries. The 5.4V provides lots of extra
voltage to light the blue LED. Long before C3 is discharged
the circuit switches back and C3 starts charging again.
+3 +3,
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C3
Q2 CRI
Figure 2
In the LM3909 CRI1 was a resistor. The diode was used to
minimize current, by allowing R4 to be its maximum value.
You may notice a dim blue glow in the blue LED when it is off.
This demonstrates the difference between theory and
practice, 3V is enough to cause some leakage through the
blue LED, even though it is not conducting. If you were to
measure this current it would be very small.
CMOS 555 LONG DURATION FLYBACK
LED FLASHER
PARTS AND MATERIALS
Two AAA Batteries
Battery Clip (Radio Shack catalog # 270-398B)
U1, U2 - CMOS TLC555 timer IC (Radio Shack catalog #
276-1718 or equivalent)
Q1 - 2N3906 PNP Transistor (Radio Shack catalog #276-
1604 (15 pack) or equivalent)
Q2 - 2N2222 NPN Transistor (Radio Shack catalog #27 6-
1617 (15 pack) or equivalent)
e D1 - Red light-emitting diode (Radio Shack catalog #
276-041 or equivalent)
e D2 - Blue light-emitting diode (Radio Shack catalog #
276-311 or equivalent)
R1-1.5 MO 1/4W 5% Resistor
R2 - 47 KQ 1/4W 5% Resistor
R3,R5 - 10 KQ 1/4W 5% Resistor
R4 -1MQ 1/4W 5% Resisto
:
e R6- 100 KQ 1/4W 5% Resistor
R7 - 1 KQ 1/4W 5% Resistor
Cl - 1 uF Tantalum Capacitor (Radio Shack catalog #
272-1025 or equivalent)
e C2 - 100 pF Ceramic Disc Capacitor (Radio Shack catalog
# 272-123)
e C3 - 100 uF Electrolytic Capacitor (Radio Shack catalog
272-1028 or equivalent)
e L1 - 200 WH Choke or Inductor (Exact value not critical,
see end of chapter)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 16: Title
"Inductor transient response"
Lessons In Electric Circuits, Volume 1, chapter 16: Title "Why
L/R and not LR?"
Lessons In Electric Circuits, Volume 3, chapter 4: Title "The
common-emitter amplifier"
Lessons In Electric Circuits, Volume 3, chapter 9: Title
"Electrostatic Discharge"
Lessons In Electric Circuits, Volume 4, chapter 10: Title
“Monostable multivibrators"
LEARNING OBJECTIVES
e Learn another mode of operation for the 555
e How to handle ESD Parts
e How to use a transistor for a simple gate (resistor
transistor inverter)
e How inductors can convert power using inductive flyback
e How to make an inductor
SCHEMATIC DIAGRAM
ILLUSTRATION
es
ESD Sensitive
INSTRUCTIONS
NOTE! This project uses a static sensitive part, the CMOS
555. If you do not use protection as described in Volume 3,
Chapter 9, ElectroStatic Discharge, you run the risk of
destroying it.
This particular experiment builds on another experiment,
“Commutating diode" (Volume 6, chapter 5). It is worth
reviewing that section before proceeding.
This is the last of the long duration LED flasher series. They
have shown how to use a CMOS 555 to flash an LED, and how
to boost the voltage of the batteries to allow an LED with
more voltage drop than the batteries to be used. Here we are
doing the same thing, but with an inductor instead of a
Capacitor.
The basic concept is adapted from another invention, the
Joule Thief. A joule thief is a simple transistor oscillator that
also uses inductive kickback to light an white light LED from
a 11/2 battery, and the LED needs at least 3.6 volts to start
conducting! Like the joule thief, it is possible to use 11/2
volts to get this circuit to work. However, since a CMOS 555
is rated for 2 volts minimum 11/2 volts is not recommended,
but we can take advantage of the extreme efficiency of this
circuit. lf you want to learn more about the joule thief plenty
of information can be found on the web.
This circuit can also drive more that 1 or 2 LEDs in series. As
the numbers of LEDs go up the ability of the batteries to last
a long duration goes down, as the amount of voltage the
inductor can generate is somewhat dependent on battery
voltage. For the purposes of this experiment two dissimilar
LEDs were used to demonstrate its independence of LED
voltage drop. The high intensity of the blue LED swamps the
red LED, but if you look closely you will find the red LED is at
its maximum brightness. You can use pretty much whatever
color of LEDs you choose for this experiment.
Generally the high voltage created by inductive kickback is
something to be eliminated. This circuit uses it, but if you
make a mistake with the polarity of the LEDs the blue LED,
which is more ESD sensitive, will likely die (this has been
verified). An uncontrolled pulse from a coil resembles an ESD
event. The transistor and the TLC555 can also be at risk.
The inductor in this circuit is probably the least critical part
in the design. The term inductor is generic, you can also find
this component called a choke or a coil. A solenoid coil would
also work, since that is also a type of inductor. So would the
coil from a relay. Of all the components | have used, this is
probably the least critical I've come across. Indeed, coils are
probably the most practical component you can make
yourself that exists. I'll cover how to make a coil that will
work in this design after the Theory of Operation, but the
part shown on the illustration is a 200UWH choke | bought from
a local electronics retailer.
THEORY OF OPERATION
Both capacitors and inductors store energy. Capacitors try to
maintain constant voltage, whereas inductors try to maintain
constant current. Both resist change to their respective
aspect. This is the basis for the flyback transformer, which is
a common circuit used in old CRT circuits and other uses
where high voltage is needed with a minimum of fuss. When
you charge a coil a magnetic field expands around it,
basically it is an electromagnet, and the magnetic field is
stored energy. When the current stops this magnetic field
collapses, created electricity as the field crosses the wires in
the coil.
This circuit uses two astable multivibrators. The first
multivibrator controls the second. Both are designed for
minimum current, as well as the inverter made using Q1.
Both the oscillators are very similar, the first has been
covered in previous experiments. The problem is it stays on,
or is high, 97% of the time. On the previous circuits we used
the low state to light the LED, in this case the high is what
turns the second multivibrator on. Using a simple transistor
inverter designed for extra low current solves this problem.
This is actually a very old logic family, RTL, which is short for
resistor transistor logic.
The second multivibrator oscillates at 68.6 KHz, witha
square wave that is around 50%. This circuit uses the exact
same principals as is shown in the Minimum Parts LED
Flasher. Again, the largest practical resistors are used to
minimize current, and this means a really small capacitor for
C2. This high frequency square wave is used to turn Q2 on
and off as a simple switch.
Figure 1 shows what happens when the Q2 is conducting,
and the coil starts to charge. If Q2 were to stay on then an
effective short across the batteries would result, but since
this is part of an oscillator this won't happen. Before the coil
can reach it's maximum current Q2 switches, and the switch
IS open.
Figure I
Figure 2 shows Q2 when it opens, and the coil is charged.
The coil tries to maintain the current, but if there is no
discharge path it can not do this. If there were no discharge
path is the coil would create a high voltage pulse, seeking to
maintain the current that was flowing through it, and this
voltage would be quite high. However, we have a couple of
LEDs in the discharge path, so the coils pulse quickly goes to
the voltage drop of the combined LEDs, then dumps the rest
of its charge as current. As a result there is no high voltage
generated, but there is a conversion to the voltage required
to light the LEDs.
Figure 2
The LEDs are pulsed, and the light curve follows the
discharge curve of the coil fairly closely. However, the human
eye averages this light output to something we perceive as
continuous light.
HOW TO MAKE AN INDUCTOR
PARTS AND MATERIALS
26 Feet (8 Meters) of 26AWG Magnet Wire (Radio Shack
catalog #27 8-1345 or equivalent)
6/32X1.5 inch screw, aM4X30mm screw, or a nail of
similar diameter cut down to size, steel or iron, but not
stainless
Matching lock nut (optional)
Transparent Tape (optional, needed if using screws)
Super Glue
Soldering Iron, Solder
As has been mentioned before, this is not a precision part.
Inductors in general can have a large variance for many
applications, and this one specifically can be off on the high
side a large amount. The target here is greater than 220uUH.
If you are using a screw, use one layer of the transparent
tape between the threads and the wire. This is to prevent the
threads of the screw from cutting into the wire and shorting
the coil out. If you are using a lock nut put it on the screw 1"
(25mm) from the head of the screw. Starting around 1" from
one end of the wire, use the glue to tack the wire on the head
of the nail or screw as shown. Let the glue set.
Super
Super
Glue Glue
Wind the wire neatly and tightly 1" the length of screw, again
tacking it in place with super glue. (Figure above). You can
use a variable speed drill to help with this, as long as you are
careful. Like all power appliances, it can bite you. Hold the
wire tight until the glue sets, then start winding a second
layer over the first. Continue this process until all of the wire
except the last 1" is used, using the glue to occasionally tack
the wire down. Arrange the wire on the last layer so the
second inductor lead is on the other end of the screw away
from the first. Tack this down for a final time with the glue.
Let dry completely.
Gently take a sharp blade and scrap the enamel off each end
of the two leads. Tin the exposed copper with the soldering
iron and the solder, and you now have a functional inductor
that can be used in this experiment.
Here is what the one | made looked like: Figure below.
: ica
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pee nee eee eee
eee eee eee =
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eee eee
eee ewe eee
i"
eee eee
The connections shown are being used to measure the
inductance, which worked out pretty close to 220UH.
CMOS 555 LONG DURATION RED LED
FLASHER
PARTS AND MATERIALS
e Two AAA Batteries
e« Battery Clip (Radio Shack catalog # 270-398B)
e A DVM or VOM
U1 - CMOS TLC555 timer IC (Radio Shack catalog # 276-
1718 or equivalent)
e Q1 - 2N3906 PNP Transistor (Radio Shack catalog #27 6-
1604 (15 pack) or equivalent)
e Q2 - 2N2222 NPN Transistor (Radio Shack catalog #276-
1617 (15 pack) or equivalent)
e D1 - Red light-emitting diode (Radio Shack catalog #
276-041 or equivalent)
R1-1.5 MO 1/4W 5% Resistor
R2 - 47 KQ 1/4W 5% Resistor
R3 - 2.2 KO 1/4W 5% Resistor
R4 - 27 Q1/4W 5% Resistor (or test select a better value)
Cl - 1 uF Tantalum Capacitor (Radio Shack catalog 27 2-
1025 or equivalent)
e C2 - 100 uF Electrolytic Capacitor (Radio Shack catalog
272-1028 or equivalent)
CROSS-REFERENCES
Lessons In Electric Circuits, Volume 1, chapter 16: “Voltage
and current calculations”
Lessons In Electric Circuits, Volume 1, chapter 16: “Solving
for unknown time”
Lessons In Electric Circuits, Volume 3, chapter 4 : “Bipolar
Junction Transistors”
Lessons In Electric Circuits, Volume 3, chapter 9 :
“ElectroStatic Discharge”
Lessons In Electric Circuits, Volume 4, chapter 10:
“Multivibrators”
LEARNING OBJECTIVES
e Learn a practical application for a RC time constant
e Learn one of several 555 timer Astable Multivibrator
Configurations
e Working knowledge of duty cycle
e How to handle ESD sensitive parts
e How to use transistors to improve current gain
e How to calculate the correct resistor for a LED
SCHEMATIC DIAGRAM
ILLUSTRATION
Remove red jumper for normal operation.
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INSTRUCTIONS
NOTE! This project uses a static sensitive part, the CMOS
555. If you do not use protection as described in Volume 3,
Chapter 9, ElectroStatic Discharge, you run the risk of
destroying it.
The circuit shown in the previous experiment, CMOS 555
Long Duration Minimum Parts Red LED Flasher, has one big
drawback, which is a lack of LED current control. This
experiment uses the same basic 555 schematic and adds
transistorized drivers to correct this.
The parts used for this transistor driver are non critical. It is
designed to load the TLC555 to an absolute minimum and
still turn on Q2 fully. This is important because as the battery
voltage approaches 2V the drive from the TLC555 is reduced
to its minimum values. Bipolar transistors can be good
switches.
Since LEDs can have so much variation R4 should be
tweaked to match the specific LED used. The current is
limited to 18.5ma with 27Q and a Vf (LED forward dropping
voltage) of 2.5V, an LED Vf of 2.1V will draw 33ma, and a LED
Vf of 1.5 will draw 56ma. The latter is too much current, not
to mention what that would do for the battery life. To correct
this use 470 if the Vf is 2.1V, and 750 if the Vf is 1.5V,
assuming the target current is 20ma.
You can measure Vf by using the jumper shown in red in the
illustration, which will turn the LED on full time. You can
calculate the value of R4 by using the equation:
R4 = (3V-Vf) /0.02A
It was mentioned in the previous experiment that capacitor
C2 extended the life of the batteries. An interesting
experiment is to remove this part periodically and see what
happens. At first you will notice a dimming of the LED, and
after a week or two the circuit will die without it, and resume
working in a couple of seconds when it is replaced. This
flasher will work for 3 months using fresh alkaline AAA
batteries.
THEORY OF OPERATION
The CMOS 555 oscillator was explained fully in the previous
experiment, so the transistor driver will be the focus of this
explanation.
The transistor driver combines elements of a common
collector configuration on Q1, along with common emitter
configuration on Q2. This allows for very high input
resistance while allowing Q2 to turn on fully. The input
resistance of the transistor is the B (gain) of the transistor
times the emitter resistor. If Q1 has a gain of 50 (a minimum
value) then the driver loads the TLC555 with more than
LOOKQ. Transistors can have large variations in gain, even
within the same family.
When Q1 turns on 1ma is sent to Q2. This is more than
enough to turn Q2 fully, which is referred to as saturation. Q2
is used as a simple switch for the LED.
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—| | +4/l—
— 4 —
Appendix 1
ABOUT THIS BOOK
Purpose
They say that necessity is the mother of invention. At least
in the case of this book, that adage is true. As an industrial
electronics instructor, | was forced to use a sub-standard
textbook during my first year of teaching. My students were
daily frustrated with the many typographical errors and
obscure explanations in this book, having spent much time
at home struggling to comprehend the material within.
Worse yet were the many incorrect answers in the back of
the book to selected problems. Adding insult to injury was
the $100+ price.
Contacting the publisher proved to be an exercise in futility.
Even though the particular text | was using had been in
print and in popular use for a couple of years, they claimed
my complaint was the first they'd ever heard. My request to
review the draft for the next edition of their book was met
with disinterest on their part, and | resolved to find an
alternative text.
Finding a Suitable alternative was more difficult than | had
imagined. Sure, there were plenty of texts in print, but the
really good books seemed a bit too heavy on the math and
the less intimidating books omitted a lot of information | felt
was important. Some of the best books were out of print, and
those that were still being printed were quite expensive.
It was out of frustration that | compiled Lessons in Electric
Circuits from notes and ideas | had been collecting for years.
My primary goal was to put readable, high-quality
information into the hands of my students, but a secondary
goal was to make the book as affordable as possible. Over
the years, | had experienced the benefit of receiving free
instruction and encouragement in my pursuit of learning
electronics from many people, including several teachers of
mine in elementary and high school. Their selfless
assistance played a key role in my own studies, paving the
way for a rewarding career and fascinating hobby. If only |
could extend the gift of their help by giving to other people
what they gavetome...
So, | decided to make the book freely available. More than
that, | decided to make it "open," following the same
development model used in the making of free software
(most notably the various UNIX utilities released by the Free
Software Foundation, and the Linux operating system,
whose fame Is growing even as | write). The goal was to
copyright the text -- so as to protect my authorship -- but
expressly allow anyone to distribute and/or modify the text
to suit their own needs with a minimum of legal
encumbrance. This willful and formal revoking of standard
distribution limitations under copyright is whimsically
termed copyleft. Anyone can "copyleft" their creative work
simply by appending a notice to that effect on their work,
but several Licenses already exist, covering the fine legal
points in great detail.
The first such License | applied to my work was the GPL --
General Public License -- of the Free Software Foundation
(GNU). The GPL, however, is intended to copyleft works of
computer software, and although its introductory language
is broad enough to cover works of text, its wording is not as
clear as it could be for that application. When other, less
specific copyleft Licenses began appearing within the free
software community, | chose one of them (the Design
Science License, or DSL) as the official notice for my project.
In "copylefting" this text, | guaranteed that no instructor
would be limited by a text insufficient for their needs, as |
had been with error-ridden textbooks from major publishers.
I'm sure this book in its initial form will not satisfy everyone,
but anyone has the freedom to change it, leveraging my
efforts to suit variant and individual requirements. For the
beginning student of electronics, learn what you can from
this book, editing it as you feel necessary if you come across
a useful piece of information. Then, if you pass it on to
someone else, you will be giving them something better
than what you received. For the instructor or electronics
professional, feel free to use this as a reference manual,
adding or editing to your heart's content. The only "catch" is
this: if you plan to distribute your modified version of this
text, you must give credit where credit is due (to me, the
Original author, and anyone else whose modifications are
contained in your version), and you must ensure that
whoever you give the text to is aware of their freedom to
similarly share and edit the text. The next chapter covers
this process in more detail.
It must be mentioned that although | strive to maintain
technical accuracy in all of this book's content, the subject
matter is broad and harbors many potential dangers.
Electricity maims and kills without provocation, and
deserves the utmost respect. | strongly encourage
experimentation on the part of the reader, but only with
circuits powered by small batteries where there is no risk of
electric shock, fire, explosion, etc. High-power electric
circuits should be left to the care of trained professionals!
The Design Science License clearly states that neither | nor
any contributors to this book bear any liability for what is
done with its contents.
The use of SPICE
One of the best ways to learn how things work is to follow
the inductive approach: to observe specific instances of
things working and derive general conclusions from those
observations. In science education, labwork is the
traditionally accepted venue for this type of learning,
although in many cases labs are designed by educators to
reinforce principles previously learned through lecture or
textbook reading, rather than to allow the student to learn
on their own through a truly exploratory process.
Having taught myself most of the electronics that | know, |
appreciate the sense of frustration students may have in
teaching themselves from books. Although electronic
components are typically inexpensive, not everyone has the
means or opportunity to set up a laboratory in their own
homes, and when things go wrong there's no one to ask for
help. Most textbooks seem to approach the task of education
from a deductive perspective: tell the student how things
are supposed to work, then apply those principles to specific
instances that the student may or may not be able to
explore by themselves. The inductive approach, as useful as
it is, is hard to find in the pages of a book.
However, textbooks don't have to be this way. | discovered
this when | started to learn a computer program called
SPICE. It is a text-based piece of software intended to model
circuits and provide analyses of voltage, current, frequency,
etc. Although nothing is quite as good as building real
circuits to gain knowledge in electronics, computer
simulation is an excellent alternative. In learning how to use
this powerful tool, | made a discovery: SPICE could be used
within a textbook to present circuit simulations to allow
students to "observe" the phenomena for themselves. This
way, the readers could learn the concepts inductively (by
interpreting SPICE's output) as well as deductively (by
interpreting my explanations). Furthermore, in seeing SPICE
used over and over again, they should be able to
understand how to use it themselves, providing a perfectly
safe means of experimentation on their own computers with
circuit simulations of their own design.
Another advantage to including computer analyses in a
textbook is the empirical verification it adds to the concepts
presented. Without demonstrations, the reader is left to take
the author's statements on faith, trusting that what has
been written is indeed accurate. The problem with faith, of
course, is that it is only as good as the authority in which it
is placed and the accuracy of interpretation through which it
is understood. Authors, like all human beings, are liable to
err and/or communicate poorly. With demonstrations,
however, the reader can immediately see for themselves
that what the author describes is indeed true.
Demonstrations also serve to clarify the meaning of the text
with concrete examples.
SPICE is introduced early in volume | (DC) of this book
series, and hopefully in a gentle enough way that it doesn't
create confusion. For those wishing to learn more, a chapter
in this volume (volume V) contains an overview of SPICE
with many example circuits. There may be more flashy
(graphic) circuit simulation programs in existence, but SPICE
is free, a virtue complementing the charitable philosophy of
this book very nicely.
Acknowledgements
First, | wish to thank my wife, whose patience during those
many and long evenings (and weekends!) of typing has
been extraordinary.
| also wish to thank those whose open-source software
development efforts have made this endeavor all the more
affordable and pleasurable. The following is a list of various
free computer software used to make this book, and the
respective programmers:
e GNU/Linux Operating System -- Linus Torvalds, Richard
Stallman, and a host of others too numerous to mention.
e Vim text editor -- Bram Moolenaar and others.
Xcircuit drafting program -- Tim Edwards.
SPICE circuit simulation program -- too many
contributors to mention.
e T-X text processing system -- Donald Knuth and others.
e Texinfo document formatting system -- Free Software
Foundation.
LAT-X document formatting system -- Leslie Lamport and
others.
Gimp image manipulation program -- too many
contributors to mention.
Winscope signal analysis software -- Dr. Constantin
Zeldovich. (Free for personal and academic use.)
Appreciation is also extended to Robert L. Boylestad, whose
first edition of Introductory Circuit Analysis taught me more
about electric circuits than any other book. Other important
texts in my electronics studies include the 1939 edition of
The "Radio" Handbook, Bernard Grob's second edition of
Introduction to Electronics I, and Forrest Mims' original
Engineer's Notebook.
Thanks to the staff of the Bellingham Antique Radio
Museum, who were generous enough to let me terrorize their
establishment with my camera and flash unit.
| wish to specifically thank Jeffrey Elkner and all those at
Yorktown High School for being willing to host my book as
part of their Open Book Project, and to make the first effort
in contributing to its form and content. Thanks also to David
Sweet (website: [*]) and Ben Crowell (website: [*]) for
providing encouragement, constructive criticism, and a
wider audience for the online version of this book.
Thanks to Michael Stutz for drafting his Design Science
License, and to Richard Stallman for pioneering the concept
of copyleft.
Last but certainly not least, many thanks to my parents and
those teachers of mine who saw in me a desire to learn
about electricity, and who kindled that flame into a passion
for discovery and intellectual adventure. | honor you by
helping others as you have helped me.
Tony Kuphaldt, July 2001
"A candle loses nothing of its light when lighting
another"
Kahlil Gibran
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—||4]l_—
—| | +]
Appendix 2
CONTRIBUTOR LIST
How to contribute to this book
As a copylefted work, this book is open to revision and expansion by
any interested parties. The only "catch" is that credit must be given
where credit is due. This /s a copyrighted work: it is notin the public
domain!
If you wish to cite portions of this book in a work of your own, you
must follow the same guidelines as for any other copyrighted work.
Here is a Sample from the Design Science License:
The Work is copyright the Author. All rights to the Work are reserved
by the Author, except as specifically described below. This License
describes the terms and conditions under which the Author permits you
to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright Law.
Your right to operate, perform, read or otherwise interpret and/or
execute the Work is unrestricted; however, you do so at your own risk,
because the Work comes WITHOUT ANY WARRANTY -- see Section 7 ("NO
WARRANTY") below.
If you wish to modify this book in any way, you must document the
nature of those modifications in the "Credits" section along with your
name, and ideally, information concerning how you may be
contacted. Again, the Design Science License:
Permission is granted to modify or sample from a copy of the Work,
producing a derivative work, and to distribute the derivative work
under the terms described in the section for distribution above,
provided that the following terms are met:
(a) The new, derivative work is published under the terms of this
License.
(b) The derivative work is given a new name, so that its name or
title can not be confused with the Work, or with a version of
the Work, in any way.
(c) Appropriate authorship credit is given: for the differences
between the Work and the new derivative work, authorship is
attributed to you, while the material sampled or used from
the Work remains attributed to the original Author; appropriate
notice must be included with the new work indicating the nature
and the dates of any modifications of the Work made by you.
Given the complexities and security issues surrounding the
maintenance of files comprising this book, it is recommended that
you submit any revisions or expansions to the original author (Tony R.
Kuphaldt). You are, of course, welcome to modify this book directly by
editing your own personal copy, but we would all stand to benefit
from your contributions if your ideas were incorporated into the
online “master copy” where all the world can see it.
Credits
All entries arranged in alphabetical order of surname. Major
contributions are listed by individual name with some detail on the
nature of the contribution(s), date, contact info, etc. Minor
contributions (typo corrections, etc.) are listed by name only for
reasons of brevity. Please understand that when | classify a
contribution as “minor,” it is in no way inferior to the effort or value of
a “major” contribution, just smaller in the sense of less text changed.
Any and all contributions are gratefully accepted. | am indebted to all
those who have given freely of their own knowledge, time, and
resources to make this a better book!
Dennis Crunkilton
« Date(s) of contribution(s): January 2006 to present
e Nature of contribution: Mini table of contents, all chapters
except appedicies; html, latex, ps, pdf; See Devel/tutorial.hAtm;
01/2006.
e Nature of contribution: CH 4, section: Induction motor,
09/2007.
e Nature of contribution: CH 4, section: Induction motor, large
02/2010.
e Contact at: dcrunkilton(at)att(dot)net
Tony R. Kuphaldt
« Date(s) of contribution(s): 1996 to present
e Nature of contribution: Original author.
e Contact at: liec0@lycos.com
Bill Marsden
« Date(s) of contribution(s): August 2008
e Nature of contribution: Original author: “555 Schmidt trigger”
Section, Chapter 7.
¢ Contact at: bill _marsden2(at) hotmail (dot) com
Forrest M. Mims lll
Date(s) of contribution(s):February 2008
Nature of contribution:Ch 5; Clarification concerning LEDs as
photosensors.
Contact at: FMims(at)aol.com
Your name here
Date(s) of contribution(s): Month and year of contribution
Nature of contribution: Insert text here, describing how you
contributed to the book.
Contact at: my email@provider.net
Typo corrections and other “minor” contributions
line-allaboutcircuits.com (June 2005) Typographical error
correction in Volumes 1,2,3,5, various chapters ,(:s/visa-versa/vice
versa/).
The students of Bellingham Technical College's Instrumentation
program.
Colin Creitz (May 2007) Chapters: several, s/it's/its.
Jeff DeFreitas (March 2006)Improve appearance: replace “/" and
”/" Chapters: Al, A2.
Don Stalkowski (June 2002) Technical help with PostScript-to-
PDF file format conversion.
Joseph Teichman (June 2002) Suggestion and technical help
regarding use of PNG images instead of JPEG.
Michael Warner (April 2002) Suggestions for a section
describing home laboratory setup.
jut@allaboutcircuits.com (August 2007) Chl,
s/starting/started .
Unregistered@allaboutcircuits.com (August 2007) Ch 6,
s/and and off/on and off/ .
Timothy Unregistered@allaboutcircuits.com (Feb 2008)
Changed default roman font to newcent.
Imranullah Syed (Feb 2008) Suggested centering of
uncaptioned schematics.
Sylverce@allaboutcircuits.com,
Caveman@allaboutcircuits.com (May 2008) Changed image
05320.png to agree with image
05321.pngsarwiz@allaboutcircuits.com (April 2009) Ch4,
s/Try changed/Try changing/jrap@allaboutcircuits.com
(August 2009) added <section> tags to "555 Schmitt trigger",
d_ic.sml .
« Heavydoody@allaboutcircuits.com (August 2009) correction
to image 05198.eps &.png.
¢ Dcrunkilton@allaboutcircuits.com (January 2010) added
<proofread> tag to "555 Schmitt trigger".
« Bereahorn@allaboutcircuits.com (January 2010) Ch2, s/The
less ressistance/ The more resistance.
¢ Bill Marsden@allaboutcircuits.com (April 2010) added new
CROSS-REFERENCE to "555 Schmitt trigger”.
¢ Dcrunkilton@allaboutcircuits.com (September 2010) Ch6,
s/useable/usable/ .
e D. Crunkilton (June 2011) hi.latex, header file; updated link to
openbookproject.net .
¢ hillshaveeyes57 (January 2013) Ch8, Hysteretic Oscillator,
Swap R3 and R4 with description in parts list.
¢« Bill Marsden@allaboutcircuits.com (January 2014) Ch8,
s/circuits operation/circuit's operation.
¢ Dennis Crunkilton (January 2014) Ch8, many yu instances
corrected.
Lessons In Electric Circuits copyright (C) 2002-2020 Tony R. Kuphaldt,
under the terms and conditions of the Design Science License.
—|/]|+4|l\—
—/ | 4]
Appendix 3
DESIGN SCIENCE LICENSE
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[ $Id: dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]
— 4 —
a
a
Copyright (C) 2000-2020, Tony R.
Kuphaldt
See the Design Science License (Appendix 3)
for details regarding copying and distribution
Revised January 18, 2006
Master Index
Chapter 1: NUMERATION SYSTEMS
Chapter 2: BINARY ARITHMETIC
Chapter 3: LOGIC GATES
Chapter 4: SWITCHES
Chapter 5: ELECTROMECHANICAL RELAYS
Chapter 6: LADDER LOGIC
Chapter 7: BOOLEAN ALGEBRA
Chapter 8: KARNAUGH MAPPING
Chapter 9: COMBINATIONAL LOGIC FUNCTIONS
Chapter 10: MULTIVIBRATORS
Chapter 11; SEQUENTIAL CIRCUITS ***INCOMPLETE***
Chapter 12: SHIFT REGISTERS
Chapter 13: DIGITAL-ANALOG CONVERSION
Chapter 14: DIGITAL COMMUNICATION
Chapter 15: DIGITAL STORAGE (MEMORY)
Chapter 16: PRINCIPLES OF DIGITAL COMPUTING
Appendix 1: ABOUT THIS BOOK
Appendix 2: CONTRIBUTOR LIST
Appendix 3: DESIGN SCIENCE LICENSE
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Back to Master Index
=| L4) _
Lessons In Electric Circuits --
Volume IV
Chapter 1
NUMERATION SYSTEMS
e Numbers and symbols
e Systems of numeration
¢« Decimal versus binary numeration
e Octal and hexadecimal numeration
e Octal and hexadecimal to decimal conversion
e Conversion from decimal numeration
"There are three types of people: those who can count, and
those who can't."
Anonymous
Numbers and symbols
The expression of numerical quantities is something we tend to take
for granted. This is both a good and a bad thing in the study of
electronics. It is good, in that we're accustomed to the use and
manipulation of numbers for the many calculations used in
analyzing electronic circuits. On the other hand, the particular
system of notation we've been taught from grade school onward is
not the system used internally in modern electronic computing
devices, and learning any different system of notation requires some
re-examination of deeply ingrained assumptions.
First, we have to distinguish the difference between numbers and
the symbols we use to represent numbers. A number is a
mathematical quantity, usually correlated in electronics to a physical
quantity such as voltage, current, or resistance. There are many
different types of numbers. Here are just a few types, for example:
WHOLE NUMBERS:
Lig ep oe Ae! Og! Tag Oe
INTEGERS:
a4 335 22, =1y.-0y Dy Ze 3% A 4%
IRRATIONAL NUMBERS:
TM (approx. 3.1415927), e (approx. 2.718281828),
Square root of any prime
REAL NUMBERS:
(ALL one-dimensional numerical values, negative and positive,
including zero, whole, integer, and irrational numbers)
COMPLEX NUMBERS:
3.- j4, 34.5 z 20°
Different types of numbers find different application in the physical
world. Whole numbers work well for counting discrete objects, such
as the number of resistors in a circuit. Integers are needed when
negative equivalents of whole numbers are required. Irrational
numbers are numbers that cannot be exactly expressed as the ratio
of two integers, and the ratio of a perfect circle's circumference to its
diameter (tt) is a good physical example of this. The non-integer
quantities of voltage, current, and resistance that we're used to
dealing with in DC circuits can be expressed as real numbers, in
either fractional or decimal form. For AC circuit analysis, however,
real numbers fail to capture the dual essence of magnitude and
phase angle, and so we turn to the use of complex numbers in either
rectangular or polar form.
If we are to use numbers to understand processes in the physical
world, make scientific predictions, or balance our checkbooks, we
must have a way of symbolically denoting them. In other words, we
may know how much money we have in our checking account, but to
keep record of it we need to have some system worked out to
symbolize that quantity on paper, or in some other kind of form for
record-keeping and tracking. There are two basic ways we can do
this: analog and digital. With analog representation, the quantity is
symbolized in a way that is infinitely divisible. With digital
representation, the quantity is symbolized in a way that is discretely
packaged.
You're probably already familiar with an analog representation of
money, and didn't realize it for what it was. Have you ever seen a
fund-raising poster made with a picture of a thermometer on it,
where the height of the red column indicated the amount of money
collected for the cause? The more money collected, the taller the
column of red ink on the poster.
An analog representation
of a numerical quantity
— $50,000
— $40,000
— $30,000
— $20,000
— $10,000
— $0
This is an example of an analog representation of a number. There is
no real limit to how finely divided the height of that column can be
made to symbolize the amount of money in the account. Changing
the height of that column is something that can be done without
changing the essential nature of what it is. Length is a physical
quantity that can be divided as small as you would like, with no
practical limit. The slide rule is a mechanical device that uses the
very same physical quantity -- length -- to represent numbers, and to
help perform arithmetical operations with two or more numbers at a
time. It, too, is an analog device.
On the other hand, a digita/ representation of that same monetary
figure, written with standard symbols (sometimes called ciphers),
looks like this:
$35,955.38
Unlike the "thermometer" poster with its red column, those symbolic
characters above cannot be finely divided: that particular
combination of ciphers stand for one quantity and one quantity only.
If more money is added to the account (+ $40.12), different symbols
must be used to represent the new balance ($35,995.50), or at least
the same symbols arranged in different patterns. This is an example
of digital representation. The counterpart to the slide rule (analog) is
also a digital device: the abacus, with beads that are moved back
and forth on rods to symbolize numerical quantities:
Slide rule (an analog device)
Slide
Numerical quantities are represented by
the positioning of the slide.
Abacus (a digital device)
Numerical quantities are represented by
the discrete positions of the beads.
Let's contrast these two methods of numerical representation:
ANALOG DIGITAL
Intuitively understood ----------- Requires training to interpret
Infinitely divisible -------------- Discrete
Prone to errors of precision ------ Absolute precision
Interpretation of numerical symbols is something we tend to take for
granted, because it has been taught to us for many years. However,
if you were to try to communicate a quantity of something toa
person ignorant of decimal numerals, that person could still
understand the simple thermometer chart!
The infinitely divisible vs. discrete and precision comparisons are
really flip-sides of the same coin. The fact that digital representation
is composed of individual, discrete symbols (decimal digits and
abacus beads) necessarily means that it will be able to symbolize
quantities in precise steps. On the other hand, an analog
representation (such as a slide rule's length) is not composed of
individual steps, but rather a continuous range of motion. The ability
for a slide rule to characterize a numerical quantity to infinite
resolution is a trade-off for imprecision. If a slide rule is bumped, an
error will be introduced into the representation of the number that
was "entered" into it. However, an abacus must be bumped much
harder before its beads are completely dislodged from their places
(sufficient to represent a different number).
Please don't misunderstand this difference in precision by thinking
that digital representation is necessarily more accurate than analog.
Just because a clock is digital doesn't mean that it will always read
time more accurately than an analog clock, it just means that the
interpretation of its display is less ambiguous.
Divisibility of analog versus digital representation can be further
illuminated by talking about the representation of irrational
numbers. Numbers such as 71 are called irrational, because they
cannot be exactly expressed as the fraction of integers, or whole
numbers. Although you might have learned in the past that the
fraction 22/7 can be used for min calculations, this is just an
approximation. The actual number "pi" cannot be exactly expressed
by any finite, or limited, number of decimal places. The digits of m go
on forever:
3.1415926535897932384 .....
It is possible, at least theoretically, to set a slide rule (or even a
thermometer column) so as to perfectly represent the number m1,
because analog symbols have no minimum limit to the degree that
they can be increased or decreased. If my slide rule shows a figure of
3.141593 instead of 3.141592654, | can bump the slide just a bit
more (or less) to get it closer yet. However, with digital
representation, such as with an abacus, | would need additional rods
(place holders, or digits) to represent m to further degrees of
precision. An abacus with 10 rods simply cannot represent any more
than 10 digits worth of the number tt, no matter how | set the beads.
To perfectly represent m, an abacus would have to have an infinite
number of beads and rods! The tradeoff, of course, is the practical
limitation to adjusting, and reading, analog symbols. Practically
Speaking, one cannot read a slide rule's scale to the 10th digit of
precision, because the marks on the scale are too coarse and human
vision is too limited. An abacus, on the other hand, can be set and
read with no interpretational errors at all.
Furthermore, analog symbols require some kind of standard by which
they can be compared for precise interpretation. Slide rules have
markings printed along the length of the slides to translate length
into standard quantities. Even the thermometer chart has numerals
written along its height to show how much money (in dollars) the red
column represents for any given amount of height. Imagine if we all
tried to communicate simple numbers to each other by spacing our
hands apart varying distances. The number 1 might be signified by
holding our hands 1 inch apart, the number 2 with 2 inches, and so
on. If someone held their hands 17 inches apart to represent the
number 17, would everyone around them be able to immediately
and accurately interpret that distance as 17? Probably not. Some
would guess short (15 or 16) and some would guess long (18 or 19).
Of course, fishermen who brag about their catches don't mind
overestimations in quantity!
Perhaps this is why people have generally settled upon digital
symbols for representing numbers, especially whole numbers and
integers, which find the most application in everyday life. Using the
fingers on our hands, we have a ready means of symbolizing integers
from 0 to 10. We can make hash marks on paper, wood, or stone to
represent the same quantities quite easily:
5 +5 +3 =13
dat det I
For large numbers, though, the "hash mark" numeration system is
too inefficient.
Systems of numeration
The Romans devised a system that was a substantial improvement
over hash marks, because it used a variety of symbols (or ciphers) to
represent increasingly large quantities. The notation for 1 is the
capital letter I. The notation for 5 is the capital letter v. Other
ciphers possess increasing values:
X = 10
L = 50
C = 100
D = 500
M = 1000
If a cipher is accompanied by another cipher of equal or lesser value
to the immediate right of it, with no ciphers greater than that other
cipher to the right of that other cipher, that other cipher's value is
added to the total quantity. Thus, vIII symbolizes the number 8, and
CLVII symbolizes the number 157. On the other hand, if a cipher is
accompanied by another cipher of lesser value to the immediate left,
that other cipher's value is subtracted from the first. Therefore, Iv
symbolizes the number 4 (v minus I), and cm symbolizes the number
900 (M minus c). You might have noticed that ending credit
sequences for most motion pictures contain a notice for the date of
production, in Roman numerals. For the year 1987, it would read:
MCMLXXXVII. Let's break this numeral down into its constituent parts,
from left to right:
1000
M = 900
H+<+Rtr+otrs
no So
wo ow
{fo}
WwW
fo)
ke
ll
N
Aren't you glad we don't use this system of numeration? Large
numbers are very difficult to denote this way, and the left vs. right /
subtraction vs. addition of values can be very confusing, too.
Another major problem with this system is that there is no provision
for representing the number zero or negative numbers, both very
important concepts in mathematics. Roman culture, however, was
more pragmatic with respect to mathematics than most, choosing
only to develop their numeration system as far as it was necessary
for use in daily life.
We owe one of the most important ideas in numeration to the
ancient Babylonians, who were the first (as far as we know) to
develop the concept of cipher position, or place value, in
representing larger numbers. Instead of inventing new ciphers to
represent larger numbers, as the Romans did, they re-used the same
ciphers, placing them in different positions from right to left. Our
own decimal numeration system uses this concept, with only ten
ciphers (0, 1, 2, 3, 4, 5, 6,7, 8, and 9) used in "weighted" positions
to represent very large and very small numbers.
Each cipher represents an integer quantity, and each place from
right to left in the notation represents a multiplying constant, or
weight, for each integer quantity. For example, if we see the decimal
notation "1206", we known that this may be broken down into its
constituent weight-products as such:
1206 = 1000 + 200 + 6
1206 = (1x 1000) + (2 x 100) + (0 x 10) + (6 x 1)
Each cipher is called a digit in the decimal numeration system, and
each weight, or place value, is ten times that of the one to the
immediate right. So, we have a ones place, a tens place, a hundreds
place, a thousands place, and so on, working from right to left.
Right about now, you're probably wondering why I'm laboring to
describe the obvious. Who needs to be told how decimal numeration
works, after you've studied math as advanced as algebra and
trigonometry? The reason is to better understand other numeration
systems, by first Knowing the how's and why's of the one you're
already used to.
The decimal numeration system uses ten ciphers, and place-weights
that are multiples of ten. What if we made a numeration system with
the same strategy of weighted places, except with fewer or more
ciphers?
The binary numeration system is such a system. Instead of ten
different cipher symbols, with each weight constant being ten times
the one before it, we only have two cipher symbols, and each weight
constant is twice as much as the one before it. The two allowable
cipher symbols for the binary system of numeration are "1" and "0,"
and these ciphers are arranged right-to-left in doubling values of
weight. The rightmost place is the ones place, just as with decimal
notation. Proceeding to the left, we have the twos place, the fours
place, the e/ghts place, the sixteens place, and so on. For example,
the following binary number can be expressed, just like the decimal
number 1206, as a sum of each cipher value times its respective
weight constant:
11010
11010
2 +8 + 16 = 26
(1 x 16) + (1 x 8) + (0 x 4) + (1 xX 2) + (0 x 1)
This can get quite confusing, as I've written a number with binary
numeration (11010), and then shown its place values and total in
standard, decimal numeration form (16 + 8 + 2 = 26). In the above
example, we're mixing two different kinds of numerical notation. To
avoid unnecessary confusion, we have to denote which form of
numeration we're using when we write (or type!). Typically, this is
done in subscript form, with a "2" for binary and a "10" for decimal,
so the binary number 11010, is equal to the decimal number 2640.
The subscripts are not mathematical operation symbols like
superscripts (exponents) are. All they do is indicate what system of
numeration we're using when we write these symbols for other
people to read. If you see "339", all this means is the number three
written using decima/ numeration. However, if you see "310", this
means something completely different: three to the tenth power
(59,049). As usual, if no subscript is shown, the cipher(s) are
assumed to be representing a decimal number.
Commonly, the number of cipher types (and therefore, the place-
value multiplier) used in a numeration system is called that system's
base. Binary is referred to as "base two" numeration, and decimal as
"base ten." Additionally, we refer to each cipher position in binary as
a bit rather than the familiar word digit used in the decimal system.
Now, why would anyone use binary numeration? The decimal
system, with its ten ciphers, makes a lot of sense, being that we have
ten fingers on which to count between our two hands. (It is
interesting that some ancient central American cultures used
numeration systems with a base of twenty. Presumably, they used
both fingers and toes to count!!). But the primary reason that the
binary numeration system is used in modern electronic computers is
because of the ease of representing two cipher states (0 and 1)
electronically. With relatively simple circuitry, we can perform
mathematical operations on binary numbers by representing each
bit of the numbers by a circuit which is either on (current) or off (no
current). Just like the abacus with each rod representing another
decimal digit, we simply add more circuits to give us more bits to
symbolize larger numbers. Binary numeration also lends itself well to
the storage and retrieval of numerical information: on magnetic tape
(spots of iron oxide on the tape either being magnetized for a binary
"L" or demagnetized for a binary "0"), optical disks (a laser-burned
pit in the aluminum foil representing a binary "1" and an unburned
Spot representing a binary "0"), or a variety of other media types.
Before we go on to learning exactly how all this is done in digital
circuitry, we need to become more familiar with binary and other
associated systems of numeration.
Decimal versus binary numeration
Let's count from zero to twenty using four different kinds of
numeration systems: hash marks, Roman numerals, decimal, and
binary:
System: Hash Marks Roman Decimal Binary
Zero n/a n/a 0 0
One | I 1 1
Two | | II 2 10
Three I || III 3 11
Four II] IV 4 100
Five /\\\/ V 5 101
Six J\II7 | VI 6 110
Seven JI II7 I VII 7 111
Eight J\II7 I VIII 8 1000
Nine J\II7 IIA IX 9 1001
Ten JIIIZ ZI IZ X 10 1010
Eleven IVE AAT XI 11 1011
Twelve AVE AL tl XII 12 1100
Thirteen /|||/ /||I7 [II XIII 13 1101
Fourteen /|||/ /|II7 [I] XIV 14 1110
Fifteen AVN AE A XV 15 1111
Sixteen JIIIZ ZIIIZ ZILIZ I XVI 16 10000
Seventeen /|||/ /|||/7 /|||/7 |] XVII 17 10001
Eighteen /|||/ /III7 ZII17 |1] XVIII Lo 10010
Nineteen /|||/ /||I/7 /|I1|7 JI] XIX 19 10011
Twenty JIIIZ ZUIIZ ZTIAZ ZU LNZ XX 20 10100
Neither hash marks nor the Roman system are very practical for
symbolizing large numbers. Obviously, place-weighted systems such
as decimal and binary are more efficient for the task. Notice, though,
how much shorter decimal notation is over binary notation, for the
Same number of quantities. What takes five bits in binary notation
only takes two digits in decimal notation.
This raises an interesting question regarding different numeration
systems: how large of a number can be represented with a limited
number of cipher positions, or places? With the crude hash-mark
system, the number of places IS the largest number that can be
represented, since one hash mark "place" is required for every
integer step. For place-weighted systems of numeration, however,
the answer is found by taking base of the numeration system (10 for
decimal, 2 for binary) and raising it to the power of the number of
places. For example, 5 digits in a decimal numeration system can
represent 100,000 different integer number values, from 0 to 99,999
(10 to the 5th power = 100,000). 8 bits in a binary numeration
system can represent 256 different integer number values, from 0 to
11111111 (binary), or 0 to 255 (decimal), because 2 to the 8th
power equals 256. With each additional place position to the number
field, the capacity for representing numbers increases by a factor of
the base (10 for decimal, 2 for binary).
An interesting footnote for this topic is the one of the first electronic
digital computers, the Eniac. The designers of the Eniac chose to
represent numbers in decimal form, digitally, using a series of
circuits called "ring counters" instead of just going with the binary
numeration system, in an effort to minimize the number of circuits
required to represent and calculate very large numbers. This
approach turned out to be counter-productive, and virtually all
digital computers since then have been purely binary in design.
To convert a number in binary numeration to its equivalent in
decimal form, all you have to do is calculate the sum of all the
products of bits with their respective place-weight constants. To
illustrate:
Convert 11001101, to decimal form:
bits = 11003131 #0éi421
weight = 163 1 8 4 2 #1
(in decimal 2 4 2 6
notation) 8
The bit on the far right side is called the Least Significant Bit (LSB),
because it stands in the place of the lowest weight (the one's place).
The bit on the far left side is called the Most Significant Bit (MSB),
because it stands in the place of the highest weight (the one
hundred twenty-eight's place). Remember, a bit value of "1" means
that the respective place weight gets added to the total value, anda
bit value of "0" means that the respective place weight does not get
added to the total value. With the above example, we have:
12816 + 6419 + 810 + 419 + lio = 20516
If we encounter a binary number with a dot (.), called a "binary
point" instead of a decimal point, we follow the same procedure,
realizing that each place weight to the right of the point is one-half
the value of the one to the left of it (just as each place weight to the
right of a decimal! point is one-tenth the weight of the one to the left
of it). For example:
Convert 101.011, to decimal form:
bits = 1031. 0 11
weight = 4 2 1 111
(in decimal / / f
notation) 2 4 8
410 + lio + 0.2519 + 0.12546 = 5.37546
Octal and hexadecimal numeration
Because binary numeration requires so many bits to represent
relatively small numbers compared to the economy of the decimal
system, analyzing the numerical states inside of digital electronic
circuitry can be a tedious task. Computer programmers who design
sequences of number codes instructing a computer what to do would
have a very difficult task if they were forced to work with nothing but
long strings of 1's and 0's, the "native language" of any digital
circuit. To make it easier for human engineers, technicians, and
programmers to "Speak" this language of the digital world, other
systems of place-weighted numeration have been made which are
very easy to convert to and from binary.
One of those numeration systems is called octa/, because it is a
place-weighted system with a base of eight. Valid ciphers include the
symbols 0, 1, 2, 3, 4, 5, 6, and 7. Each place weight differs from the
one next to it by a factor of eight.
Another system is called hexadecimal, because it is a place-weighted
system with a base of sixteen. Valid ciphers include the normal
decimal symbols 0, 1, 2, 3, 4, 5, 6,7, 8, and 9, plus six alphabetical
characters A, B, C, D, E, and F, to make a total of sixteen. As you
might have guessed already, each place weight differs from the one
before it by a factor of sixteen.
Let's count again from zero to twenty using decimal, binary, octal,
and hexadecimal to contrast these systems of numeration:
Number Decimal Binary Octal Hexadecimal
Zero 0 0 0 0
One 1 1 1 1
Two 2 10 2 2
Three 3 11 3 3
Four 4 100 4 4
Five 5 101 5 5
Six 6 110 6 6
Seven 7 111 7 7
Eight 8 1000 10 8
Nine 9 1001 11 9
Ten 10 1010 12 A
Eleven 11 1011 13 B
Twelve 12 1100 14 C
Thirteen 13 1101 15 D
Fourteen 14 1110 16 E
Fifteen 15 1111 17 F
Sixteen 16 10000 20 10
Seventeen 17 10001 21 11
Eighteen 18 10010 22 12
Nineteen 19 10011 23 13
Twenty 20 10100 24 14
Octal and hexadecimal numeration systems would be pointless if not
for their ability to be easily converted to and from binary notation.
Their primary purpose in being is to serve as a "Shorthand" method
of denoting a number represented electronically in binary form.
Because the bases of octal (eight) and hexadecimal (sixteen) are
even multiples of binary's base (two), binary bits can be grouped
together and directly converted to or from their respective octal or
hexadecimal digits. With octal, the binary bits are grouped in three's
(because 23 = 8), and with hexadecimal, the binary bits are grouped
in four's (because 24 = 16):
BINARY TO OCTAL CONVERSION
Convert 10110111.1, to octal:
implied zero implied zeros
|
010 110 111 100
Convert each group of bits HHH #HH ##H . HHH
to its octal equivalent: 2 6 7 4
Answer: 10110111.1, = 267.4,
We had to group the bits in three's, from the binary point left, and
from the binary point right, adding (implied) zeros as necessary to
make complete 3-bit groups. Each octal digit was translated from the
3-bit binary groups. Binary-to-Hexadecimal conversion is much the
same:
BINARY TO HEXADECIMAL CONVERSION
Convert 10110111.1, to hexadecimal:
implied zeros
|| |
; 1011 0111 1000
Convert each group of bits -- ---- yo rere
to its hexadecimal equivalent: B 7 8
Answer: 10110111.1, = B7.8i¢
Here we had to group the bits in four's, from the binary point left,
and from the binary point right, adding (implied) zeros as necessary
to make complete 4-bit groups:
Likewise, the conversion from either octal or hexadecimal to binary is
done by taking each octal or hexadecimal digit and converting it to
its equivalent binary (3 or 4 bit) group, then putting all the binary
bit groups together.
Incidentally, hexadecimal notation is more popular, because binary
bit groupings in digital equipment are commonly multiples of eight
(8, 16, 32, 64, and 128 bit), which are also multiples of 4. Octal,
being based on binary bit groups of 3, doesn't work out evenly with
those common bit group sizings.
Octal and hexadecimal to decimal
conversion
Although the prime intent of octal and hexadecimal numeration
systems is for the "shorthand" representation of binary numbers in
digital electronics, we sometimes have the need to convert from
either of those systems to decimal form. Of course, we could simply
convert the hexadecimal or octal format to binary, then convert from
binary to decimal, since we already know how to do both, but we can
also convert directly.
Because octal is a base-eight numeration system, each place-weight
value differs from either adjacent place by a factor of eight. For
example, the octal number 245.37 can be broken down into place
values as such:
octal
digits = 245 . 3 7
weight = 6 8 1 1 #1
(in decimal 4 / f/f
notation) 8 6
. 4
The decimal value of each octal place-weight times its respective
cipher multiplier can be determined as follows:
(2 xX 6449) + (4X 819) + (5 X lig) + (3 X 0.12549) +
(7 x @.01562519) = 165.48437549
The technique for converting hexadecimal notation to decimal is the
same, except that each successive place-weight changes by a factor
of sixteen. Simply denote each digit's weight, multiply each
hexadecimal digit value by its respective weight (in decimal form),
then add up all the decimal values to get a total. For example, the
hexadecimal number 30F.A91.¢ can be converted like this:
hexadecimal
digits = 3 0 F .A QY
weight = 2 1 £1 1 1
(in decimal 5 6 / f/f
notation) 6 1 2
(3 xX 256,59) + (0 X 1639) + (15 X 1y9) + (10 X 0.062535) +
(9 x 0.0039062519) = 783.6601562519
These basic techniques may be used to convert a numerical notation
of any base into decimal form, if you know the value of that
numeration system's base.
Conversion from decimal numeration
Because octal and hexadecimal numeration systems have bases that
are multiples of binary (base 2), conversion back and forth between
either hexadecimal or octal and binary is very easy. Also, because we
are so familiar with the decimal system, converting binary, octal, or
hexadecimal to decimal form is relatively easy (simply add up the
products of cipher values and place-weights). However, conversion
from decimal to any of these "strange" numeration systems is a
different matter.
The method which will probably make the most sense is the "trial-
and-fit" method, where you try to "fit" the binary, octal, or
hexadecimal notation to the desired value as represented in decimal
form. For example, let's say that | wanted to represent the decimal
value of 87 in binary form. Let's start by drawing a binary number
field, complete with place-weight values:
weight =
(in decimal
notation)
CONF !
RO
NW!
Or!
Well, we know that we won't have a "1" bit in the 128's place,
because that would immediately give us a value greater than 87.
However, since the next weight to the right (64) is less than 87, we
know that we must have a "1" there.
1
: - = - Decimal value so far = 64j9
weight = 6 3 1 8 4 2 1
(in decimal 4 2 6
notation)
If we were to make the next place to the right a "1" as well, our total
value would be 64,9 + 3249, Or 96j0. This is greater than 8719, So we
know that this bit must be a "0". If we make the next (16's) place bit
equal to "1," this brings our total value to 64,9 + 1640, or 8040,
which is closer to our desired value (873,) without exceeding it:
101
P - = eee Decimal value so far = 809
weight = 6 3 1 8 4 2 1
(in decimal 4 2 6
notation)
By continuing in this progression, setting each lesser-weight bit as
we need to come up to our desired total value without exceeding it,
we will eventually arrive at the correct figure:
F Decimal value so far = 87j9
weight = 6
(in decimal 4
notation)
This trial-and-fit strategy will work with octal and hexadecimal
conversions, too. Let's take the same decimal figure, 8749, and
convert it to octal numeration:
weight = 6 <B> 4
(in decimal 4
notation)
If we put a cipher of "1" in the 64's place, we would have a total
value of 6449 (less than 87,0). If we put a cipher of "2" in the 64's
place, we would have a total value of 128) (greater than 8749). This
tells us that our octal numeration must start with a"1" in the 64's
place:
1
; “ Decimal value so far = 64y9
weight = 6) 38u. FL
(in decimal 4
notation)
Now, we need to experiment with cipher values in the 8's place to try
and get a total (decimal) value as close to 87 as possible without
exceeding it. Trying the first few cipher options, we get:
aed By 6446 + 816 = 72146
nae 6419 + 1619 8019
"3" = 6410 + 2410 = 8819
A cipher value of "3" in the 8's place would put us over the desired
total of 8745, so "2" it is!
1 2
weight = 6 8 1
(in decimal 4
notation)
Decimal value so far = 809
Now, all we need to make a total of 87 is a cipher of "7" in the L's
place:
12 7
weight = 6 8 1
(in decimal 4
notation)
Decimal value so far = 87 49
Of course, if you were paying attention during the last section on
octal/binary conversions, you will realize that we can take the binary
representation of (decimal) 8739, which we previously determined to
be 10101113, and easily convert from that to octal to check our
work:
Implied zeros
| |
001 010 111 Binary
1 Z 7 Octal
Answer: 1010111, = 127,
Can we do decimal-to-hexadecimal conversion the same way? Sure,
but who would want to? This method is simple to understand, but
laborious to carry out. There is another way to do these conversions,
which is essentially the same (mathematically), but easier to
accomplish.
This other method uses repeated cycles of division (using decimal
notation) to break the decimal numeration down into multiples of
binary, octal, or hexadecimal place-weight values. In the first cycle
of division, we take the original decimal number and divide it by the
base of the numeration system that we're converting to (binary=2
octal=8, hex=16). Then, we take the whole-number portion of
division result (quotient) and divide it by the base value again, and
so on, until we end up with a quotient of less than 1. The binary,
octal, or hexadecimal digits are determined by the "remainders" left
over by each division step. Let's see how this works for binary, with
the decimal example of 870:
87 Divide 87 by 2, to get a quotient of 43.5
— = 43.5 Division "remainder" = 1, or the < 1 portion
2 of the quotient times the divisor (0.5 x 2)
43 Take the whole-number portion of 43.5 (43)
— = 21.5 and divide it by 2 to get 21.5, or 21 with
2 a remainder of 1
21 And so on... remainder = 1 (0.5 x 2)
— = 10.5
2
10 And so on... . remainder = 0
— = 5.0
2
5 And so on. . .. remainder = 1 (0.5 x 2)
—-=2.5
2
2 And so on... . remainder = 0
—-= 1.0
2
A . . . until we get a quotient of less than 1
—= 0.5 remainder = 1 (0.5 x 2)
2
The binary bits are assembled from the remainders of the successive
division steps, beginning with the LSB and proceeding to the MSB. In
this case, we arrive at a binary notation of 10101115. When we
divide by 2, we will always get a quotient ending with either ".0" or
"5", i.e. a remainder of either 0 or 1. As was said before, this repeat-
division technique for conversion will work for numeration systems
other than binary. If we were to perform successive divisions using a
different number, such as 8 for conversion to octal, we will
necessarily get remainders between 0 and 7. Let's try this with the
same decimal number, 87 40:
87 Divide 87 by 8, to get a quotient of 10.875
— = 10.875 Division "remainder" = 7, or the < 1 portion
8 of the quotient times the divisor (.875 x 8)
— = 1.25 Remainder = 2
8
1
— = 0.125 Quotient is less than 1, so we'll stop here.
8 Remainder = 1
RESULT: 8719 = 127.8
We can use a similar technique for converting numeration systems
dealing with quantities less than 1, as well. For converting a decimal
number less than 1 into binary, octal, or hexadecimal, we use
repeated multiplication, taking the integer portion of the product in
each step as the next digit of our converted number. Let's use the
decimal number 0.812549 as an example, converting to binary:
0.8125 x 2 = 1.625 Integer portion of product = 1
0.625 x 2 = 1.25 Take < 1 portion of product and remultiply
Integer portion of product = 1
0.25 x 2 = 0.5 Integer portion of product = 0
0.5 x 2 = 1.0 Integer portion of product = 1
Stop when product is a pure integer
(ends with .0)
RESULT: 0.8125, = 0.1101;
As with the repeat-division process for integers, each step gives us
the next digit (or bit) further away from the "point." With integer
(division), we worked from the LSB to the MSB (right-to-left), but
with repeated multiplication, we worked from the left to the right. To
convert a decimal number greater than 1, with a < 1 component, we
must use both techniques, one at a time. Take the decimal example
of 54.40625,9, converting to binary:
REPEATED DIVISION FOR THE INTEGER PORTION:
54
— = 27.0 Remainder = 0
2
27
-— = 13.5 Remainder = 1 (0.5 x 2)
2
13
—-=6.5 Remainder = 1 (0.5 x 2)
2
6
—- = 3.0 Remainder = 0
2
3
—-=1.5 Remainder = 1 (0.5 x 2)
2
1
—-=0.5 Remainder = 1 (0.5 x 2)
2
PARTIAL ANSWER: 54,9 = 1101105
REPEATED MULTIPLICATION FOR THE < 1 PORTION:
0.40625 x 2 = 0.8125 Integer portion of product
0.8125 x 2 = 1.625 Integer portion of product
0.625 x 2 = 1.25 Integer portion of product
0.25 x 2 = 0.5 Integer portion of product
0.5 x 2 = 1.0 Integer portion of product
PARTIAL ANSWER: 0.406259 = 0.01101;
COMPLETE ANSWER: 5419 + 0.4062519 = 54.4062519
1101105 + 0.01101, = 110110.01101,
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design Science
License.
a 4 —>
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 2
BINARY ARITHMETIC
e Numbers versus numeration
Binary addition
Negative binary numbers
¢ Subtraction
e Overflow
¢ Bit groupings
Numbers versus numeration
It is imperative to understand that the type of numeration
system used to represent numbers has no impact upon the
outcome of any arithmetical function (addition, subtraction,
multiplication, division, roots, powers, or logarithms). A
number is a number is a number; one plus one will always
equal two (so long as we're dealing with rea/numbers), no
matter how you symbolize one, one, and two. A prime
number in decimal form is still prime if its shown in binary
form, or octal, or hexadecimal. tt is still the ratio between the
circumference and diameter of a circle, no matter what
symbol(s) you use to denote its value. The essential
functions and interrelations of mathematics are unaffected
by the particular system of symbols we might choose to
represent quantities. This distinction between numbers and
systems of numeration is critical to understand.
The essential distinction between the two is much like that
between an object and the spoken word(s) we associate with
it. A house is still a house regardless of whether we call it by
its English name house or its Spanish name casa. The first is
the actual thing, while the second is merely the symbol for
the thing.
That being said, performing a simple arithmetic operation
such as addition (longhand) in binary form can be confusing
to a person accustomed to working with decimal numeration
only. In this lesson, we'll explore the techniques used to
perform simple arithmetic functions on binary numbers,
since these techniques will be employed in the design of
electronic circuits to do the same. You might take longhand
addition and subtraction for granted, having used a
calculator for so long, but deep inside that calculator's
circuitry all those operations are performed "longhand,"
using binary numeration. To understand how that's
accomplished, we need to review to the basics of arithmetic.
Binary addition
Adding binary numbers is a very simple task, and very
similar to the longhand addition of decimal numbers. As
with decimal numbers, you start by adding the bits (digits)
one column, or place weight, at a time, from right to left.
Unlike decimal addition, there is little to memorize in the
way of rules for the addition of binary bits:
PrOFrS®e
+++ 4+
PrrROO®
+ oH oi ot oll
PrrPro®
Just as with decimal addition, when the sum in one column
is a two-bit (two-digit) number, the least significant figure is
written as part of the total sum and the most significant
figure is "carried" to the next left column. Consider the
following examples:
11 1 <--- Carry bits ----- > 11
1001101 1001001 1000111
+ 0010010 + 0011001 + 0010110
1011111 1100010 1011101
The addition problem on the left did not require any bits to
be carried, since the sum of bits in each column was either 1
or 0, not 10 or 11. In the other two problems, there definitely
were bits to be carried, but the process of addition is still
quite simple.
As we'll see later, there are ways that electronic circuits can
be built to perform this very task of addition, by
representing each bit of each binary number as a voltage
signal (either "high," for a 1; or "low" for a OQ). This is the
very foundation of all the arithmetic which modern digital
computers perform.
Negative binary numbers
With addition being easily accomplished, we can perform
the operation of subtraction with the same technique simply
by making one of the numbers negative. For example, the
subtraction problem of 7 - 5 is essentially the same as the
addition problem 7 + (-5). Since we already know how to
represent positive numbers in binary, all we need to know
now is how to represent their negative counterparts and
we'll be able to subtract.
Usually we represent a negative decimal number by placing
a minus sign directly to the left of the most significant digit,
just as in the example above, with -5. However, the whole
purpose of using binary notation is for constructing on/off
circuits that can represent bit values in terms of voltage (2
alternative values: either "high" or "low"). In this context, we
don't have the luxury of a third symbol such as a "minus"
sign, since these circuits can only be on or off (two possible
states). One solution is to reserve a bit (circuit) that does
nothing but represent the mathematical sign:
101, = 5i9 (positive)
Extra bit, representing sign (0=positive, l=negative)
01015 = 549 (positive)
Extra bit, representing sign (0=positive, l=negative)
1101, = -5i9 (negative)
As you can see, we have to be careful when we start using
bits for any purpose other than standard place-weighted
values. Otherwise, 1101, could be misinterpreted as the
number thirteen when in fact we mean to represent negative
five. To keep things straight here, we must first decide how
many bits are going to be needed to represent the largest
numbers we'll be dealing with, and then be sure not to
exceed that bit field length in our arithmetic operations. For
the above example, I've limited myself to the representation
of numbers from negative seven (1111,) to positive seven
(0111;5), and no more, by making the fourth bit the "sign"
bit. Only by first establishing these limits can | avoid
confusion of a negative number with a larger, positive
number.
Representing negative five as 1101; is an example of the
sign-magnitude system of negative binary numeration. By
using the leftmost bit as a sign indicator and not a place-
weighted value, | am sacrificing the "pure" form of binary
notation for something that gives me a practical advantage:
the representation of negative numbers. The leftmost bit is
read as the sign, either positive or negative, and the
remaining bits are interpreted according to the standard
binary notation: left to right, place weights in multiples of
two.
As simple as the sign-magnitude approach is, it is not very
practical for arithmetic purposes. For instance, how do | add
a negative five (11015) to any other number, using the
standard technique for binary addition? I'd have to invent a
new way of doing addition in order for it to work, and if | do
that, | might as well just do the job with longhand
subtraction; there's no arithmetical advantage to using
negative numbers to perform subtraction through addition if
we have to do it with sign-magnitude numeration, and that
was our goal!
There's another method for representing negative numbers
which works with our familiar technique of longhand
addition, and also happens to make more sense from a
place-weighted numeration point of view, called
complementation. With this strategy, we assign the leftmost
bit to serve a special purpose, just as we did with the sign-
magnitude approach, defining our number limits just as
before. However, this time, the leftmost bit is more than just
a sign bit; rather, it possesses a negative place-weight
value. For example, a value of negative five would be
represented as such:
Extra bit, place weight = negative eight
|
10115 = 549 (negative)
(1 x -839) + (0 X 449) + (1 X 239) + (1 X IQ)
With the right three bits being able to represent a
magnitude from zero through seven, and the leftmost bit
representing either zero or negative eight, we can
successfully represent any integer number from negative
seven (1001, = -839 + 119 = -719) to positive seven (0111,
= 010 + 710 = 710):
Representing positive numbers in this scheme (with the
fourth bit designated as the negative weight) is no different
from that of ordinary binary notation. However, representing
negative numbers is not quite as straightforward:
zero 0000
positive one 0001 negative one 1111
positive two 0010 negative two 1110
positive three 0011 negative three 1101
positive four 0100 negative four 1100
positive five 0101 negative five 1011
positive six 0110 negative six 1010
positive seven 0111 negative seven 1001
negative eight 1000
Note that the negative binary numbers in the right column,
being the sum of the right three bits' total plus the negative
eight of the leftmost bit, don't "count" in the same
progression as the positive binary numbers in the left
column. Rather, the right three bits have to be set at the
proper value to equal the desired (negative) total when
summed with the negative eight place value of the leftmost
bit.
Those right three bits are referred to as the two's
complement of the corresponding positive number. Consider
the following comparison:
positive number two's complement
001 Lit
010 110
011 101
100 100
101 011
110 010
In this case, with the negative weight bit being the fourth bit
(place value of negative eight), the two's complement for
any positive number will be whatever value is needed to add
to negative eight to make that positive value's negative
equivalent. Thankfully, there's an easy way to figure out the
two's complement for any binary number: simply invert all
the bits of that number, changing all 1's to 0's and vice
versa (to arrive at what is called the one's complement) and
then add one! For example, to obtain the two's complement
of five (1015), we would first invert all the bits to obtain
010, (the "one's complement"), then add one to obtain
0115, or -5,9 in three-bit, two's complement form.
Interestingly enough, generating the two's complement of a
binary number works the same if you manipulate a// the bits,
including the leftmost (sign) bit at the same time as the
magnitude bits. Let's try this with the former example,
converting a positive five to a negative five, but performing
the complementation process on all four bits. We must be
sure to include the O (positive) sign bit on the original
number, five (01013). First, inverting all bits to obtain the
one's complement: 10105. Then, adding one, we obtain the
final answer: 10115, or -5;9 expressed in four-bit, two's
complement form.
It is critically important to remember that the place of the
negative-weight bit must be already determined before any
two's complement conversions can be done. If our binary
numeration field were such that the eighth bit was
designated as the negative-weight bit (10000000>;), we'd
have to determine the two's complement based on all seven
of the other bits. Here, the two's complement of five
(0000101-,) would be 1111011,. A positive five in this
system would be represented as 00000101;, and a negative
five as 11111011).
Subtraction
We can subtract one binary number from another by using
the standard techniques adapted for decimal numbers
(subtraction of each bit pair, right to left, "borrowing" as
needed from bits to the left). However, if we can leverage
the already familiar (and easier) technique of binary
addition to subtract, that would be better. As we just
learned, we can represent negative binary numbers by using
the "two's complement" method and a negative place-
weight bit. Here, we'll use those negative binary numbers to
subtract through addition. Here's a sample problem:
Subtraction: 719 - 549 Addition equivalent: 7j9 +
(-519)
If all we need to do is represent seven and negative five in
binary (two's complemented) form, all we need is three bits
plus the negative-weight bit:
positive seven
negative five
0111,
1011,
Now, let's add them together:
1111 <--- Carry bits
Discard extra bit
Answer = 00105
Since we've already defined our number bit field as three
bits plus the negative-weight bit, the fifth bit in the answer
(1) will be discarded to give us a result of 00105, or positive
two, which is the correct answer.
Another way to understand why we discard that extra bit is
to remember that the leftmost bit of the lower number
possesses a negative weight, in this case equal to negative
eight. When we add these two binary numbers together,
what we're actually doing with the MSBs is subtracting the
lower number's MSB from the upper number's MSB. In
subtraction, one never "carries" a digit or bit on to the next
left place-weight.
Let's try another example, this time with larger numbers. If
we want to add -25,, to 18,5, we must first decide how large
our binary bit field must be. To represent the largest
(absolute value) number in our problem, which is twenty-
five, we need at least five bits, plus a sixth bit for the
negative-weight bit. Let's start by representing positive
twenty-five, then finding the two's complement and putting
it all together into one numeration:
+2539 = 011001, (Showing all six bits)
One's complement of 110015, = 100110,
One's complement + 1 = two's complement = 100111,
-2519 = 100111,
Essentially, we're representing negative twenty-five by
using the negative-weight (sixth) bit with a value of
negative thirty-two, plus positive seven (binary 111,).
Now, let's represent positive eighteen in binary form,
showing all six bits:
1819 = 0100105
Now, let's add them together and see what we get:
11 <--- Carry bits
100111
+ 010010
111001
Since there were no "extra" bits on the left, there are no bits
to discard. The leftmost bit on the answer is a 1, which
means that the answer is negative, in two's complement
form, as it should be. Converting the answer to decimal form
by summing all the bits times their respective weight values,
we get:
(1 xX - 3219) + (1 xX 1619) + (1 x 819) + (1 x lio) = -716
Indeed -719 is the proper sum of -25 9 and 18).
Overflow
One caveat with signed binary numbers is that of overflow,
where the answer to an addition or subtraction problem
exceeds the magnitude which can be represented with the
alloted number of bits. Remember that the place of the sign
bit is fixed from the beginning of the problem. With the last
example problem, we used five binary bits to represent the
magnitude of the number, and the left-most (sixth) bit as
the negative-weight, or sign, bit. With five bits to represent
magnitude, we have a representation range of 2°, or thirty-
two integer steps from 0 to maximum. This means that we
can represent a number as high as +31j,9 (011111,), or as
low as -323, (1000003). If we set up an addition problem
with two binary numbers, the sixth bit used for sign, and the
result either exceeds +3149 or is less than -32,,9, our answer
will be incorrect. Let's try adding 17,9 and 19, to see how
this overflow condition works for excessive positive
numbers:
1716 = 10001, 1949 = 10011,
1 11 <--- Carry bits
(Showing sign bits) 010001
+ 010011
100100
The answer (1001003), interpreted with the sixth bit as the
-3210 place, is actually equal to -28 5, not +369 as we
should get with +17) and +19, added together!
Obviously, this is not correct. What went wrong? The answer
lies in the restrictions of the six-bit number field within
which we're working Since the magnitude of the true and
proper sum (36,9) exceeds the allowable limit for our
designated bit field, we have an overflow error. Simply put,
six places doesn't give enough bits to represent the correct
sum, so whatever figure we obtain using the strategy of
discarding the left-most "carry" bit will be incorrect.
A similar error will occur if we add two negative numbers
together to produce a sum that is too low for our six-bit
binary field. Let's try adding -17 9 and -19,, together to see
how this works (or doesn't work, as the case may be!):
-1749 = 101111, -1949 = 101101,
11111 <--- Carry bits
(Showing sign bits) 101111
+ 101101
1011100
Discard extra bit
FINAL ANSWER: 011100, = +2849
The (incorrect) answer is a positive twenty-eight. The fact
that the real sum of negative seventeen and negative
nineteen was too low to be properly represented with a five
bit magnitude field and a sixth sign bit is the root cause of
this difficulty.
Let's try these two problems again, except this time using
the seventh bit for a sign bit, and allowing the use of 6 bits
for representing the magnitude:
1719 + 1916 (-1719) + (-1949)
1 141 11 1111
0010001 1101111
+ 0010011 + 1101101
0100100, 11011100,
. ANSWERS: 0100100,
1011100,
+3619
- 3619
Discard extra bit
By using bit fields sufficiently large to handle the magnitude
of the sums, we arrive at the correct answers.
In these sample problems we've been able to detect
overflow errors by performing the addition problems in
decimal form and comparing the results with the binary
answers. For example, when adding +17 j9 and +1949
together, we knew that the answer was supposed to be
+3619, so when the binary sum checked out to be -28)9, we
knew that something had to be wrong. Although this is a
valid way of detecting overflow, it is not very efficient. After
all, the whole idea of complementation is to be able to
reliably add binary numbers together and not have to
double-check the result by adding the same numbers
together in decimal form! This is especially true for the
purpose of building electronic circuits to add binary
quantities together: the circuit has to be able to check itself
for overflow without the supervision of a human being who
already knows what the correct answer is.
What we need is a simple error-detection method that
doesn't require any additional arithmetic. Perhaps the most
elegant solution is to check for the sign of the sum and
compare it against the signs of the numbers added.
Obviously, two positive numbers added together should give
a positive result, and two negative numbers added together
should give a negative result. Notice that whenever we had
a condition of overflow in the example problems, the sign of
the sum was always opposite of the two added numbers:
+1749 plus +1949 giving -28 0, or -1749 plus -1939 giving
+281). By checking the signs alone we are able to tell that
something is wrong.
But what about cases where a positive number is added toa
negative number? What sign should the sum be in order to
be correct. Or, more precisely, what sign of sum would
necessarily indicate an overflow error? The answer to this is
equally elegant: there will never be an overflow error when
two numbers of opposite signs are added together! The
reason for this is apparent when the nature of overflow is
considered. Overflow occurs when the magnitude of a
number exceeds the range allowed by the size of the bit
field. The sum of two identically-signed numbers may very
well exceed the range of the bit field of those two numbers,
and so in this case overflow is a possibility. However, if a
positive number is added to a negative number, the sum will
always be closer to zero than either of the two added
numbers: its magnitude must be less than the magnitude of
either original number, and so overflow is impossible.
Fortunately, this technique of overflow detection is easily
implemented in electronic circuitry, and it is a standard
feature in digital adder circuits: a subject for a later chapter.
Bit groupings
The singular reason for learning and using the binary
numeration system in electronics is to understand how to
design, build, and troubleshoot circuits that represent and
process numerical quantities in digital form. Since the
bivalent (two-valued) system of binary bit numeration lends
itself so easily to representation by "on" and "off" transistor
states (saturation and cutoff, respectively), it makes sense
to design and build circuits leveraging this principle to
perform binary calculations.
If we were to build a circuit to represent a binary number, we
would have to allocate enough transistor circuits to
represent as many bits as we desire. In other words, in
designing a digital circuit, we must first decide how many
bits (maximum) we would like to be able to represent, since
each bit requires one on/off circuit to represent it. This is
analogous to designing an abacus to digitally represent
decimal numbers: we must decide how many digits we wish
to handle in this primitive "calculator" device, for each digit
requires a separate rod with its own beads.
A 10-rod abacus
NATE
Each rod represents
a single decimal digit
A ten-rod abacus would be able to represent a ten-digit
decimal number, or a maxmium value of 9,999,999,999. If
we wished to represent a larger number on this abacus, we
would be unable to, unless additional rods could be added to
it.
In digital, electronic computer design, it is common to
design the system for a common "bit width:" a maximum
number of bits allocated to represent numerical quantities.
Early digital computers handled bits in groups of four or
eight. More modern systems handle numbers in clusters of
32 bits or more. To more conveniently express the "bit
width" of such clusters in a digital computer, specific labels
were applied to the more common groupings.
Eight bits, grouped together to form a single binary
quantity, is known as a byte. Four bits, grouped together as
one binary number, is known by the humorous title of
nibble, often spelled as nybble.
A multitude of terms have followed byte and nibble for
labeling specfiic groupings of binary bits. Most of the terms
shown here are informal, and have not been made
“authoritative" by any standards group or other sanctioning
body. However, their inclusion into this chapter is warranted
by their occasional appearance in technical literature, as
well as the levity they add to an otherwise dry subject:
e Bit: A single, bivalent unit of binary notation. Equivalent
to a decimal "digit."
Crumb, Tydbit, or Tayste: Two bits.
Nibble, or Nybble: Four bits.
Nickle: Five bits.
Byte: Eight bits.
Deckle: Ten bits.
Playte: Sixteen bits.
Dynner: Thirty-two bits.
Word: (system dependent).
The most ambiguous term by far is word, referring to the
standard bit-grouping within a particular digital system. For
a computer system using a 32 bit-wide "data path," a "word"
would mean 32 bits. If the system used 16 bits as the
standard grouping for binary quantities, a "word" would
mean 16 bits. The terms playte and dynner, by contrast,
always refer to 16 and 32 bits, respectively, regardless of the
system context in which they are used.
Context dependence is likewise true for derivative terms of
word, such as double word and longword (both meaning
twice the standard bit-width), ha/f-word (half the standard
bit-width), and quad (meaning four times the standard bit-
width). One humorous addition to this somewhat boring
collection of word-derivatives is the term chawmp, which
means the same as hal/f-word. For example, a chawmp would
be 16 bits in the context of a 32-bit digital system, and 18
bits in the context of a 36-bit system. Also, the term gawble
is sometimes synonymous with word.
Definitions for bit grouping terms were taken from Eric S.
Raymond's "Jargon Lexicon," an indexed collection of terms -
- both common and obscure -- germane to the world of
computer programming.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+4/l—
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 3
LOGIC GATES
Digital signals and gates
The NOT gate
The "buffer" gate
Multiple-input gates
o The AND gate
o The NAND gate
o The OR gate
The NOR gate
The Negative-AND gate
The Negative-OR gate
The Exclusive-OR gate
The Exclusive-NOR gate
TTL NAND and AND gates
TTL NOR and OR gates
CMOS gate circuitry
Special-output gates
Gate universality
o Constructing the NOT function
o Constructing the "buffer" function
o Constructing the AND function
o Constructing the NAND function
o Oo 0 0 O
Constructing the OR function
Constructing the NOR function
Logic signal voltage levels
DIP gate packaging
Contributors
Digital signals and gates
While the binary numeration system is an interesting
mathematical abstraction, we haven't yet seen its practical
application to electronics. This chapter is devoted to just
that: practically applying the concept of binary bits to
circuits. What makes binary numeration so important to the
application of digital electronics is the ease in which bits
may be represented in physical terms. Because a binary bit
can only have one of two different values, either 0 or 1, any
physical medium capable of switching between two
saturated states may be used to represent a bit.
Consequently, any physical system capable of representing
binary bits is able to represent numerical quantities, and
potentially has the ability to manipulate those numbers.
This is the basic concept underlying digital computing.
Electronic circuits are physical systems that lend themselves
well to the representation of binary numbers. Transistors,
when operated at their bias limits, may be in one of two
different states: either cutoff (no controlled current) or
saturation (maximum controlled current). If a transistor
circuit is designed to maximize the probability of falling into
either one of these states (and not operating in the linear, or
active, mode), it can serve as a physical representation of a
binary bit. A voltage signal measured at the output of sucha
circuit may also serve as a representation of a single bit, a
low voltage representing a binary "0" and a (relatively) high
voltage representing a binary "1." Note the following
transistor circuit:
Transistor in saturation
"high" input > "low" output
0 V = "low" logic level (0)
5 V ="high" logic level (1)
In this circuit, the transistor is in a state of saturation by
virtue of the applied input voltage (5 volts) through the two-
position switch. Because its saturated, the transistor drops
very little voltage between collector and emitter, resulting in
an output voltage of (practically) O volts. If we were using
this circuit to represent binary bits, we would say that the
input signal is a binary "L" and that the output signal is a
binary "0." Any voltage close to full supply voltage
(measured in reference to ground, of course) is considered a
"L" and a lack of voltage is considered a "0." Alternative
terms for these voltage levels are high (same as a binary
"L") and /jow (same as a binary "0"). A general term for the
representation of a binary bit by a circuit voltage is /ogic
level.
Moving the switch to the other position, we apply a binary
"0" to the input and receive a binary "1" at the output:
Transistor in cutoff
"low" input = "high" output
0 V = "low" logic level (0)
5 V ="high" logic level (1)
What we've created here with a single transistor is a circuit
generally known as a /ogic gate, or simply gate. A gateisa
special type of amplifier circuit designed to accept and
generate voltage signals corresponding to binary 1's and
O's. As such, gates are not intended to be used for
amplifying analog signals (voltage signals between 0 and
full voltage). Used together, multiple gates may be applied
to the task of binary number storage (memory circuits) or
manipulation (computing circuits), each gate's output
representing one bit of a multi-bit binary number. Just how
this is done is a subject for a later chapter. Right now it is
important to focus on the operation of individual gates.
The gate shown here with the single transistor is known as
an inverter, or NOT gate, because it outputs the exact
opposite digital signal as what is input. For convenience,
gate circuits are generally represented by their own symbols
rather than by their constituent transistors and resistors. The
following is the symbol for an inverter:
Inverter, or NOT gate
Input > Output
An alternative symbol for an inverter is shown here:
Input —l>— Output
Notice the triangular shape of the gate symbol, much like
that of an operational amplifier. As was stated before, gate
circuits actually are amplifiers. The small circle, or "bubble"
shown on either the input or output terminal is standard for
representing the inversion function. As you might suspect, if
we were to remove the bubble from the gate symbol, leaving
only a triangle, the resulting symbol would no longer
indicate inversion, but merely direct amplification. Such a
symbol and such a gate actually do exist, and it is called a
buffer, the subject of the next section.
Like an operational amplifier symbol, input and output
connections are shown as single wires, the implied reference
point for each voltage signal being "ground." In digital gate
circuits, ground is almost always the negative connection of
a single voltage source (power supply). Dual, or "split,"
power supplies are seldom used in gate circuitry. Because
gate circuits are amplifiers, they require a source of power to
operate. Like operational amplifiers, the power supply
connections for digital gates are often omitted from the
symbol for simplicity's sake. If we were to show a// the
necessary connections needed for operating this gate, the
schematic would look something like this:
— Ground
Power supply conductors are rarely shown in gate circuit
schematics, even if the power supply connections at each
gate are. Minimizing lines in our schematic, we get this:
Vv C ba
ce
TL
"V.¢" stands for the constant voltage supplied to the
collector of a bipolar junction transistor circuit, in reference
to ground. Those points in a gate circuit marked by the label
"Vcc" are all connected to the same point, and that point is
the positive terminal of a DC voltage source, usually 5 volts.
As we will see in other sections of this chapter, there are
quite a few different types of logic gates, most of which have
multiple input terminals for accepting more than one signal.
The output of any gate is dependent on the state of its
input(s) and its logical function.
One common way to express the particular function of a
gate circuit is called a truth table. Truth tables show all
combinations of input conditions in terms of logic level
states (either "high" or "low," "1" or "0," for each input
terminal of the gate), along with the corresponding output
logic level, either "high" or "low." For the inverter, or NOT,
circuit just illustrated, the truth table is very simple indeed:
NOT gate truth table
Input {>e Output
eae ee
ie a ae:
Truth tables for more complex gates are, of course, larger
than the one shown for the NOT gate. A gate's truth table
must have as many rows as there are possibilities for unique
input combinations. For a single-input gate like the NOT
gate, there are only two possibilities, 0 and 1. For a two
input gate, there are four possibilities (00, 01,10, and 11),
and thus four rows to the corresponding truth table. Fora
three-input gate, there are eight possibilities (000, 001, 010,
011, 100, 101, 110, and 111), and thus a truth table with
eight rows are needed. The mathematically inclined will
realize that the number of truth table rows needed for a gate
iS equal to 2 raised to the power of the number of input
terminals.
e REVIEW:
e In digital circuits, binary bit values of 0 and 1 are
represented by voltage signals measured in reference to
a common circuit point called ground. An absence of
voltage represents a binary "0" and the presence of full
DC supply voltage represents a binary "1."
e A logic gate, or simply gate, is a special form of amplifier
circuit designed to input and output /ogic leve/ voltages
(voltages intended to represent binary bits). Gate
circuits are most commonly represented in a schematic
by their own unique symbols rather than by their
constituent transistors and resistors.
e Just as with operational amplifiers, the power supply
connections to gates are often omitted in schematic
diagrams for the sake of simplicity.
e A truth table is a standard way of representing the
input/output relationships of a gate circuit, listing all the
possible input logic level combinations with their
respective output logic levels.
The NOT gate
The single-transistor inverter circuit illustrated earlier is
actually too crude to be of practical use as a gate. Real
inverter circuits contain more than one transistor to
maximize voltage gain (so as to ensure that the final output
transistor is either in full cutoff or full saturation), and other
components designed to reduce the chance of accidental
damage.
Shown here is a schematic diagram for a real inverter circuit,
complete with all necessary components for efficient and
reliable operation:
Practical inverter (NOT) circuit
Input
Output
This circuit is composed exclusively of resistors, diodes and
bipolar transistors. Bear in mind that other circuit designs
are capable of performing the NOT gate function, including
designs substituting field-effect transistors for bipolar
(discussed later in this chapter).
Let's analyze this circuit for the condition where the input is
"high," or in a binary "1" state. We can simulate this by
showing the input terminal connected to V,, through a
switch:
V.c = 5 volts
In this case, diode D, will be reverse-biased, and therefore
not conduct any current. In fact, the only purpose for having
D, in the circuit is to prevent transistor damage in the case
of a negative voltage being impressed on the input (a
voltage that is negative, rather than positive, with respect to
ground). With no voltage between the base and emitter of
transistor Q,, we would expect no current through it, either.
However, as strange as it may seem, transistor Q, is not
being used as is customary for a transistor. In reality, Q, is
being used in this circuit as nothing more than a back-to-
back pair of diodes. The following schematic shows the real
function of Q;:
V.c = 5 volts
The purpose of these diodes is to "steer" current to or away
from the base of transistor Q5, depending on the logic level
of the input. Exactly how these two diodes are able to
"steer" current isn't exactly obvious at first inspection, soa
short example may be necessary for understanding.
Suppose we had the following diode/resistor circuit,
representing the base-emitter junctions of transistors Q5 and
Q, as single diodes, stripping away all other portions of the
circuit so that we can concentrate on the current "steered"
through the two back-to-back diodes:
With the input switch in the "up" position (connected to V,,),
it should be obvious that there will be no current through
the left steering diode of Q,, because there isn't any voltage
in the switch-diode-R,-switch loop to motivate electrons to
flow. However, there wi// be current through the right
steering diode of Q), as well as through Q>'s base-emitter
diode junction and Q,'s base-emitter diode junction:
This tells us that in the real gate circuit, transistors Q> and
Q, will have base current, which will turn them on to
conduct collector current. The total voltage dropped
between the base of Q, (the node joining the two back-to-
back steering diodes) and ground will be about 2.1 volts,
equal to the combined voltage drops of three PN junctions:
the right steering diode, Q's base-emitter diode, and Q,'s
base-emitter diode.
Now, let's move the input switch to the "down" position and
see what happens:
If we were to measure current in this circuit, we would find
that a// of the current goes through the left steering diode of
Q, and none of it through the right diode. Why is this? It still
appears as though there is a complete path for current
through Q,'s diode, Q,'s diode, the right diode of the pair,
and Rj, so why will there be no current through that path?
Remember that PN junction diodes are very nonlinear
devices: they do not even begin to conduct current until the
forward voltage applied across them reaches a certain
minimum quantity, approximately 0.7 volts for silicon and
0.3 volts for germanium. And then when they begin to
conduct current, they will not drop substantially more than
0.7 volts. When the switch in this circuit is in the "down"
position, the left diode of the steering diode pair is fully
conducting, and so it drops about 0.7 volts across it and no
more.
Recall that with the switch in the "up" position (transistors
Q, and Q, conducting), there was about 2.1 volts dropped
between those same two points (Q's base and ground),
which also happens to be the minimum voltage necessary to
forward-bias three series-connected silicon PN junctions into
a state of conduction. The 0.7 volts provided by the left
diode's forward voltage drop is simply insufficient to allow
any electron flow through the series string of the right diode,
Q,'s diode, and the R3//Q, diode parallel subcircuit, and so
no electrons flow through that path. With no current through
the bases of either transistor Q> or Qy, neither one will be
able to conduct collector current: transistors Q> and Q, will
both be in a state of cutoff.
Consequently, this circuit configuration allows 100 percent
switching of Q> base current (and therefore control over the
rest of the gate circuit, including voltage at the output) by
diversion of current through the left steering diode.
In the case of our example gate circuit, the input is held
"high" by the switch (connected to V,,), making the left
steering diode (zero voltage dropped across it). However,
the right steering diode is conducting current through the
base of Q5, through resistor Rj:
volts
Veo
I
wi
Output
With base current provided, transistor Q> will be turned "on."
More specifically, it will be saturated by virtue of the more-
than-adequate current allowed by R, through the base. With
Q, saturated, resistor R3 will be dropping enough voltage to
forward-bias the base-emitter junction of transistor Q,, thus
saturating it as well:
With Q, saturated, the output terminal will be almost
directly shorted to ground, leaving the output terminal at a
voltage (in reference to ground) of almost 0 volts, or a binary
"0" ("low") logic level. Due to the presence of diode D>, there
will not be enough voltage between the base of Q3 and its
emitter to turn it on, so it remains in cutoff.
Let's see now what happens if we reverse the input's logic
level to a binary "O" by actuating the input switch:
Now there will be current through the left steering diode of
Q, and no current through the right steering diode. This
eliminates current through the base of Q35, thus turning it off.
With Q> off, there is no longer a path for Q, base current, so
Q, goes into cutoff as well. Q3, on the other hand, now has
sufficient voltage dropped between its base and ground to
forward-bias its base-emitter junction and saturate it, thus
raising the output terminal voltage to a "high" state. In
actuality, the output voltage will be somewhere around 4
volts depending on the degree of saturation and any load
current, but still high enough to be considered a "high" (1)
logic level.
With this, our simulation of the inverter circuit is complete: a
"L" in gives a "O" out, and vice versa.
The astute observer will note that this inverter circuit's input
will assume a "high" state of left floating (not connected to
either V.. or ground). With the input terminal left
unconnected, there will be no current through the left
steering diode of Q,, leaving all of R's current to go through
Q,'s base, thus saturating Q> and driving the circuit output
to a "low" state:
Vi =5 volts
Q;
Input
(floating)
The tendency for such a circuit to assume a high input state
if left floating is one shared by all gate circuits based on this
type of design, known as Transistor-to-Transistor Logic, or
TTL. This characteristic may be taken advantage of in
simplifying the design of a gate's output circuitry, knowing
that the outputs of gates typically drive the inputs of other
gates. If the input of a TTL gate circuit assumes a high state
when floating, then the output of any gate driving a TTL
input need only provide a path to ground for a low state and
be floating for a high state. This concept may require further
elaboration for full understanding, so | will explore it in
detail here.
A gate circuit as we have just analyzed has the ability to
handle output current in two directions: in and out.
Technically, this is Known as sourcing and sinking current,
respectively. When the gate output Is high, there is
continuity from the output terminal to V,, through the top
output transistor (Q3), allowing electrons to flow from
ground, through a load, into the gate's output terminal,
through the emitter of Q3, and eventually up to the V,,
power terminal (positive side of the DC power supply):
V.. =5 volts
&
Inverter gate sourcing current
To simplify this concept, we may show the output of a gate
circuit as being a double-throw switch, capable of
connecting the output terminal either to V., or ground,
depending on its state. For a gate outputting a "high" logic
level, the combination of Q3 saturated and Q, cutoff is
analogous to a double-throw switch in the "V,." position,
providing a path for current through a grounded load:
Simplified gate circuit sourcing current
Vic
Input Th Output
T Load
Please note that this two-position switch shown inside the
gate symbol is representative of transistors Q3 and Qy
alternately connecting the output terminal to V,, or ground,
not of the switch previously shown sending an input signal
to the gate!
Conversely, when a gate circuit is outputting a "low" logic
level to a load, it is analogous to the double-throw switch
being set in the "ground" position. Current will then be
going the other way if the load resistance connects to V,,:
from ground, through the emitter of Q,, out the output
terminal, through the load resistance, and back to V¢<. In
this condition, the gate is said to be sinking current:
V.. =5 volts
Inverter gate sinking current
Simplified gate circuit sinking current
The combination of Q3 and Qy working as a "push-pull"
transistor pair (otherwise known as a totem pole output) has
the ability to either source current (draw in current to V,,) or
sink current (output current from ground) to a load.
However, a standard TTL gate /nput never needs current to
be sourced, only sunk. That is, since a TTL gate input
naturally assumes a high state if left floating, any gate
output driving a TTL input need only sink current to provide
a "0" or "low" input, and need not source current to provide
a"L" ora "high" logic level at the input of the receiving
gate:
A direct connection to V... is not
V necessary to drive the TTL gate
«
Pie high!
Input =a TTL xy
gate
Vo An output that "floats" when high
ai we sufficient.
Input a ie TTL
’ gate
— Any gate driving a TTL
input must sink some
current in the low state.
This means we have the option of simplifying the output
stage of a gate circuit so as to eliminate Q3 altogether. The
result is known as an open-collector output:
Inverter circuit with open-collector output
Input
Output
To designate open-collector output circuitry within a
standard gate symbol, a special marker is used. Shown here
is the symbol for an inverter gate with open-collector
output:
Inverter with open-
collector output
|d>-
Please keep in mind that the "high" default condition of a
floating gate input is only true for TTL circuitry, and not
necessarily for other types, especially for logic gates
constructed of field-effect transistors.
REVIEW:
An inverter, or NOT, gate is one that outputs the
opposite state as what is input. That is, a "low" input (0)
gives a "high" output (1), and vice versa.
Gate circuits constructed of resistors, diodes and bipolar
transistors as illustrated in this section are called 77L.
TTLis an acronym standing for Transistor-to-Transistor
Logic. There are other design methodologies used in
gate circuits, some which use field-effect transistors
rather than bipolar transistors.
A gate is said to be sourcing current when it provides a
path for current between the output terminal and the
positive side of the DC power supply (V,,). In other
words, it is connecting the output terminal to the power
source (+V).
A gate is said to be sinking current when it provides a
path for current between the output terminal and
ground. In other words, it is grounding (sinking) the
output terminal.
Gate circuits with totem pole output stages are able to
both source and sink current. Gate circuits with open-
collector output stages are only able to sink current, and
not source current. Open-collector gates are practical
when used to drive TTL gate inputs because TTL inputs
don't require current sourcing.
The "buffer" gate
If we were to connect two inverter gates together so that the
output of one fed into the input of another, the two inversion
functions would "cancel" each other out so that there would
be no inversion from input to final output:
Double inversion
Logic state re-inverted
to original status
! f
0 sa ?
O inverted into a 1
While this may seem like a pointless thing to do, it does
have practical application. Remember that gate circuits are
signal amplifiers, regardless of what logic function they may
perform. A weak signal source (one that is not capable of
sourcing or sinking very much current to a load) may be
boosted by means of two inverters like the pair shown in the
previous illustration. The logic level is unchanged, but the
full current-sourcing or -sinking capabilities of the final
inverter are available to drive a load resistance if needed.
For this purpose, a special logic gate called a bufferis
manufactured to perform the same function as two inverters.
Its symbol is simply a triangle, with no inverting "bubble" on
the output terminal:
"Buffer" gate
Input —>— Output
0
fe
a ao
The internal schematic diagram for a typical open-collector
buffer is not much different from that of a simple inverter:
only one more common-emitter transistor stage is added to
re-invert the output signal.
Buffer circuit with open-collector output
Input Output
—— Inverter —-> ~— Inverter —~-
Let's analyze this circuit for two conditions: an input logic
level of "1" and an input logic level of "0." First, a "high" (1)
input:
Output
As before with the inverter circuit, the "high" input causes
no conduction through the left steering diode of Q, (emitter-
to-base PN junction). All of Ry's current goes through the
base of transistor Q>, saturating it:
Output
Having Q> saturated causes Q3 to be saturated as well,
resulting in very little voltage dropped between the base
and emitter of the final output transistor Q,. Thus, Q, will be
in cutoff mode, conducting no current. The output terminal
will be floating (neither connected to ground nor V,,), and
this will be equivalent to a "high" state on the input of the
next TTL gate that this one feeds in to. Thus, a "high" input
gives a "high" output.
With a "low" input signal (input terminal grounded), the
analysis looks something like this:
Output
All of R's current is now diverted through the input switch,
thus eliminating base current through Q>. This forces
transistor Q, into cutoff so that no base current goes
through Q3 either. With Q3 cutoff as well, Q, is will be
saturated by the current through resistor Ry, thus
connecting the output terminal to ground, making it a "low"
logic level. Thus, a "low" input gives a "low" output.
The schematic diagram for a buffer circuit with totem pole
output transistors is a bit more complex, but the basic
principles, and certainly the truth table, are the same as for
the open-collector circuit:
Buffer circuit with totem pole output
Output
—+— Inverter —-> ~— Inverter —~-
e REVIEW:
e Two inverter, or NOT, gates connected in "series" so as to
invert, then re-invert, a binary bit perform the function
of a buffer. Buffer gates merely serve the purpose of
signal amplification: taking a "weak" signal source that
isn't capable of sourcing or sinking much current, and
boosting the current capacity of the signal so as to be
able to drive a load.
e Buffer circuits are symbolized by a triangle symbol with
no inverter "bubble."
e Buffers, like inverters, may be made in open-collector
output or totem pole output forms.
Multiple-input gates
Inverters and buffers exhaust the possibilities for single-
input gate circuits. What more can be done with a single
logic signal but to buffer it or invert it? To explore more logic
gate possibilities, we must add more input terminals to the
circuit(s).
Adding more input terminals to a logic gate increases the
number of input state possibilities. With a single-input gate
such as the inverter or buffer, there can only be two possible
input states: either the input is "high" (1) or it is "low" (0).
As was mentioned previously in this chapter, a two input
gate has four possibilities (00, 01, 10, and 11). A three-input
gate has e/ght possibilities (000, 001, 010, 011, 100, 101,
110, and 111) for input states. The number of possible input
states is equal to two to the power of the number of inputs:
Number of possible input states = 2"
Where,
n = Number of inputs
This increase in the number of possible input states
obviously allows for more complex gate behavior. Now,
instead of merely inverting or amplifying (buffering) a single
"high" or "low" logic level, the output of the gate will be
determined by whatever combination of 1's and O's is
present at the input terminals.
Since so many combinations are possible with just a few
input terminals, there are many different types of multiple-
input gates, unlike single-input gates which can only be
inverters or buffers. Each basic gate type will be presented
in this section, showing its standard symbol, truth table, and
practical operation. The actual TTL circuitry of these
different gates will be explored in subsequent sections.
The AND gate
One of the easiest multiple-input gates to understand is the
AND gate, so-called because the output of this gate will be
"high" (1) if and only if a//inputs (first input and the second
input and...) are "high" (1). If any input(s) are "low" (0),
the output is guaranteed to be in a "low" state as well.
2-input AND gate 3-input AND gate
Input
Input ‘
: “TOR Output Inputs—| Output
Input, Input,
In case you might have been wondering, AND gates are
made with more than three inputs, but this is less common
than the simple two-input variety.
A two-input AND gate's truth table looks like this:
2-input AND gate
Input
P TOR Output
Input,
FAB] Ontpat |
fofof o
o}i} o |
jtjo| o |
yt}
What this truth table means in practical terms is shown in
the following sequence of illustrations, with the 2-input AND
gate subjected to all possibilities of input logic levels. An
LED (Light-Emitting Diode) provides visual indication of the
output logic level:
Output
Input,
Input, = 0
Input, = 0
Output = 0 (no light)
Output
Input,
Input, = L
Input, = 0
Output = 0 (no light)
Output
Input,
Input, = 0
Input, = L
Output = 0 (no light)
Input,
Input, = L
Input, = L
Output= 1 (Jight!)
It is only with all inputs raised to "high" logic levels that the
AND gate's output goes "high," thus energizing the LED for
only one out of the four input combination states.
The NAND gate
A variation on the idea of the AND gate is called the NAND
gate. The word "NAND" is a verbal contraction of the words
NOT and AND. Essentially, a NAND gate behaves the same
as an AND gate with a NOT (inverter) gate connected to the
output terminal. To symbolize this output signal inversion,
the NAND gate symbol has a bubble on the output line. The
truth table for a NAND gate is as one might expect, exactly
opposite as that of an AND gate:
2-input NAND gate
In nD Output
Input,
Equivalent gate circuit
Input, Output
Input,
As with AND gates, NAND gates are made with more than
two inputs. In such cases, the same general principle
applies: the output will be "low" (0) if and only if all inputs
are "high" (1). If any input is "low" (0), the output will go
"high" (1).
The OR gate
Our next gate to investigate is the OR gate, so-called
because the output of this gate will be "high" (1) if any of
the inputs (first input orthe second input or...) are "high"
(1). The output of an OR gate goes "low" (0) if and only if all
inputs are "low" (0).
2-input OR gate 3-input OR gate
Input, Input,
Output Input, Output
Input, Input,
A two-input OR gate's truth table looks like this:
2-input OR gate
Input
‘i —) >> Output
Input,
FAB] Outpar |
fofo[ o
ofi} a
jtfof 1
eh
The following sequence of illustrations demonstrates the OR
gate's function, with the 2-inputs experiencing all possible
logic levels. An LED (Light-Emitting Diode) provides visual
indication of the gate's output logic level:
Output
Input,
Input, = 0
Input, = 0
Output = 0 (no light)
Input, = 1
Input, = 0
Output= 1 (light!)
Input, = 0
Input, = L .
Output= 1 (light!)
Inputs = L
Input, = 1
Output= 1 (light!)
A condition of any input being raised to a "high" logic level
makes the OR gate's output go "high," thus energizing the
LED for three out of the four input combination states.
The NOR gate
As you might have suspected, the NOR gate is an OR gate
with its output inverted, just like a NAND gate is an AND
gate with an inverted output.
2-input NOR gate
In Oe Output
Input,
Equivalent gate circuit
Input,
‘ ) >be Output
Input,
NOR gates, like all the other multiple-input gates seen thus
far, can be manufactured with more than two inputs. Still,
the same logical principle applies: the output goes "low" (0)
if any of the inputs are made "high" (1). The output is "high"
(1) only when all inputs are "low" (0).
The Negative-AND gate
A Negative-AND gate functions the same as an AND gate
with all its inputs inverted (connected through NOT gates).
In keeping with standard gate symbol convention, these
inverted inputs are signified by bubbles. Contrary to most
peoples' first instinct, the logical behavior of a Negative-
AND gate is not the same as a NAND gate. Its truth table,
actually, is identical to a NOR gate:
2-input Negative-AND gate
mis Output
Input,
ATE] Oupar
ofof 1
ofif 0
Equivalent gate circuits
Input,
Output
Input,
Input
npu ) ee Output
Input,
The Negative-OR gate
Following the same pattern, a Negative-OR gate functions
the same as an OR gate with all its inputs inverted. In
keeping with standard gate symbol convention, these
inverted inputs are signified by bubbles. The behavior and
truth table of a Negative-OR gate is the same as for a NAND
gate:
2-input Negative-OR gate
die = Suita
Input,
ATE] Oupar
ofof 1
On
Equivalent gate circuits
Input,
Output
Input,
Input
‘ TT pe Output
Input,
The Exclusive-OR gate
The last six gate types are all fairly direct variations on three
basic functions: AND, OR, and NOT. The Exclusive-OR gate,
however, is something quite different.
Exclusive-OR gates output a "high" (1) logic level if the
inputs are at different logic levels, either 0 and 1 or 1 and O.
Conversely, they output a "low" (0) logic level if the inputs
are at the same logic levels. The Exclusive-OR (Sometimes
called XOR) gate has both a symbol and a truth table
pattern that is unique:
Exclusive-OR gate
Input
aie ) > Output
Input,
FAB Outpar |
fofof o
oft} a
jtjo} 1
ti} 0 |
There are equivalent circuits for an Exclusive-OR gate made
up of AND, OR, and NOT gates, just as there were for NAND,
NOR, and the negative-input gates. A rather direct approach
to simulating an Exclusive-OR gate is to start with a regular
OR gate, then add additional gates to inhibit the output
from going "high" (1) when both inputs are "high" (1):
Exclusive-OR equivalent circuit
Input, Output
Input,
In this circuit, the final AND gate acts as a buffer for the
output of the OR gate whenever the NAND gate's output is
high, which it is for the first three input state combinations
(00, 01, and 10). However, when both inputs are "high" (1),
the NAND gate outputs a "low" (0) logic level, which forces
the final AND gate to produce a "low" (0) output.
Another equivalent circuit for the Exclusive-OR gate uses a
strategy of two AND gates with inverters, set up to generate
"high" (1) outputs for input conditions 01 and 10. A final OR
gate then allows either of the AND gates' "high" outputs to
create a final "high" output:
Exclusive-OR equivalent circuit
Input, Output
Input,
Exclusive-OR gates are very useful for circuits where two or
more binary numbers are to be compared bit-for-bit, and also
for error detection (parity check) and code conversion
(binary to Grey and vice versa).
The Exclusive-NOR gate
Finally, our last gate for analysis is the Exclusive-NOR gate,
otherwise known as the XNOR gate. It is equivalent to an
Exclusive-OR gate with an inverted output. The truth table
for this gate is exactly opposite as for the Exclusive-OR gate:
Exclusive-NOR gate
In put, —)) >> Output
Input,
[AT Output
fofo| 1
oft] 0
Equivalent gate circuit
Input
eke ) > Output
Input,
As indicated by the truth table, the purpose of an Exclusive-
NOR gate is to output a "high" (1) logic level whenever both
inputs are at the same logic levels (either 00 or 11).
REVIEW:
Rule for an AND gate: output is "high" only if first input
and second input are both "high."
Rule for an OR gate: output is "high" if input A orinput B
are "high."
Rule for a NAND gate: output is not "high" if both the
first input and the second input are "high."
Rule for a NOR gate: output is not "high" if either the
first input orthe second input are "high."
A Negative-AND gate behaves like a NOR gate.
A Negative-OR gate behaves like a NAND gate.
Rule for an Exclusive-OR gate: output is "high" if the
input logic levels are different.
e Rule for an Exclusive-NOR gate: output is "high" if the
input logic levels are the same.
TTL NAND and AND gates
Suppose we altered our basic open-collector inverter circuit,
adding a second input terminal just like the first:
A two-input inverter circuit
;
V cc
This schematic illustrates a real circuit, but it isn't called a
“two-input inverter." Through analysis we will discover what
this circuit's logic function is and correspondingly what it
should be designated as.
Just as in the case of the inverter and buffer, the "steering"
diode cluster marked "Q," is actually formed like a
transistor, even though it isn't used in any amplifying
capacity. Unfortunately, a simple NPN transistor structure is
inadequate to simulate the three PN junctions necessary in
this diode network, so a different transistor (and symbol) is
needed. This transistor has one collector, one base, and two
emitters, and in the circuit it looks like this:
V
cc
Output
In the single-input (inverter) circuit, grounding the input
resulted in an output that assumed the "high" (1) state. In
the case of the open-collector output configuration, this
"high" state was simply "floating." Allowing the input to float
(or be connected to V,,) resulted in the output becoming
grounded, which is the "low" or O state. Thus, a 1 in resulted
in a O out, and vice versa.
Since this circuit bears so much resemblance to the simple
inverter circuit, the only difference being a second input
terminal connected in the same way to the base of transistor
Q>, we can Say that each of the inputs will have the same
effect on the output. Namely, if either of the inputs are
grounded, transistor Q> will be forced into a condition of
cutoff, thus turning Q3 off and floating the output (output
goes "high"). The following series of illustrations shows this
for three input states (00, 01, and 10):
Input, = 0
Input, = 0
Output= 1
Input, = 0
Input, = L
Output= 1
1
Output
Q, Cutoff
Input, = L
Input, = 0
Output= L
In any case where there is a grounded ("low") input, the
output is guaranteed to be floating ("high"). Conversely, the
only time the output will ever go "low" is if transistor Q3
turns on, which means transistor Q, must be turned on
(saturated), which means neither input can be diverting R,
current away from the base of Q>. The only condition that
will satisfy this requirement is when both inputs are "high"
(1):
Output
Q, Saturation
Input, = L
Input, = L
Output = 0
Collecting and tabulating these results into a truth table, we
see that the pattern matches that of the NAND gate:
NAND gate
nett T > output
Input,
FAB] Outpar |
fofof 1
l
o}i} i
jtjo} 1
ti} 0 |
In the earlier section on NAND gates, this type of gate was
created by taking an AND gate and increasing its complexity
by adding an inverter (NOT gate) to the output. However,
when we examine this circuit, we see that the NAND
function is actually the simplest, most natural mode of
operation for this TTL design. To create an AND function
using TTL circuitry, we need to increase the complexity of
this circuit by adding an inverter stage to the output, just
like we had to add an additional transistor stage to the TTL
inverter circuit to turn it into a buffer:
AND gate with open-collector output
Output
~— NAND gate —-~~ ~— Inverter —~
The truth table and equivalent gate circuit (an inverted-
output NAND gate) are shown here:
AND gate
nett — cua
Input,
A[B] Outpat_
i ae
jofi} o
jtjo| o |
i
Equivalent circuit
Input
Input,
Of course, both NAND and AND gate circuits may be
designed with totem-pole output stages rather than open-
collector. | am opting to show the open-collector versions for
the sake of simplicity.
e REVIEW:
e A TTL NAND gate can be made by taking a TTL inverter
circuit and adding another input.
e An AND gate may be created by adding an inverter
stage to the output of the NAND gate circuit.
TTL NOR and OR gates
Let's examine the following TTL circuit and analyze its
operation:
Output
Transistors Q; and Q> are both arranged in the same manner
that we've seen for transistor Q, in all the other TTL circuits.
Rather than functioning as amplifiers, Q; and Q> are both
being used as two-diode "steering" networks. We may
replace Q, and Q> with diode sets to help illustrate:
Output
If input A is left floating (or connected to V,,), current will go
through the base of transistor Q3, saturating it. If input A is
grounded, that current is diverted away from Q3's base
through the left steering diode of "Q,," thus forcing Q3 into
cutoff. The same can be said for input B and transistor Q,:
the logic level of input B determines Q,'s conduction: either
saturated or cutoff.
Notice how transistors Q3 and Q, are paralleled at their
collector and emitter terminals. In essence, these two
transistors are acting as paralleled switches, allowing
current through resistors R3 and Ry according to the logic
levels of inputs A and B. If anyinput is at a "high" (1) level,
then at least one of the two transistors (Q3 and/or Q,) will be
saturated, allowing current through resistors R3 and Ry, and
turning on the final output transistor Qs. for a "low" (0) logic
level output. The only way the output of this circuit can ever
assume a "high" (1) state is if both Q3 and Q,y are cutoff,
which means both inputs would have to be grounded, or
"low" (0).
This circuit's truth table, then, is equivalent to that of the
NOR gate:
NOR gate
mite) Output
Input,
FAB] Ontpar |
fofo[ 1
fol} o__|
jtjo} o |
pe tO
In order to turn this NOR gate circuit into an OR gate, we
would have to invert the output logic level with another
transistor stage, just like we did with the NAND-to-AND gate
example:
OR gate with open-collector output
Vv
cc
~— NOR gate —»~=— Inverter —>
The truth table and equivalent gate circuit (an inverted-
output NOR gate) are shown here:
OR gate
mie) Output
Input,
ATE] Oupar
ofof o
(a a
jtjo} i
i
Equivalent circuit
Input, ~) >to Output
Input,
Of course, totem-pole output stages are also possible in both
NOR and OR TTL logic circuits.
e REVIEW:
e An OR gate may be created by adding an inverter stage
to the output of the NOR gate circuit.
CMOS gate circuitry
Up until this point, our analysis of transistor logic circuits
has been limited to the 77L design paradigm, whereby
bipolar transistors are used, and the general strategy of
floating inputs being equivalent to "high" (connected to V,,)
inputs -- and correspondingly, the allowance of "open-
collector" output stages -- is maintained. This, however, is
not the only way we can build logic gates.
Field-effect transistors, particularly the insulated-gate
variety, may be used in the design of gate circuits. Being
voltage-controlled rather than current-controlled devices,
IGFETs tend to allow very simple circuit designs. Take for
instance, the following inverter circuit built using P- and N-
channel IGFETs:
Inverter circuit using IGFETs
Vdd (+5 volts)
Input i: Output
Notice the "Vy," label on the positive power supply terminal.
This label follows the same convention as "V,," in TTL
Circuits: it stands for the constant voltage applied to the
drain of a field effect transistor, in reference to ground.
Let's connect this gate circuit to a power source and input
switch, and examine its operation. Please note that these
IGFET transistors are E-type (Enhancement-mode), and so
are normally-off devices. It takes an applied voltage between
gate and drain (actually, between gate and substrate) of the
correct polarity to bias them on.
Input = "low" (0)
Output = "high” (1)
The upper transistor is a P-channel IGFET. When the channel
(substrate) is made more positive than the gate (gate
negative in reference to the substrate), the channel is
enhanced and current is allowed between source and drain.
So, in the above illustration, the top transistor is turned on.
The lower transistor, having zero voltage between gate and
substrate (source), is in its normal mode: off. Thus, the
action of these two transistors are such that the output
terminal of the gate circuit has a solid connection to Vgg and
a very high resistance connection to ground. This makes the
output "high" (1) for the "low" (0) state of the input.
Next, we'll move the input switch to its other position and
see what happens:
Cutoftt
Output
— 5V
Saturated
Input = "high" (1)
Output = "low" (0)
Now the lower transistor (N-channel) is saturated because it
has sufficient voltage of the correct polarity applied between
gate and substrate (channel) to turn it on (positive on gate,
negative on the channel). The upper transistor, having zero
voltage applied between its gate and substrate, is in its
normal mode: off. Thus, the output of this gate circuit is now
"low" (0). Clearly, this circuit exhibits the behavior of an
inverter, or NOT gate.
Using field-effect transistors instead of bipolar transistors
has greatly simplified the design of the inverter gate. Note
that the output of this gate never floats as is the case with
the simplest TTL circuit: it has a natural "totem-pole"
configuration, capable of both sourcing and sinking load
current. Key to this gate circuit's elegant design is the
complementary use of both P- and N-channel IGFETs. Since
IGFETs are more commonly known as MOSFETs (Metal-
Oxide-Semiconductor Field Effect Transistor), and this
circuit uses both P- and N-channel transistors together, the
general classification given to gate circuits like this one is
CMOS: Complementary Metal Oxide Semiconductor.
CMOS circuits aren't plagued by the inherent nonlinearities
of the field-effect transistors, because as digital circuits their
transistors always operate in either the saturated or cutoff
modes and never in the active mode. Their inputs are,
however, sensitive to high voltages generated by
electrostatic (static electricity) sources, and may even be
activated into "high" (1) or "low" (0) states by spurious
voltage sources if left floating. For this reason, it is
inadvisable to allow a CMOS logic gate input to float under
any circumstances. Please note that this is very different
from the behavior of a TTL gate where a floating input was
safely interpreted as a "high" (1) logic level.
This may cause a problem if the input to a CMOS logic gate
is driven by a single-throw switch, where one state has the
input solidly connected to either Vyg or ground and the
other state has the input floating (not connected to
anything):
CMOS gate
et Output
When switch is closed, the gate sees a
definite "low” (0) input. However, when
switch is open, the input logic level will
be uncertain because it’s floating.
Also, this problem arises if a CMOS gate input is being
driven by an open-collector TTL gate. Because such a TTL
gate's output floats when it goes "high" (1), the CMOS gate
input will be left in an uncertain state:
Open-collector
TAL: de CMOS gate
Vad
ae Input
an 1
Input
When the open-collector TTL gate’s output
is "hi Ae (1), the CMOS gate’s input will be
left floating and in an uncertain logic state.
Fortunately, there is an easy solution to this dilemma, one
that is used frequently in CMOS logic circuitry. Whenever a
single-throw switch (or any other sort of gate output
incapable of both sourcing and sinking current) is being
used to drive a CMOS input, a resistor connected to either
Vag Or ground may be used to provide a stable logic level for
the state in which the driving device's output is floating.
This resistor's value is not critical: 10 kQ is usually sufficient.
When used to provide a "high" (1) logic level in the event of
a floating signal source, this resistor is Known as a pullup
resistor.
Vdd
R CMOS gate
pullup
Dek input eve Output
When switch is closed, the gate sees a
definite "low” (0) input. When the switch
is open, Pyynypy Will provide the connection
to Vdd needed to secure a reliable "high"
logic level for the CMOS gate input.
When such a resistor is used to provide a "low" (0) logic
level in the event of a floating signal source, it is Known as a
pulldown resistor. Again, the value for a pulldown resistor is
not critical:
CMOS gate
2 Input
R pulldown
Output
When switch is closed, the gate sees a
definite "high" (1) input. When the switch
is open, Rouidown Will provide the connection
to ground needed to secure a reliable "low"
logic level for the CMOS gate input.
Because open-collector TTL outputs always sink, never
source, Current, pullup resistors are necessary when
interfacing such an output to a CMOS gate input:
Open-collector
TTL gate Vea CMOS gate
Voc Vad
al. Routtup
a
Although the CMOS gates used in the preceding examples
were all inverters (single-input), the same principle of pullup
and pulldown resistors applies to multiple-input CMOS
gates. Of course, a separate pullup or pulldown resistor will
be required for each gate input:
Pullup resistors for a 3-input
CMOS AND gate
Vdd
Input,
Input,
Input,
This brings us to the next question: how do we design
multiple-input CMOS gates such as AND, NAND, OR, and
NOR? Not surprisingly, the answer(s) to this question reveal
a simplicity of design much like that of the CMOS inverter
over its TTL equivalent.
For example, here is the schematic diagram for a CMOS
NAND gate:
CMOS NAND gate
Vdd
Output
Notice how transistors Q, and Q3 resemble the series-
connected complementary pair from the inverter circuit.
Both are controlled by the same input signal (input A), the
upper transistor turning off and the lower transistor turning
on when the input is "high" (1), and vice versa. Notice also
how transistors Q> and Q, are similarly controlled by the
same input signal (input B), and how they will also exhibit
the same on/off behavior for the same input logic levels. The
upper transistors of both pairs (Q, and Q;) have their source
and drain terminals paralleled, while the lower transistors
(Q3 and Q,) are series-connected. What this means is that
the output will go "high" (1) if e/thertop transistor saturates,
and will go "low" (0) only if both lower transistors saturate.
The following sequence of illustrations shows the behavior of
this NAND gate for all four possibilities of input logic levels
(00, 01, 10, and 11):
Output
As with the TTL NAND gate, the CMOS NAND gate circuit
may be used as the starting point for the creation of an AND
gate. All that needs to be added is another stage of
transistors to invert the output signal:
CMOS AND gate
Vdd
~— NAND gate —-~~— Inverter —~-
A CMOS NOR gate circuit uses four MOSFETs just like the
NAND gate, except that its transistors are differently
arranged. Instead of two paralleled sourcing (upper)
transistors connected to Vyg and two series-connected
sinking (lower) transistors connected to ground, the NOR
gate uses two series-connected sourcing transistors and two
parallel-connected sinking transistors like this:
CMOS NOR gate
Vdd
Output
As with the NAND gate, transistors Q, and Q3 work asa
complementary pair, as do transistors Q5 and Q,. Each pair
is controlled by a single input signal. If e/therinput A or
input B are "high" (1), at least one of the lower transistors
(Q3 or Q,) will be saturated, thus making the output "low"
(0). Only in the event of both inputs being "low" (0) will both
lower transistors be in cutoff mode and both upper
transistors be saturated, the conditions necessary for the
output to go "high" (1). This behavior, of course, defines the
NOR logic function.
The OR function may be built up from the basic NOR gate
with the addition of an inverter stage on the output:
CMOS OR gate
Vdd
~«— NOR gate —-++~—_ Inverter —+
Since it appears that any gate possible to construct using
TTL technology can be duplicated in CMOS, why do these
two "families" of logic design still coexist? The answer is that
both TTL and CMOS have their own unique advantages.
First and foremost on the list of comparisons between TTL
and CMOS is the issue of power consumption. In this
measure of performance, CMOS is the unchallenged victor.
Because the complementary P- and N-channel MOSFET pairs
of a CMOS gate circuit are (ideally) never conducting at the
same time, there is little or no current drawn by the circuit
from the Vgg power supply except for what current is
necessary to source current to a load. TTL, on the other
hand, cannot function without some current drawn at all
times, due to the biasing requirements of the bipolar
transistors from which it is made.
There is a caveat to this advantage, though. While the power
dissipation of a TTL gate remains rather constant regardless
of its operating state(s), a CMOS gate dissipates more power
as the frequency of its input signal(s) rises. If a CMOS gate is
operated in a static (unchanging) condition, it dissipates
zero power (ideally). However, CMOS gate circuits draw
transient current during every output state switch from
"low" to "high" and vice versa. So, the more often a CMOS
gate switches modes, the more often it will draw current
from the Vygg supply, hence greater power dissipation at
greater frequencies.
A CMOS gate also draws much less current from a driving
gate output than a TTL gate because MOSFETs are voltage-
controlled, not current-controlled, devices. This means that
one gate can drive many more CMOS inputs than TTL inputs.
The measure of how many gate inputs a single gate output
can drive is called fanout.
Another advantage that CMOS gate designs enjoy over TTL
is a much wider allowable range of power supply voltages.
Whereas TTL gates are restricted to power supply (V,,)
voltages between 4.75 and 5.25 volts, CMOS gates are
typically able to operate on any voltage between 3 and 15
volts! The reason behind this disparity in power supply
voltages is the respective bias requirements of MOSFET
versus bipolar junction transistors. MOSFETs are controlled
exclusively by gate voltage (with respect to substrate),
whereas BJTs are current-controlled devices. TTL gate circuit
resistances are precisely calculated for proper bias currents
assuming a 5 volt regulated power supply. Any significant
variations in that power supply voltage will result in the
transistor bias currents being incorrect, which then results in
unreliable (unpredictable) operation. The only effect that
variations in power supply voltage have on a CMOS gate is
the voltage definition of a "high" (1) state. For a CMOS gate
operating at 15 volts of power supply voltage (Vgq), an input
signal must be close to 15 volts in order to be considered
"high" (1). The voltage threshold for a "low" (0) signal
remains the same: near 0 volts.
One decided disadvantage of CMOS is slow speed, as
compared to TTL. The input capacitances of a CMOS gate are
much, much greater than that of a comparable TTL gate --
owing to the use of MOSFETs rather than BJTs -- and soa
CMOS gate will be slower to respond to a signal transition
(low-to-high or vice versa) than a TTL gate, all other factors
being equal. The RC time constant formed by circuit
resistances and the input capacitance of the gate tend to
impede the fast rise- and fall-times of a digital logic level,
thereby degrading high-frequency performance.
A strategy for minimizing this inherent disadvantage of
CMOS gate circuitry is to "buffer" the output signal with
additional transistor stages, to increase the overall voltage
gain of the device. This provides a faster-transitioning
output voltage (high-to-low or low-to-high) for an input
voltage slowly changing from one logic state to another.
Consider this example, of an "unbuffered" NOR gate versus a
"buffered," or B-series, NOR gate:
"Unbuffered" NOR gate
Output
"B-series" (buffered) NOR gate
ie Output
In essence, the B-series design enhancement adds two
inverters to the output of a simple NOR circuit. This serves
no purpose as far as digital logic is concerned, since two
cascaded inverters simply cancel:
Dp
(same as)
(same as)
|
“a
However, adding these inverter stages to the circuit does
serve the purpose of increasing overall voltage gain, making
the output more sensitive to changes in input state, working
to overcome the inherent slowness caused by CMOS gate
input capacitance.
REVIEW:
CMOS logic gates are made of IGFET (MOSFET)
transistors rather than bipolar junction transistors.
CMOS gate inputs are sensitive to static electricity. They
may be damaged by high voltages, and they may
assume any logic level if left floating.
Pullup and pulldown resistors are used to prevent a
CMOS gate input from floating if being driven by a
signal source capable only of sourcing or sinking
current.
e CMOS gates dissipate far less power than equivalent TTL
gates, but their power dissipation increases with signal
frequency, whereas the power dissipation of a TTL gate
IS approximately constant over a wide range of
operating conditions.
e CMOS gate inputs draw far less current than TTL inputs,
because MOSFETs are voltage-controlled, not current-
controlled, devices.
e CMOS gates are able to operate on a much wider range
of power supply voltages than TTL: typically 3 to 15
volts versus 4.75 to 5.25 volts for TTL.
e CMOS gates tend to have a much lower maximum
operating frequency than TTL gates due to input
Capacitances caused by the MOSFET gates.
B-series CMOS gates have "buffered" outputs to increase
voltage gain from input to output, resulting in faster
output response to input signal changes. This helps
overcome the inherent slowness of CMOS gates due to
MOSFET input capacitance and the RC time constant
thereby engendered.
Special-output gates
It is sometimes desirable to have a logic gate that provides
both inverted and non-inverted outputs. For example, a
single-input gate that is both a buffer and an inverter, with a
separate output terminal for each function. Or, a two-input
gate that provides both the AND and the NAND functions in
a single circuit. Such gates do exist and they are referred to
as complementary output gates.
The general symbology for such a gate is the basic gate
figure with a bar and two output lines protruding from it. An
array of complementary gate symbols is shown in the
following illustration:
Complementary buffer
Ph
Complementary AND gate
=i
Complementary OR gate
=»
Complementary XOR gate
=x
Complementary gates are especially useful in "crowded"
circuits where there may not be enough physical room to
mount the additional integrated circuit chips necessary to
provide both inverted and noninverted outputs using
standard gates and additional inverters. They are also useful
in applications where a complementary output is necessary
from a gate, but the addition of an inverter would introduce
an unwanted time lag in the inverted output relative to the
noninverted output. The internal circuitry of complemented
gates is such that both inverted and noninverted outputs
change state at almost exactly the same time:
Complemented gate Standard gate with inverter added
Time delay introduced
by the inverter
Another type of special gate output is called tristate,
because it has the ability to provide three different output
modes: current sinking ("low" logic level), current sourcing
("high"), and floating ("high-Z," or high-impedance). Tristate
outputs are usually found as an optional feature on buffer
gates. Such gates require an extra input terminal to control
the "high-Z" mode, and this input is usually called the
enable.
Tristate buffer gate
Enable
Output
With the enable input held "high" (1), the buffer acts like an
ordinary buffer with a totem pole output stage: it is capable
of both sourcing and sinking current. However, the output
terminal floats (goes into "high-Z" mode) if ever the enable
input is grounded ("low"), regardless of the data signal's
logic level. In other words, making the enable input terminal
"low" (0) effectively disconnects the gate from whatever its
output is wired to so that it can no longer have any effect.
Tristate buffers are marked in schematic diagrams by a
triangle character within the gate symbol like this:
Tristate buffer symbol
Enable (B)
Input > Output
(A)
Truth table
TAB] Ourpat_
[0 [o | High-Z__
fofif o
ro | High-Z_
oft 1
Tristate buffers are also made with inverted enable inputs.
Such a gate acts normal when the enable input is "low" (0)
and goes into high-Z output mode when the enable input is
"high" (1):
Tristate buffer with
inverted enable input
Enable (B)
Input fy Output
(A)
Truth table
FAB] Outpat |
ofol o
fo 1 | High-Z._
rifof 1
aft [High
One special type of gate known as the bilateral switch uses
gate-controlled MOSFET transistors acting as on/off switches
to switch electrical signals, analog or digital. The "on"
resistance of such a switch Is in the range of several
hundred ohms, the "off" resistance being in the range of
several hundred mega-ohms.
Bilateral switches appear in schematics as SPST (Single-Pole,
Single-Throw) switches inside of rectangular boxes, with a
control terminal on one of the box's long sides:
CMOS bilateral switch
Control
In/Out et In/Out
A bilateral switch might be best envisioned as a solid-state
(semiconductor) version of an electromechanical relay: a
signal-actuated switch contact that may be used to conduct
virtually any type of electric signal. Of course, being solid-
state, the bilateral switch has none of the undesirable
characteristics of electromechanical relays, such as contact
"bouncing," arcing, slow speed, or susceptibility to
mechanical vibration. Conversely, though, they are rather
limited in their current-carrying ability. Additionally, the
signal conducted by the "contact" must not exceed the
power supply "rail" voltages powering the bilateral switch
Circuit.
Four bilateral switches are packaged inside the popular
model "4066" integrated circuit:
Quad CMOS bilateral switch
4066
e REVIEW:
e Complementary gates provide both inverted and
noninverted output signals, in such a way that neither
one is delayed with respect to the other.
e Tristate gates provide three different output states: high,
low, and floating (High-Z). Such gates are commanded
into their high-impedance output modes by a separate
input terminal called the enable.
Bilateral switches are MOSFET circuits providing on/off
switching for a variety of electrical signal types (analog
and digital), controlled by logic level voltage signals. In
essence, they are solid-state relays with very low
current-handling ability.
Gate universality
NAND and NOR gates possess a special property: they are
universal. That is, given enough gates, either type of gate is
able to mimic the operation of any other gate type. For
example, it is possible to build a circuit exhibiting the OR
function using three interconnected NAND gates. The ability
for a single gate type to be able to mimic any other gate
type is one enjoyed only by the NAND and the NOR. In fact,
digital control systems have been designed around nothing
but either NAND or NOR gates, all the necessary logic
functions being derived from collections of interconnected
NANDs or NORs.
As proof of this property, this section will be divided into
subsections showing how all the basic gate types may be
formed using only NANDs or only NORs.
Constructing the NOT function
Input
Input
Output TL) > Output
wie ME ia
+V
Input
Output Output
Input
As you Can see, there are two ways to use a NAND gate as an
inverter, and two ways to use a NOR gate as an inverter.
Either method works, although connecting TTL inputs
together increases the amount of current loading to the
driving gate. For CMOS gates, common input terminals
decreases the switching speed of the gate due to increased
input capacitance.
Inverters are the fundamental tool for transforming one type
of logic function into another, and so there will be many
inverters shown in the illustrations to follow. In those
diagrams, | will only show one method of inversion, and that
will be where the unused NAND gate input is connected to
+V (either V.. Or Vgg, depending on whether the circuit is
TTL or CMOS) and where the unused input for the NOR gate
is connected to ground. Bear in mind that the other
inversion method (connecting both NAND or NOR inputs
together) works just as well from a logical (1's and O's) point
of view, but is undesirable from the practical perspectives of
increased current loading for TTL and increased input
Capacitance for CMOS.
Constructing the "buffer" function
Being that it is quite easy to employ NAND and NOR gates to
perform the inverter (NOT) function, it stands to reason that
two such stages of gates will result in a buffer function,
where the output is the same logical state as the input.
Input +> Output
ro fo |
+V
+V
Output
Input
Input
=Pab- Output
Constructing the AND function
To make the AND function from NAND gates, all that is
needed is an inverter (NOT) stage on the output of a NAND
gate. This extra inversion "cancels out" the first NV in NAND,
leaving the AND function. It takes a little more work to
wrestle the same functionality out of NOR gates, but it can
be done by inverting ("NOT") all of the inputs to a NOR gate.
2-input AND gate
an Output
Input,
FATB] Ourpat_
jojo} o |
jofif oO
jifo} oO
+V
Input,
Input,
= Output
Input,
Constructing the NAND function
It would be pointless to show you how to "construct" the
NAND function using a NAND gate, since there is nothing to
do. To make a NOR gate perform the NAND function, we
must invert all inputs to the NOR gate as well as the NOR
gate's output. For a two-input gate, this requires three more
NOR gates connected as inverters.
2-input NAND gate
ian Output
Input,
ra BT Ourpar |
fofof 1
l
ofa} a
fifo} 1
ENE ae
Output
Constructing the OR function
Inverting the output of a NOR gate (with another NOR gate
connected as an inverter) results in the OR function. The
NAND gate, on the other hand, requires inversion of all
inputs to mimic the OR function, just as we needed to invert
all inputs of a NOR gate to obtain the AND function.
Remember that inversion of all inputs to a gate results in
changing that gate's essential function from AND to OR (or
vice versa), plus an inverted output. Thus, with all inputs
inverted, a NAND behaves as an OR, a NOR behaves as an
AND, an AND behaves as a NOR, and an OR behaves as a
NAND. In Boolean algebra, this transformation is referred to
as DeMorgan's Theorem, covered in more detail in a later
chapter of this book.
2-input OR gate
rea) Output
Input,
[A[B | Ouepat |
jojo} o |
fofif i
pifof i |
+V
Inputs
+V Output
Input,
Input,
rout) J > Output
Constructing the NOR function
Much the same as the procedure for making a NOR gate
behave as a NAND, we must invert all inputs and the output
to make a NAND gate function as a NOR.
2-input NOR gate
Input
‘ ) Output
Input,
[AB] Ourpat
fofot i
fofi[ 0
+V
Input,
4 Output
Input,
e REVIEW:
e NAND and NOR gates are universal: that is, they have
the ability to mimic any type of gate, if interconnected
in sufficient numbers.
Logic signal voltage levels
Logic gate circuits are designed to input and output only
two types of signals: "high" (1) and "low" (0), as represented
by a variable voltage: full power supply voltage for a "high"
state and zero voltage for a "low" state. In a perfect world,
all logic circuit signals would exist at these extreme voltage
limits, and never deviate from them (i.e., less than full
voltage for a "high," or more than zero voltage for a "low").
However, in reality, logic signal voltage levels rarely attain
these perfect limits due to stray voltage drops in the
transistor circuitry, and so we must understand the signal
level limitations of gate circuits as they try to interpret
signal voltages lying somewhere between full supply
voltage and zero.
TTL gates operate on a nominal power supply voltage of 5
volts, +/- 0.25 volts. Ideally, a TTL "high" signal would be
5.00 volts exactly, and a TTL "low" signal 0.00 volts exactly.
However, real TTL gate circuits cannot output such perfect
voltage levels, and are designed to accept "high" and "low"
signals deviating substantially from these ideal values.
"Acceptable" input signal voltages range from 0 volts to 0.8
volts for a "low" logic state, and 2 volts to 5 volts fora "high"
logic state. "Acceptable" output signal voltages (voltage
levels guaranteed by the gate manufacturer over a specified
range of load conditions) range from 0 volts to 0.5 volts fora
"low" logic state, and 2.7 volts to 5 volts for a "high" logic
state:
Acceptable TTL gate Acceptable TTL gate
input signal levels output signal levels
5 V 5V
Y= 35: ¥ j
High cc High |
2.1V
2V
0.8V — e
Low 0.5V
ae - Low — OV
If a voltage signal ranging between 0.8 volts and 2 volts
were to be sent into the input of a TTL gate, there would be
no certain response from the gate. Such a signal would be
considered uncertain, and no logic gate manufacturer would
guarantee how their gate circuit would interpret such a
signal.
As you can see, the tolerable ranges for output signal levels
are narrower than for input signal levels, to ensure that any
TTL gate outputting a digital signal into the input of another
TTL gate will transmit voltages acceptable to the receiving
gate. The difference between the tolerable output and input
ranges is called the no/se margin of the gate. For TTL gates,
the low-level noise margin is the difference between 0.8
volts and 0.5 volts (0.3 volts), while the high-level noise
margin is the difference between 2.7 volts and 2 volts (0.7
volts). Simply put, the noise margin is the peak amount of
spurious or "noise" voltage that may be superimposed on a
weak gate output voltage signal before the receiving gate
might interpret it wrongly:
Acceptable TTL gate Acceptable TTL gate
input signal levels output signal levels
5 V a¥
High high-level noise = High |
SSS a
Low 0.5 V
aa OV
low-level noise margin
CMOS gate circuits have input and output signal
specifications that are quite different from TTL. For a CMOS
gate operating at a power supply voltage of 5 volts, the
acceptable input signal voltages range from O volts to 1.5
volts for a "low" logic state, and 3.5 volts to 5 volts fora
"high" logic state. "Acceptable" output signal voltages
(voltage levels guaranteed by the gate manufacturer over a
specified range of load conditions) range from O volts to
0.05 volts for a "low" logic state, and 4.95 volts to 5 volts for
a "high" logic state:
Acceptable CMOS gate Acceptable CMOS gate
input signal levels output signal levels
e . -¥
xe High — = 4 95 -v
High Va=5V
ga ¥
Lov
on + 7 0.05 V
.05
OV Low —="= ov
It should be obvious from these figures that CMOS gate
circuits have far greater noise margins than TTL: 1.45 volts
for CMOS low-level and high-level margins, versus a
maximum of 0.7 volts for TTL. In other words, CMOS circuits
can tolerate over twice the amount of superimposed "noise"
voltage on their input lines before signal interpretation
errors will result.
CMOS noise margins widen even further with higher
operating voltages. Unlike TTL, which is restricted to a power
supply voltage of 5 volts, CMOS may be powered by
voltages as high as 15 volts (some CMOS circuits as high as
18 volts). Shown here are the acceptable "high" and "low"
states, for both input and output, of CMOS integrated
circuits operating at 10 volts and 15 volts, respectively:
Acceptable CMOS gate Acceptable CMOS gate
input signal levels output signal levels
10 V High — 10 V
9.95 V
High
7V Vai = 10V
3V a
Low
fae 0.05 V
OV OV
Acceptable CMOS gate Acceptable CMOS gate
input signal levels output signal levels
5 High — I5V
ay : 14.95 V
High
11 V
4V
Low
0.05 V
OV Low —="=ov
The margins for acceptable "high" and "low" signals may be
greater than what is shown in the previous illustrations.
What is shown represents "worst-case" input signal
performance, based on manufacturer's specifications. In
practice, it may be found that a gate circuit will tolerate
"high" signals of considerably less voltage and "low" signals
of considerably greater voltage than those specified here.
Conversely, the extremely small output margins shown --
guaranteeing output states for "high" and "low" signals to
within 0.05 volts of the power supply "rails" -- are optimistic.
Such "solid" output voltage levels will be true only for
conditions of minimum loading. If the gate is sourcing or
sinking substantial current to a load, the output voltage will
not be able to maintain these optimum levels, due to
internal channel resistance of the gate's final output
MOSFETs.
Within the "uncertain" range for any gate input, there will be
some point of demarcation dividing the gate's actual "low"
input signal range from its actual "high" input signal range.
That is, somewhere between the lowest "high" signal voltage
level and the highest "low" signal voltage level guaranteed
by the gate manufacturer, there is a threshold voltage at
which the gate will actually switch its interpretation of a
signal from "low" or "high" or vice versa. For most gate
circuits, this unspecified voltage is a single point:
Typical response of a logic gate
to a variable (analog) input voltage
5V Vag = 5 V
OV - -
Time —>
In the presence of AC "noise" voltage superimposed on the
DC input signal, a single threshold point at which the gate
alters its interpretation of logic level will result in an erratic
output:
Slowly-changing DC signal with
AC noise superimposed
5V Vag = 5 V
threshold Vv
OV : ; =
Time —>
If this scenario looks familiar to you, its because you
remember a similar problem with (analog) voltage
comparator op-amp circuits. With a single threshold point at
which an input causes the output to switch between "high"
and "low" states, the presence of significant noise will cause
erratic changes in the output:
Square wave
output voltage
AC input
voltage
The solution to this problem is a bit of positive feedback
introduced into the amplifier circuit. With an op-amp, this is
done by connecting the output back around to the
noninverting (+) input through a resistor. In a gate circuit,
this entails redesigning the internal gate circuitry,
establishing the feedback inside the gate package rather
than through external connections. A gate so designed is
called a Schmitt trigger. Schmitt triggers interpret varying
input voltages according to two threshold voltages: a
positive-going threshold (V7), and a negative-going
threshold (V+.):
Schmitt trigger response to a
"noisy" input signal
OV
Time —>
Schmitt trigger gates are distinguished in schematic
diagrams by the small "hysteresis" symbol drawn within
them, reminiscent of the B-H curve for a ferromagnetic
material. Hysteresis engendered by positive feedback within
the gate circuitry adds an additional level of noise immunity
to the gate's performance. Schmitt trigger gates are
frequently used in applications where noise is expected on
the input signal line(s), and/or where an erratic output
would be very detrimental to system performance.
The differing voltage level requirements of TTL and CMOS
technology present problems when the two types of gates
are used in the same system. Although operating CMOS
gates on the same 5.00 volt power supply voltage required
by the TTL gates is no problem, TTL output voltage levels
will not be compatible with CMOS input voltage
requirements.
Take for instance a TTL NAND gate outputting a signal into
the input of a CMOS inverter gate. Both gates are powered
by the same 5.00 volt supply (V_,). If the TTL gate outputs a
"low" signal (guaranteed to be between 0 volts and 0.5
volts), it will be properly interpreted by the CMOS gate's
input as a "low" (expecting a voltage between 0 volts and
1.5 volts):
SV 5V
TTL CMOS
output input
L5V
O05 V gum --------------
Oy ae Sea sSes==sS55 Ov
TTL output falls within
acceptable limits for
MOS input
However, if the TTL gate outputs a "high" signal (guaranteed
to be between 5 volts and 2.7 volts), it might not be properly
interpreted by the CMOS gate's input as a "high" (expecting
a voltage between 5 volts and 3.5 volts):
27V¥ =e ..-.--.-.-----
CMOS
TTL input
output
OV OV
TTL output falls outside of
acceptable limits for
MOS input
Given this mismatch, it is entirely possible for the TTL gate
to output a valid "high" signal (valid, that is, according to
the standards for TTL) that lies within the "uncertain" range
for the CMOS input, and may be (falsely) interpreted as a
"low" by the receiving gate. An easy "fix" for this problem is
to augment the TTL gate's "high" signal voltage level by
means of a pullup resistor:
Vm 5V
3.5V
TAL
output CMOS
input
OV OV
TTL "high" output voltage
assisted bY R yutivp
Something more than this, though, is required to interface a
TTL output with a CMOS input, if the receiving CMOS gate is
powered by a greater power supply voltage:
2 CMOS
SV mc input
TTL 57y MM __-__ _--e
output ae
ir ay cape apse at OV
The TTL "high" signal will
definitely not fall within the
CMOS gate’s acceptable limits
There will be no problem with the CMOS gate interpreting
the TTL gate's "low" output, of course, but a "high" signal
from the TTL gate is another matter entirely. The guaranteed
output voltage range of 2.7 volts to 5 volts from the TTL gate
output is nowhere near the CMOS gate's acceptable range of
7 volts to 10 volts for a "high" signal. If we use an open-
collector TTL gate instead of a totem-pole output gate,
though, a pullup resistor to the 10 volt Vgg supply rail will
raise the TTL gate's "high" output voltage to the full power
supply voltage supplying the CMOS gate. Since an open-
collector gate can only sink current, not source current, the
"high" state voltage level is entirely determined by the
power supply to which the pullup resistor is attached, thus
neatly solving the mismatch problem:
: 10 V =yo ----------- lov
7V
TTL CMOS
output input
3V
ony Sseac5522-> OV
Now, both "low" and "high"
TTL signals are acceptable
to the CMOS gate input
Due to the excellent output voltage characteristics of CMOS
gates, there is typically no problem connecting a CMOS
output to a TTL input. The only significant issue is the
current loading presented by the TTL inputs, since the CMOS
output must sink current for each of the TTL inputs while in
the "low" state.
When the CMOS gate in question is powered by a voltage
source in excess of 5 volts (V,,), though, a problem will
result. The "high" output state of the CMOS gate, being
greater than 5 volts, will exceed the TTL gate's acceptable
input limits for a "high" signal. A solution to this problem is
to create an "open-collector" inverter circuit using a discrete
NPN transistor, and use it to interface the two gates
together:
The "Roullup | resistor is optional, since TTL inputs
automatically assume a "high" state when left floating,
which is what will happen when the CMOS gate output is
"low" and the transistor cuts off. Of course, one very
important consequence of implementing this solution is the
logical inversion created by the transistor: when the CMOS
gate outputs a "low" signal, the TTL gate sees a "high" input;
and when the CMOS gate outputs a "high" signal, the
transistor saturates and the TTL gate sees a "low" input. So
long as this inversion is accounted for in the logical scheme
of the system, all will be well.
DIP gate packaging
Digital logic gate circuits are manufactured as integrated
circuits: all the constituent transistors and resistors built on
a single piece of semiconductor material. The engineer,
technician, or hobbyist using small numbers of gates will
likely find what he or she needs enclosed in a DIP (Dual
Inline Package) housing. DIP-enclosed integrated circuits
are available with even numbers of pins, located at 0.100
inch intervals from each other for standard circuit board
layout compatibility. Pin counts of 8, 14, 16, 18, and 24 are
common for DIP "chips."
Part numbers given to these DIP packages specify what type
of gates are enclosed, and how many. These part numbers
are industry standards, meaning that a "74LS02"
manufactured by Motorola will be identical in function to a
"7 4LS02" manufactured by Fairchild or by any other
manufacturer. Letter codes prepended to the part number
are unique to the manufacturer, and are not industry-
standard codes. For instance, a SN74LSO02 is a quad 2-input
TTL NOR gate manufactured by Motorola, while a DM74LS02
is the exact same circuit manufactured by Fairchild.
Logic circuit part numbers beginning with "74" are
commercial-grade TTL. If the part number begins with the
number "54", the chip is a military-grade unit: having a
greater operating temperature range, and typically more
robust in regard to allowable power supply and signal
voltage levels. The letters "LS" immediately following the
74/54 prefix indicate "Low-power Schottky" circuitry, using
Schottky-barrier diodes and transistors throughout, to
decrease power dissipation. Non-Schottky gate circuits
consume more power, but are able to operate at higher
frequencies due to their faster switching times.
A few of the more common TTL "DIP" circuit packages are
shown here for reference:
5400/7400 5402/7402
Quad NAND gate Quad NOR gate
5408/7408 5432/7432
Quad AND gate
5486/7486 5404/7404
4011 4001
Quad NAND gate Quad NOR gate
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Jan-Willem Rensman (May 2, 2002): Suggested the
inclusion of Schmitt triggers and gate hysteresis to this
chapter.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
|| 4] l_—
—| | +4/l—
Lessons In Electric Circuits
-- Volume IV
Chapter 4
SWITCHES
e Switch types
e Switch contact design
e Contact "normal" state and make/break sequence
e Contact "bounce"
Switch types
An electrical switch is any device used to interrupt the flow of
electrons in a circuit. Switches are essentially binary devices:
they are either completely on ("closed") or completely off
("open"). There are many different types of switches, and we
will explore some of these types in this chapter.
Though it may seem strange to cover this elementary
electrical topic at such a late stage in this book series, | do so
because the chapters that follow explore an older realm of
digital technology based on mechanical switch contacts
rather than solid-state gate circuits, and a thorough
understanding of switch types is necessary for the
undertaking. Learning the function of switch-based circuits at
the same time that you learn about solid-state logic gates
makes both topics easier to grasp, and sets the stage for an
enhanced learning experience in Boolean algebra, the
mathematics behind digital logic circuits.
The simplest type of switch is one where two electrical
conductors are brought in contact with each other by the
motion of an actuating mechanism. Other switches are more
complex, containing electronic circuits able to turn on or off
depending on some physical stimulus (such as light or
magnetic field) sensed. In any case, the final output of any
switch will be (at least) a pair of wire-connection terminals
that will either be connected together by the switch's
internal contact mechanism ("closed"), or not connected
together ("open").
Any switch designed to be operated by a person is generally
called a hand switch, and they are manufactured in several
varieties:
Toggle switch
saad sat
Toggle switches are actuated by a lever angled in one of two
or more positions. The common light switch used in
household wiring is an example of a toggle switch. Most
toggle switches will come to rest in any of their lever
positions, while others have an internal spring mechanism
returning the lever to a certain normal position, allowing for
what is called "momentary" operation.
Pushbutton switch
=i i
—@e eo
Pushbutton switches are two-position devices actuated with a
button that is pressed and released. Most pushbutton
switches have an internal spring mechanism returning the
button to its "out," or "Uunpressed," position, for momentary
operation. Some pushbutton switches will latch alternately
on or off with every push of the button. Other pushbutton
switches will stay in their "in," or "pressed," position until the
button is pulled back out. This last type of pushbutton
switches usually have a mushroom-shaped button for easy
push-pull action.
Selector switch
—2)le—
—e o—
Selector switches are actuated with a rotary knob or lever of
some sort to select one of two or more positions. Like the
toggle switch, selector switches can either rest in any of their
positions or contain spring-return mechanisms for
momentary operation.
Joystick switch
=
—®
A joystick switch is actuated by a lever free to move in more
than one axis of motion. One or more of several switch
contact mechanisms are actuated depending on which way
the lever is pushed, and sometimes by how far it is pushed.
The circle-and-dot notation on the switch symbol represents
the direction of joystick lever motion required to actuate the
contact. Joystick hand switches are commonly used for crane
and robot control.
Some switches are specifically designed to be operated by
the motion of a machine rather than by the hand of a human
operator. These motion-operated switches are commonly
called /imit switches, because they are often used to limit the
motion of a machine by turning off the actuating power to a
component if it moves too far. As with hand switches, limit
switches come in several varieties:
Lever actuator limit switch
ge
These limit switches closely resemble rugged toggle or
selector hand switches fitted with a lever pushed by the
machine part. Often, the levers are tipped with a small roller
bearing, preventing the lever from being worn off by
repeated contact with the machine part.
Proximity switch
prox
a
Proximity switches sense the approach of a metallic machine
part either by a magnetic or high-frequency electromagnetic
field. Simple proximity switches use a permanent magnet to
actuate a sealed switch mechanism whenever the machine
part gets close (typically 1 inch or less). More complex
proximity switches work like a metal detector, energizing a
coil of wire with a high-frequency current, and electronically
monitoring the magnitude of that current. If a metallic part
(not necessarily magnetic) gets close enough to the coil, the
current will increase, and trip the monitoring circuit. The
symbol shown here for the proximity switch is of the
electronic variety, as indicated by the diamond-shaped box
surrounding the switch. A non-electronic proximity switch
would use the same symbol as the lever-actuated limit
switch.
Another form of proximity switch is the optical switch,
comprised of a light source and photocell. Machine position is
detected by either the interruption or reflection of a light
beam. Optical switches are also useful in safety applications,
where beams of light can be used to detect personnel entry
into a dangerous area.
In many industrial processes, it is necessary to monitor
various physical quantities with switches. Such switches can
be used to sound alarms, indicating that a process variable
has exceeded normal parameters, or they can be used to
shut down processes or equipment if those variables have
reached dangerous or destructive levels. There are many
different types of process switches:
Speed switch
r
—ao—
>
These switches sense the rotary speed of a shaft either by a
centrifugal weight mechanism mounted on the shaft, or by
some kind of non-contact detection of shaft motion such as
optical or magnetic.
Pressure switch
Gas or liquid pressure can be used to actuate a switch
mechanism if that pressure is applied to a piston, diaphragm,
or bellows, which converts pressure to mechanical force.
Temperature switch
An inexpensive temperature-sensing mechanism is the
“bimetallic strip:" a thin strip of two metals, joined back-to-
back, each metal having a different rate of thermal
expansion. When the strip heats or cools, differing rates of
thermal expansion between the two metals causes it to bend.
The bending of the strip can then be used to actuate a switch
contact mechanism. Other temperature switches use a brass
bulb filled with either a liquid or gas, with a tiny tube
connecting the bulb to a pressure-sensing switch. As the bulb
is heated, the gas or liquid expands, generating a pressure
increase which then actuates the switch mechanism.
Liquid level switch
on
A floating object can be used to actuate a switch mechanism
when the liquid level in an tank rises past a certain point. If
the liquid is electrically conductive, the liquid itself can be
used as a conductor to bridge between two metal probes
inserted into the tank at the required depth. The conductivity
technique is usually implemented with a special design of
relay triggered by a small amount of current through the
conductive liquid. In most cases it is impractical and
dangerous to switch the full load current of the circuit
through a liquid.
Level switches can also be designed to detect the level of
solid materials such as wood chips, grain, coal, or animal
feed in a storage silo, bin, or hopper. A common design for
this application is a small paddle wheel, inserted into the bin
at the desired height, which is slowly turned by a small
electric motor. When the solid material fills the bin to that
height, the material prevents the paddle wheel from turning.
The torque response of the small motor than trips the switch
mechanism. Another design uses a "tuning fork" shaped
metal prong, inserted into the bin from the outside at the
desired height. The fork is vibrated at its resonant frequency
by an electronic circuit and magnet/electromagnet coil
assembly. When the bin fills to that height, the solid material
dampens the vibration of the fork, the change in vibration
amplitude and/or frequency detected by the electronic
circuit.
Liquid flow switch
a
Inserted into a pipe, a flow switch will detect any gas or
liquid flow rate in excess of a certain threshold, usually with
a small paddle or vane which is pushed by the flow. Other
flow switches are constructed as differential pressure
switches, measuring the pressure drop across a restriction
built into the pipe.
Another type of level switch, suitable for liquid or solid
material detection, is the nuclear switch. Composed of a
radioactive source material and a radiation detector, the two
are mounted across the diameter of a storage vessel for
either solid or liquid material. Any height of material beyond
the level of the source/detector arrangement will attenuate
the strength of radiation reaching the detector. This decrease
in radiation at the detector can be used to trigger a relay
mechanism to provide a switch contact for measurement,
alarm point, or even control of the vessel level.
Nuclear level switch
(for solid or liquid material)
source detector
source __] detector
Both source and detector are outside of the vessel, with no
intrusion at all except the radiation flux itself. The
radioactive sources used are fairly weak and pose no
immediate health threat to operations or maintenance
personnel.
As usual, there is usually more than one way to implement a
switch to monitor a physical process or serve as an operator
control. There is usually no single "perfect" switch for any
application, although some obviously exhibit certain
advantages over others. Switches must be intelligently
matched to the task for efficient and reliable operation.
e REVIEW:
e A switch is an electrical device, usually
electromechanical, used to control continuity between
two points.
e Hand switches are actuated by human touch.
e Limit switches are actuated by machine motion.
e Process switches are actuated by changes in some
physical process (temperature, level, flow, etc.).
Switch contact design
A switch can be constructed with any mechanism bringing
two conductors into contact with each other in a controlled
manner. This can be as simple as allowing two copper wires
to touch each other by the motion of a lever, or by directly
pushing two metal strips into contact. However, a good
switch design must be rugged and reliable, and avoid
presenting the operator with the possibility of electric shock.
Therefore, industrial switch designs are rarely this crude.
The conductive parts in a switch used to make and break the
electrical connection are called contacts. Contacts are
typically made of silver or silver-cadmium alloy, whose
conductive properties are not significantly compromised by
surface corrosion or oxidation. Gold contacts exhibit the best
corrosion resistance, but are limited in current-carrying
Capacity and may "cold weld" if brought together with high
mechanical force. Whatever the choice of metal, the switch
contacts are guided by a mechanism ensuring square and
even contact, for maximum reliability and minimum
resistance.
Contacts such as these can be constructed to handle
extremely large amounts of electric current, up to thousands
of amps in some cases. The limiting factors for switch contact
ampacity are as follows:
e Heat generated by current through metal contacts (while
closed).
e Sparking caused when contacts are opened or closed.
e The voltage across open switch contacts (potential of
current jumping across the gap).
One major disadvantage of standard switch contacts is the
exposure of the contacts to the surrounding atmosphere. In a
nice, clean, control-room environment, this is generally not a
problem. However, most industrial environments are not this
benign. The presence of corrosive chemicals in the air can
cause contacts to deteriorate and fail prematurely. Even more
troublesome is the possibility of regular contact sparking
causing flammable or explosive chemicals to ignite.
When such environmental concerns exist, other types of
contacts can be considered for small switches. These other
types of contacts are sealed from contact with the outside
air, and therefore do not suffer the same exposure problems
that standard contacts do.
A common type of sealed-contact switch is the mercury
switch. Mercury is a metallic element, liquid at room
temperature. Being a metal, it possesses excellent
conductive properties. Being a liquid, it can be brought into
contact with metal probes (to close a circuit) inside of a
sealed chamber simply by tilting the chamber so that the
probes are on the bottom. Many industrial switches use small
glass tubes containing mercury which are tilted one way to
close the contact, and tilted another way to open. Aside from
the problems of tube breakage and spilling mercury (which is
a toxic material), and susceptibility to vibration, these
devices are an excellent alternative to open-air switch
contacts wherever environmental exposure problems are a
concern.
Here, a mercury switch (often called a t//E switch) is shown in
the open position, where the mercury Is out of contact with
the two metal contacts at the other end of the glass bulb:
Here, the same switch is shown in the closed position.
Gravity now holds the liquid mercury in contact with the two
metal contacts, providing electrical continuity from one to
the other:
Mercury switch contacts are impractical to build in large
sizes, and so you will typically find such contacts rated at no
more than a few amps, and no more than 120 volts. There are
exceptions, of course, but these are common limits.
Another sealed-contact type of switch is the magnetic reed
switch. Like the mercury switch, a reed switch's contacts are
located inside a sealed tube. Unlike the mercury switch
which uses liquid metal as the contact medium, the reed
switch is simply a pair of very thin, magnetic, metal strips
(hence the name "reed") which are brought into contact with
each other by applying a strong magnetic field outside the
sealed tube. The source of the magnetic field in this type of
switch is usually a permanent magnet, moved closer to or
further away from the tube by the actuating mechanism. Due
to the small size of the reeds, this type of contact is typically
rated at lower currents and voltages than the average
mercury switch. However, reed switches typically handle
vibration better than mercury contacts, because there is no
liquid inside the tube to splash around.
It is common to find general-purpose switch contact voltage
and current ratings to be greater on any given switch or relay
if the electric power being switched is AC instead of DC. The
reason for this is the self-extinguishing tendency of an
alternating-current arc across an air gap. Because 60 Hz
power line current actually stops and reverses direction 120
times per second, there are many opportunities for the
ionized air of an arc to lose enough temperature to stop
conducting current, to the point where the arc will not re-
start on the next voltage peak. DC, on the other hand, is a
continuous, uninterrupted flow of electrons which tends to
maintain an arc across an air gap much better. Therefore,
switch contacts of any kind incur more wear when switching
a given value of direct current than for the same value of
alternating current. The problem of switching DC is
exaggerated when the load has a significant amount of
inductance, as there will be very high voltages generated
across the switch's contacts when the circuit is opened (the
inductor doing its best to maintain circuit current at the
Same magnitude as when the switch was closed).
With both AC and DC, contact arcing can be minimized with
the addition of a "snubber" circuit (a capacitor and resistor
wired in series) in parallel with the contact, like this:
"Snubber"
R C
EL
A sudden rise in voltage across the switch contact caused by
the contact opening will be tempered by the capacitor's
charging action (the capacitor opposing the increase in
voltage by drawing current). The resistor limits the amount of
current that the capacitor will discharge through the contact
when it closes again. If the resistor were not there, the
Capacitor might actually make the arcing during contact
closure worse than the arcing during contact opening
without a capacitor! While this addition to the circuit helps
mitigate contact arcing, it is not without disadvantage: a
prime consideration is the possibility of a failed (shorted)
Ccapacitor/resistor combination providing a path for electrons
to flow through the circuit at all times, even when the
contact is open and current is not desired. The risk of this
failure, and the severity of the resulting consequences must
be considered against the increased contact wear (and
inevitable contact failure) without the snubber circuit.
The use of snubbers in DC switch circuits is nothing new:
automobile manufacturers have been doing this for years on
engine ignition systems, minimizing the arcing across the
switch contact "points" in the distributor with a small
Capacitor called a condenser. As any mechanic can tell you,
the service life of the distributor's "points" is directly related
to how well the condenser is functioning.
With all this discussion concerning the reduction of switch
contact arcing, one might be led to think that less current is
always better for a mechanical switch. This, however, is not
necessarily so. It has been found that a small amount of
periodic arcing can actually be good for the switch contacts,
because it keeps the contact faces free from small amounts
of dirt and corrosion. If a mechanical switch contact is
operated with too little current, the contacts will tend to
accumulate excessive resistance and may fail prematurely!
This minimum amount of electric current necessary to keep a
mechanical switch contact in good health is called the
wetting current.
Normally, a switch's wetting current rating is far below its
maximum current rating, and well below its normal operating
current load in a properly designed system. However, there
are applications where a mechanical switch contact may be
required to routinely handle currents below normal wetting
current limits (for instance, if a mechanical selector switch
needs to open or close a digital logic or analog electronic
circuit where the current value is extremely small). In these
applications, is it highly recommended that gold-plated
switch contacts be specified. Gold is a "noble" metal and
does not corrode as other metals will. Such contacts have
extremely low wetting current requirements as a result.
Normal silver or copper alloy contacts will not provide
reliable operation if used in such low-current service!
e REVIEW:
e The parts of a switch responsible for making and
breaking electrical continuity are called the "contacts."
Usually made of corrosion-resistant metal alloy, contacts
are made to touch each other by a mechanism which
helps maintain proper alignment and spacing.
e Mercury switches use a slug of liquid mercury metal as a
moving contact. Sealed in a glass tube, the mercury
contact's spark is sealed from the outside environment,
making this type of switch ideally suited for atmospheres
potentially harboring explosive vapors.
e Reed switches are another type of sealed-contact device,
contact being made by two thin metal "reeds" inside a
glass tube, brought together by the influence of an
external magnetic field.
e Switch contacts suffer greater duress switching DC than
AC. This is primarily due to the self-extinguishing nature
of an AC arc.
e A resistor-capacitor network called a "snubber" can be
connected in parallel with a switch contact to reduce
contact arcing.
e Wetting currentis the minimum amount of electric
current necessary for a switch contact to carry in order
for it to be self-cleaning. Normally this value is far below
the switch's maximum current rating.
Contact "normal" state and
make/break sequence
Any kind of switch contact can be designed so that the
contacts "close" (establish continuity) when actuated, or
"open" (interrupt continuity) when actuated. For switches
that have a spring-return mechanism in them, the direction
that the spring returns it to with no applied force is called the
normal position. Therefore, contacts that are open in this
position are called normally open and contacts that are
closed in this position are called normally closed.
For process switches, the normal position, or state, is that
which the switch is in when there is no process influence on
it. An easy way to figure out the normal condition of a
process switch is to consider the state of the switch as it sits
on a storage shelf, uninstalled. Here are some examples of
"normal" process switch conditions:
e Speed switch: Shaft not turning
e Pressure switch: Zero applied pressure
e Temperature switch: Ambient (room) temperature
e Level switch: Empty tank or bin
e Flow switch: Zero liquid flow
It is important to differentiate between a switch's "normal"
condition and its "normal" use in an operating process.
Consider the example of a liquid flow switch that serves as a
low-flow alarm in a cooling water system. The normal, or
properly-operating, condition of the cooling water system is
to have fairly constant coolant flow going through this pipe.
If we want the flow switch's contact to close in the event of a
loss of coolant flow (to complete an electric circuit which
activates an alarm siren, for example), we would want to use
a flow switch with normally-closed rather than normally-open
contacts. When there's adequate flow through the pipe, the
switch's contacts are forced open; when the flow rate drops
to an abnormally low level, the contacts return to their
normal (closed) state. This is confusing if you think of
"normal" as being the regular state of the process, so be sure
to always think of a switch's "normal" state as that which its
in as it sits on a shelf.
The schematic symbology for switches vary according to the
switch's purpose and actuation. A normally-open switch
contact is drawn in such a way as to Signify an open
connection, ready to close when actuated. Conversely, a
normally-closed switch is drawn as a closed connection which
will be opened when actuated. Note the following symbols:
Pushbutton switch
Normally-open Normally-closed
There is also a generic symbology for any switch contact,
using a pair of vertical lines to represent the contact points in
a switch. Normally-open contacts are designated by the lines
not touching, while normally-closed contacts are designated
with a diagonal line bridging between the two lines. Compare
the two:
Generic switch contact designation
Normally-open Normally-closed
4h a
The switch on the left will close when actuated, and will be
open while in the "normal" (unactuated) position. The switch
on the right will open when actuated, and is closed in the
"normal" (unactuated) position. If switches are designated
with these generic symbols, the type of switch usually will be
noted in text immediately beside the symbol. Please note
that the symbol on the left is not to be confused with that of
a capacitor. If a capacitor needs to be represented in a
control logic schematic, it will be shown like this:
Capacitor
io
In standard electronic symbology, the figure shown above is
reserved for polarity-sensitive capacitors. In control logic
symbology, this capacitor symbol is used for any type of
capacitor, even when the capacitor is not polarity sensitive,
so as to Clearly distinguish it from a normally-open switch
contact.
With multiple-position selector switches, another design
factor must be considered: that is, the sequence of breaking
old connections and making new connections as the switch is
moved from position to position, the moving contact
touching several stationary contacts in sequence.
-——_ l
=< 3
common____, _,,
3
, 4
L__ 5
The selector switch shown above switches a common contact
lever to one of five different positions, to contact wires
numbered 1 through 5. The most common configuration of a
multi-position switch like this is one where the contact with
one position is broken before the contact with the next
position is made. This configuration is called break-before-
make. To give an example, if the switch were set at position
number 3 and slowly turned clockwise, the contact lever
would move off of the number 3 position, opening that
circuit, move to a position between number 3 and number 4
(both circuit paths open), and then touch position number 4,
closing that circuit.
There are applications where it is unacceptable to completely
open the circuit attached to the "common" wire at any point
in time. For such an application, a make-before-break switch
design can be built, in which the movable contact lever
actually bridges between two positions of contact (between
number 3 and number 4, in the above scenario) as it travels
between positions. The compromise here is that the circuit
must be able to tolerate switch closures between adjacent
position contacts (1 and 2, 2 and 3, 3 and 4, 4 and 5) as the
selector knob is turned from position to position. Such a
switch is shown here:
p—!
———— 9
common i ae :
3
ne |
er
When movable contact(s) can be brought into one of several
positions with stationary contacts, those positions are
sometimes called throws. The number of movable contacts Is
sometimes called po/es. Both selector switches shown above
with one moving contact and five stationary contacts would
be designated as "single-pole, five-throw" switches.
If two identical single-pole, five-throw switches were
mechanically ganged together so that they were actuated by
the same mechanism, the whole assembly would be called a
"double-pole, five-throw" switch:
Double-pole, 5-throw switch
assembly
Here are a few common switch configurations and their
abbreviated designations:
Single-pole, single-throw
(SPST)
fe
Double-pole, single-throw
(DPST)
—e—
==
Single-pole, double-throw
(SPDT)
H
Double-pole , double-throw
(DPDT)
Four-pole , double-throw
(4PDT
—
fi id
e REVIEW:
e The norma! state of a switch is that where it is
unactuated. For process switches, this is the condition its
in when sitting on a shelf, uninstalled.
e A switch that is open when unactuated is called
normally-open. A switch that is closed when unactuated
is called normally-closed. Sometimes the terms
“normally-open" and "normally-closed" are abbreviated
N.O. and N.C., respectively.
e The generic symbology for N.O. and N.C. switch contacts
is as follows:
Generic switch contact designation
Normally-open Normally-closed
_ db a
e Multiposition switches can be either break-before-make
(most common) or make-before-break.
e The "poles" of a switch refers to the number of moving
contacts, while the "throws" of a switch refers to the
number of stationary contacts per moving contact.
Contact "bounce"
When a switch is actuated and contacts touch one another
under the force of actuation, they are supposed to establish
continuity in a single, crisp moment. Unfortunately, though,
switches do not exactly achieve this goal. Due to the mass of
the moving contact and any elasticity inherent in the
mechanism and/or contact materials, contacts will "bounce"
upon closure for a period of milliseconds before coming toa
full rest and providing unbroken contact. In many
applications, switch bounce is of no consequence: it matters
little if a switch controlling an incandescent lamp "bounces"
for a few cycles every time it is actuated. Since the lamp's
warm-up time greatly exceeds the bounce period, no
irregularity in lamp operation will result.
However, if the switch is used to send a signal to an
electronic amplifier or some other circuit with a fast response
time, contact bounce may produce very noticeable and
undesired effects:
Switch
actuated
I Kis
S;
A closer look at the oscilloscope display reveals a rather ugly
set of makes and breaks when the switch is actuated a single
time:
Close-up view of oscilloscope display:
Switch is actuated
Contacts bouncing
If, for example, this switch is used to provide a "clock" signal
to a digital counter circuit, so that each actuation of the
pushbutton switch is supposed to increment the counter by a
value of 1, what will happen instead is the counter will
increment by several counts each time the switch is
actuated. Since mechanical switches often interface with
digital electronic circuits in modern systems, switch contact
bounce Is a frequent design consideration. Somehow, the
"chattering" produced by bouncing contacts must be
eliminated so that the receiving circuit sees a clean, crisp
off/on transition:
"Bounceless" switch operation
Switch is actuated
Switch contacts may be debounced several different ways.
The most direct means is to address the problem at its
source: the switch itself. Here are some suggestions for
designing switch mechanisms for minimum bounce:
Reduce the kinetic energy of the moving contact. This
will reduce the force of impact as it comes to rest on the
stationary contact, thus minimizing bounce.
Use "buffer springs" on the stationary contact(s) so that
they are free to recoil and gently absorb the force of
impact from the moving contact.
Design the switch for "wiping" or "sliding" contact rather
than direct impact. "Knife" switch designs use sliding
contacts.
Dampen the switch mechanism's movement using an air
or oil "shock absorber" mechanism.
Use sets of contacts in parallel with each other, each
Slightly different in mass or contact gap, so that when
one is rebounding off the stationary contact, at least one
of the others will still be in firm contact.
"Wet" the contacts with liquid mercury in a sealed
environment. After initial contact is made, the surface
tension of the mercury will maintain circuit continuity
even though the moving contact may bounce off the
stationary contact several times.
Each one of these suggestions sacrifices some aspect of
switch performance for limited bounce, and so it is
impractical to design a// switches with limited contact
bounce in mind. Alterations made to reduce the kinetic
energy of the contact may result in a small open-contact gap
or a slow-moving contact, which limits the amount of voltage
the switch may handle and the amount of current it may
interrupt. Sliding contacts, while non-bouncing, still produce
"noise" (irregular current caused by irregular contact
resistance when moving), and suffer from more mechanical
wear than normal contacts.
Multiple, parallel contacts give less bounce, but only at
greater switch complexity and cost. Using mercury to "wet"
the contacts is a very effective means of bounce mitigation,
but it is unfortunately limited to switch contacts of low
ampacity. Also, mercury-wetted contacts are usually limited
in mounting position, as gravity may cause the contacts to
“bridge” accidently if oriented the wrong way.
If re-designing the switch mechanism is not an option,
mechanical switch contacts may be debounced externally,
using other circuit components to condition the signal. A low-
pass filter circuit attached to the output of the switch, for
example, will reduce the voltage/current fluctuations
generated by contact bounce:
Switch
actuated
Switch contacts may be debounced electronically, using
hysteretic transistor circuits (circuits that "latch" in either a
high or a low state) with built-in time delays (called "one-
shot" circuits), or two inputs controlled by a double-throw
switch. These hysteretic circuits, called mu/tivibrators, are
discussed in detail in a later chapter.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+4]—
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 5
ELECTROMECHANICAL
RELAYS
Relay_construction
Contactors
Time-delay relays
Protective relays
Solid-state relays
Relay construction
An electric current through a conductor will produce a
magnetic field at right angles to the direction of electron
flow. If that conductor is wrapped into a coil shape, the
magnetic field produced will be oriented along the length of
the coil. The greater the current, the greater the strength of
the magnetic field, all other factors being equal:
Inductors react against changes in current because of the
energy stored in this magnetic field. When we construct a
transformer from two inductor coils around a common iron
core, we use this field to transfer energy from one coil to the
other. However, there are simpler and more direct uses for
electromagnetic fields than the applications we've seen with
inductors and transformers. The magnetic field produced by
a coil of current-carrying wire can be used to exert a
mechanical force on any magnetic object, just as we can use
a permanent magnet to attract magnetic objects, except
that this magnet (formed by the coil) can be turned on or off
by switching the current on or off through the coil.
If we place a magnetic object near such a coil for the
purpose of making that object move when we energize the
coil with electric current, we have what is called a solenoid.
The movable magnetic object is called an armature, and
most armatures can be moved with either direct current (DC)
or alternating current (AC) energizing the coil. The polarity
of the magnetic field is irrelevant for the purpose of
attracting an iron armature. Solenoids can be used to
electrically open door latches, open or shut valves, move
robotic limbs, and even actuate electric switch mechanisms.
However, if a solenoid is used to actuate a set of switch
contacts, we have a device so useful it deserves its own
name: the relay.
Relays are extremely useful when we have a need to control
a large amount of current and/or voltage with a small
electrical signal. The relay coil which produces the magnetic
field may only consume fractions of a watt of power, while
the contacts closed or opened by that magnetic field may be
able to conduct hundreds of times that amount of power to a
load. In effect, a relay acts as a binary (on or off) amplifier.
Just as with transistors, the relay's ability to control one
electrical signal with another finds application in the
construction of logic functions. This topic will be covered in
greater detail in another lesson. For now, the relay's
"amplifying" ability will be explored.
relay
Load
In the above schematic, the relay's coil is energized by the
low-voltage (12 VDC) source, while the single-pole, single-
throw (SPST) contact interrupts the high-voltage (480 VAC)
circuit. It is quite likely that the current required to energize
the relay coil will be hundreds of times less than the current
rating of the contact. Typical relay coil currents are well
below 1 amp, while typical contact ratings for industrial
relays are at least 10 amps.
One relay coil/armature assembly may be used to actuate
more than one set of contacts. Those contacts may be
normally-open, normally-closed, or any combination of the
two. As with switches, the "normal" state of a relay's
contacts is that state when the coil is de-energized, just as
you would find the relay sitting on a shelf, not connected to
any circuit.
Relay contacts may be open-air pads of metal alloy, mercury
tubes, or even magnetic reeds, just as with other types of
switches. The choice of contacts in a relay depends on the
same factors which dictate contact choice in other types of
switches. Open-air contacts are the best for high-current
applications, but their tendency to corrode and spark may
cause problems in some industrial environments. Mercury
and reed contacts are sparkless and won't corrode, but they
tend to be limited in current-carrying capacity.
Shown here are three small relays (about two inches in
height, each), installed on a panel as part of an electrical
control system at a municipal water treatment plant:
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The relay units shown here are called "octal-base," because
they plug into matching sockets, the electrical connections
secured via eight metal pins on the relay bottom. The screw
terminal connections you see in the photograph where wires
connect to the relays are actually part of the socket
assembly, into which each relay is plugged. This type of
construction facilitates easy removal and replacement of the
relay(s) in the event of failure.
Aside from the ability to allow a relatively small electric
signal to switch a relatively large electric signal, relays also
offer electrical isolation between coil and contact circuits.
This means that the coil circuit and contact circuit(s) are
electrically insulated from one another. One circuit may be
DC and the other AC (such as in the example circuit shown
earlier), and/or they may be at completely different voltage
levels, across the connections or from connections to
ground.
While relays are essentially binary devices, either being
completely on or completely off, there are operating
conditions where their state may be indeterminate, just as
with semiconductor logic gates. In order for a relay to
positively "pull in" the armature to actuate the contact(s),
there must be a certain minimum amount of current through
the coil. This minimum amount is called the pu//-in current,
and it is analogous to the minimum input voltage that a
logic gate requires to guarantee a "high" state (typically 2
Volts for TTL, 3.5 Volts for CMOS). Once the armature is
pulled closer to the coil's center, however, it takes less
magnetic field flux (less coil current) to hold it there.
Therefore, the coil current must drop below a value
significantly lower than the pull-in current before the
armature "drops out" to its spring-loaded position and the
contacts resume their normal state. This current level is
called the drop-out current, and it is analogous to the
maximum input voltage that a logic gate input will allow to
guarantee a "low" state (typically 0.8 Volts for TTL, 1.5 Volts
for CMOS).
The hysteresis, or difference between pull-in and drop-out
currents, results in operation that is similar to a Schmitt
trigger logic gate. Pull-in and drop-out currents (and
voltages) vary widely from relay to relay, and are specified
by the manufacturer.
e REVIEW:
e A solenoid is a device that produces mechanical motion
from the energization of an electromagnet coil. The
movable portion of a solenoid is called an armature.
e A relay is a solenoid set up to actuate switch contacts
when its coil is energized.
e Pull-in current is the minimum amount of coil current
needed to actuate a solenoid or relay from its "normal"
(de-energized) position.
e Drop-out current is the maximum coil current below
which an energized relay will return to its "normal" state.
Contactors
When a relay is used to switch a large amount of electrical
power through its contacts, it is designated by a special
name: contactor. Contactors typically have multiple
contacts, and those contacts are usually (but not always)
normally-open, so that power to the load is shut off when the
coil is de-energized. Perhaps the most common industrial
use for contactors is the control of electric motors.
relay
A |
3-phase yd
AC power B ] (ot
ie gf,
|
ae ee ae
|
120 VAC
coil
The top three contacts switch the respective phases of the
incoming 3-phase AC power, typically at least 480 Volts for
motors 1 horsepower or greater. The lowest contact is an
"auxiliary" contact which has a current rating much lower
than that of the large motor power contacts, but is actuated
by the same armature as the power contacts. The auxiliary
contact is often used in a relay logic circuit, or for some
other part of the motor control scheme, typically switching
120 Volt AC power instead of the motor voltage. One
contactor may have several auxiliary contacts, either
normally-open or normally-closed, if required.
The three "opposed-question-mark" shaped devices in series
with each phase going to the motor are called overload
heaters. Each "heater" element is a low-resistance strip of
metal intended to heat up as the motor draws current. If the
temperature of any of these heater elements reaches a
critical point (equivalent to a moderate overloading of the
motor), a normally-closed switch contact (not shown in the
diagram) will spring open. This normally-closed contact is
usually connected in series with the relay coil, so that when
it opens the relay will automatically de-energize, thereby
shutting off power to the motor. We will see more of this
overload protection wiring in the next chapter. Overload
heaters are intended to provide overcurrent protection for
large electric motors, unlike circuit breakers and fuses which
serve the primary purpose of providing overcurrent
protection for power conductors.
Overload heater function is often misunderstood. They are
not fuses; that is, it is not their function to burn open and
directly break the circuit as a fuse is designed to do. Rather,
overload heaters are designed to thermally mimic the
heating characteristic of the particular electric motor to be
protected. All motors have thermal characteristics, including
the amount of heat energy generated by resistive
dissipation (I2R), the thermal transfer characteristics of heat
"conducted" to the cooling medium through the metal frame
of the motor, the physical mass and specific heat of the
materials constituting the motor, etc. These characteristics
are mimicked by the overload heater on a miniature scale:
when the motor heats up toward its critical temperature, so
will the heater toward /ts critical temperature, ideally at the
same rate and approach curve. Thus, the overload contact,
in sensing heater temperature with a thermo-mechanical
mechanism, will sense an analogue of the real motor. If the
overload contact trips due to excessive heater temperature,
it will be an indication that the real motor has reached its
critical temperature (or, would have done so in a short
while). After tripping, the heaters are supposed to cool down
at the same rate and approach curve as the real motor, so
that they indicate an accurate proportion of the motor's
thermal condition, and will not allow power to be re-applied
until the motor is truly ready for start-up again.
Shown here is a contactor for a three-phase electric motor,
installed on a panel as part of an electrical control system at
a municipal water treatment plant:
TS
WON J¥AOZI
g-—WiSOu
Three-phase, 480 volt AC power comes in to the three
normally-open contacts at the top of the contactor via screw
terminals labeled "L1,"""L2," and "L3" (The "L2" terminal is
hidden behind a square-shaped "sSnubber" circuit connected
across the contactor's coil terminals). Power to the motor
exits the overload heater assembly at the bottom of this
device via screw terminals labeled "T1," "T2," and "T3."
The overload heater units themselves are black, square-
Shaped blocks with the label "W34," indicating a particular
thermal response for a certain horsepower and temperature
rating of electric motor. If an electric motor of differing
power and/or temperature ratings were to be substituted for
the one presently in service, the overload heater units would
have to be replaced with units having a thermal response
suitable for the new motor. The motor manufacturer can
provide information on the appropriate heater units to use.
A white pushbutton located between the "T1" and "T2" line
heaters serves as a way to manually re-set the normally-
closed switch contact back to its normal state after having
been tripped by excessive heater temperature. Wire
connections to the "overload" switch contact may be seen at
the lower-right of the photograph, near a label reading "NC"
(normally-closed). On this particular overload unit, a small
“window" with the label "Tripped" indicates a tripped
condition by means of a colored flag. In this photograph,
there is no "tripped" condition, and the indicator appears
clear.
As a footnote, heater elements may be used as a crude
current shunt resistor for determining whether or nota
motor is drawing current when the contactor is closed. There
may be times when you're working on a motor control
circuit, where the contactor is located far away from the
motor itself. How do you know if the motor is consuming
power when the contactor coil is energized and the armature
has been pulled in? If the motor's windings are burnt open,
you could be sending voltage to the motor through the
contactor contacts, but still have zero current, and thus no
motion from the motor shaft. If a clamp-on ammeter isn't
available to measure line current, you can take your
multimeter and measure millivoltage across each heater
element: if the current is zero, the voltage across the heater
will be zero (unless the heater element itself is open, in
which case the voltage across it will be large); if there is
Current going to the motor through that phase of the
contactor, you will read a definite millivoltage across that
heater:
This is an especially useful trick to use for troubleshooting 3-
phase AC motors, to see if one phase winding is burnt open
or disconnected, which will result in a rapidly destructive
condition known as "single-phasing." If one of the lines
carrying power to the motor is open, it will not have any
current through it (as indicated by a 0.00 mV reading across
its heater), although the other two lines will (as indicated by
small amounts of voltage dropped across the respective
heaters).
¢ REVIEW:
e A contactor is a large relay, usually used to switch
current to an electric motor or other high-power load.
e Large electric motors can be protected from overcurrent
damage through the use of overload heaters and
overload contacts. |If the series-connected heaters get
too hot from excessive current, the normally-closed
overload contact will open, de-energizing the contactor
sending power to the motor.
Time-delay relays
Some relays are constructed with a kind of "shock absorber"
mechanism attached to the armature which prevents
immediate, full motion when the coil is either energized or
de-energized. This addition gives the relay the property of
time-delay actuation. Time-delay relays can be constructed
to delay armature motion on coil energization, de-
energization, or both.
Time-delay relay contacts must be specified not only as
either normally-open or normally-closed, but whether the
delay operates in the direction of closing or in the direction
of opening. The following is a description of the four basic
types of time-delay relay contacts.
First we have the normally-open, timed-closed (NOTC)
contact. This type of contact is normally open when the coil
is unpowered (de-energized). The contact is closed by the
application of power to the relay coil, but only after the coil
has been continuously powered for the specified amount of
time. In other words, the direction of the contact's motion
(either to close or to open) is identical to a regular NO
contact, but there is a delay in closing direction. Because
the delay occurs in the direction of coil energization, this
type of contact is alternatively known as a normally-open,
on-delay:
Normally-open, timed-closed
5 sec.
Closes 5 seconds after coil energization |
Opens immediately upon coil de-energization
The following is a timing diagram of this relay contact's
operation:
NOTC
as
5 sec.
on
Coil | |
power off
nee 5 S_
seconds closed
Contact | |
status open
in ——
Next we have the normally-open, timed-open (NOTO)
contact. Like the NOTC contact, this type of contact is
normally open when the coil is Uunpowered (de-energized),
and closed by the application of power to the relay coil.
However, unlike the NOTC contact, the timing action occurs
upon de-energization of the coil rather than upon
energization. Because the delay occurs in the direction of
coil de-energization, this type of contact is alternatively
known as a normally-open, off-delay:
Normally-open, timed-open
i
5 sec.
Closes immediately upon coil energization
Opens 5 seconds after coil de-energization
The following is a timing diagram of this relay contact's
operation:
NOTO
ee
5 SEC.
on
Coil | |
power off
~~ >
seconds closed
Contact | |
status open
ine ——
Next we have the normally-closed, timed-open (NCTO)
contact. This type of contact is normally closed when the coil
is unpowered (de-energized). The contact is opened with the
application of power to the relay coil, but only after the coil
has been continuously powered for the specified amount of
time. In other words, the direction of the contact's motion
(either to close or to open) is identical to a regular NC
contact, but there is a delay in the opening direction.
Because the delay occurs in the direction of coil
energization, this type of contact is alternatively known asa
normally-closed, on-delay:
Normally-closed, timed-open
5 Sec.
Opens 5 seconds after coil energization
Closes immediately upon coil de-energization
The following is a timing diagram of this relay contact's
operation:
NCTO
se
5 sec.
on
Coil | |
power off
-_—5—>
seconds closed
Contact
status open
Time —q~
Finally we have the normally-closed, timed-closed (NCTC)
contact. Like the NCTO contact, this type of contact is
normally closed when the coil is unpowered (de-energized),
and opened by the application of power to the relay coil.
However, unlike the NCTO contact, the timing action occurs
upon de-energization of the coil rather than upon
energization. Because the delay occurs in the direction of
coil de-energization, this type of contact is alternatively
known as a normally-closed, off-delay:
Normally-closed, timed-closed
—-t—
5 sec.
Opens immediately upon coil energization
Closes 5 seconds after coil de-energization
The following is a timing diagram of this relay contact's
operation:
NCTC
lr
2 Sec.
on
Coil | |
power off
~*-5O>
seconds closed
Contact | |
status open
n———
Time-delay relays are very important for use in industrial
control logic circuits. Some examples of their use include:
e Flashing light control (time on, time off): two time-delay
relays are used in conjunction with one another to
provide a constant-frequency on/off pulsing of contacts
for sending intermittent power to a lamp.
e Engine autostart control: Engines that are used to power
emergency generators are often equipped with
"autostart" controls that allow for automatic start-up if
the main electric power fails. To properly start a large
engine, certain auxiliary devices must be started first
and allowed some brief time to stabilize (fuel pumps,
pre-lubrication oil pumps) before the engine's starter
motor is energized. Time-delay relays help sequence
these events for proper start-up of the engine.
e Furnace safety purge control: Before a combustion-type
furnace can be safely lit, the air fan must be run fora
specified amount of time to "purge" the furnace
chamber of any potentially flammable or explosive
vapors. A time-delay relay provides the furnace control
logic with this necessary time element.
Motor soft-start delay control: Instead of starting large
electric motors by switching full power from a dead stop
condition, reduced voltage can be switched for a "softer"
start and less inrush current. After a prescribed time
delay (provided by a time-delay relay), full power is
applied.
e Conveyor belt sequence delay: when multiple conveyor
belts are arranged to transport material, the conveyor
belts must be started in reverse sequence (the last one
first and the first one last) so that material doesn't get
piled on to a stopped or slow-moving conveyor. In order
to get large belts up to full soeed, some time may be
needed (especially if soft-start motor controls are used).
For this reason, there is usually a time-delay circuit
arranged on each conveyor to give it adequate time to
attain full belt soeed before the next conveyor belt
feeding it is started.
The older, mechanical time-delay relays used pneumatic
dashpots or fluid-filled piston/cylinder arrangements to
provide the "shock absorbing" needed to delay the motion of
the armature. Newer designs of time-delay relays use
electronic circuits with resistor-capacitor (RC) networks to
generate a time delay, then energize a normal
(instantaneous) electromechanical relay coil with the
electronic circuit's output. The electronic-timer relays are
more versatile than the older, mechanical models, and less
prone to failure. Many models provide advanced timer
features such as "one-shot" (one measured output pulse for
every transition of the input from de-energized to
energized), "recycle" (repeated on/off output cycles for as
long as the input connection is energized) and "watchdog"
(changes state if the input signal does not repeatedly cycle
on and off).
"One-shot” normally-open relay contact
on
Coil ee ee
power off
time
~~ ee
closed
Contact | |
status open
Time ——~-
"Recycle" normally-open relay contact
on
Coil | |
power off
closed
Contact | | | | | |
status open
ine —-
"Watchdog" relay contact
on
Coil | | | | | | | |
power off
—» time ~—
closed
Contact |
status open
ine —_——_—
The "watchdog" timer is especially useful for monitoring of
computer systems. If a computer is being used to control a
critical process, it is usually recommended to have an
automatic alarm to detect computer "lockup" (an abnormal
halting of program execution due to any number of causes).
An easy way to set up such a monitoring system is to have
the computer regularly energize and de-energize the coil of
a watchdog timer relay (similar to the output of the "recycle"
timer). If the computer execution halts for any reason, the
signal it outputs to the watchdog relay coil will stop cycling
and freeze in one or the other state. A short time thereafter,
the watchdog relay will "time out" and signal a problem.
REVIEW:
Time delay relays are built in these four basic modes of
contact operation:
1: Normally-open, timed-closed. Abbreviated "NOTC",
these relays open immediately upon coil de-energization
and close only if the coil is continuously energized for
the time duration period. Also called normally-open, on-
delay relays.
2: Normally-open, timed-open. Abbreviated "NOTO",
these relays close immediately upon coil energization
and open after the coil has been de-energized for the
time duration period. Also called normally-open, off
delay relays.
3: Normally-closed, timed-open. Abbreviated "NCTO",
these relays close immediately upon coil de-energization
and open only if the coil is continuously energized for
the time duration period. Also called normally-closed,
on-delay relays.
4: Normally-closed, timed-closed. Abbreviated "NCTC",
these relays open immediately upon coil energization
and close after the coil has been de-energized for the
time duration period. Also called normally-closed, off
delay relays.
One-shot timers provide a single contact pulse of
specified duration for each coil energization (transition
from coil offto coil on).
Recycle timers provide a repeating sequence of on-off
contact pulses as long as the coil is maintained in an
energized state.
Watchdog timers actuate their contacts only if the coil
fails to be continuously sequenced on and off (energized
and de-energized) at a minimum frequency.
Protective relays
A special type of relay is one which monitors the current,
voltage, frequency, or any other type of electric power
measurement either from a generating source or to a load
for the purpose of triggering a circuit breaker to open in the
event of an abnormal condition. These relays are referred to
in the electrical power industry as protective relays.
The circuit breakers which are used to switch large
quantities of electric power on and off are actually
electromechanical relays, themselves. Unlike the circuit
breakers found in residential and commercial use which
determine when to trip (open) by means of a bimetallic strip
inside that bends when it gets too hot from overcurrent,
large industrial circuit breakers must be "told" by an
external device when to open. Such breakers have two
electromagnetic coils inside: one to close the breaker
contacts and one to open them. The "trip" coil can be
energized by one or more protective relays, as well as by
hand switches, connected to switch 125 Volt DC power. DC
power is used because it allows for a battery bank to supply
close/trip power to the breaker control circuits in the event
of a complete (AC) power failure.
Protective relays can monitor large AC currents by means of
current transformers (CT's), which encircle the current-
carrying conductors exiting a large circuit breaker,
transformer, generator, or other device. Current transformers
step down the monitored current to a secondary (output)
range of 0 to 5 amps AC to power the protective relay. The
current relay uses this 0-5 amp signal to power its internal
mechanism, closing a contact to switch 125 Volt DC power to
the breaker's trip coil if the monitored current becomes
excessive.
Likewise, (protective) voltage relays can monitor high AC
voltages by means of voltage, or potential, transformers
(PT's) which step down the monitored voltage to a
secondary range of 0 to 120 Volts AC, typically. Like
(protective) current relays, this voltage signal powers the
internal mechanism of the relay, closing a contact to switch
125 Volt DC power to the breaker's trip coil is the monitored
voltage becomes excessive.
There are many types of protective relays, some with highly
specialized functions. Not all monitor voltage or current,
either. They all, however, share the common feature of
outputting a contact closure signal which can be used to
switch power to a breaker trip coil, close coil, or operator
alarm panel. Most protective relay functions have been
categorized into an ANSI standard number code. Here area
few examples from that code list:
ANSI protective relay designation numbers
12 = Overspeed
24 = Overexcitation
25 = Syncrocheck
27 = Bus/Line undervoltage
32 = Reverse power (anti-motoring)
38 = Stator overtemp (RTD)
39 = Bearing vibration
40 = Loss of excitation
46 = Negative sequence undercurrent (phase current imbalance)
47 = Negative sequence undervoltage (phase voltage imbalance)
49 = Bearing overtemp (RTD)
50 = Instantaneous overcurrent
51 = Time overcurrent
51V = Time overcurrent -- voltage restrained
Power factor
Bus overvoltage
60FL = Voltage transformer fuse failure
67 = Phase/Ground directional current
79 = Autoreclose
81 = Bus over/underfrequency
e REVIEW:
e Large electric circuit breakers do not contain within
themselves the necessary mechanisms to automatically
trip (open) in the event of overcurrent conditions. They
must be "told" to trip by external devices.
e Protective relays are devices built to automatically
trigger the actuation coils of large electric circuit
breakers under certain conditions.
Solid-state relays
As versatile as electromechanical relays can be, they do
suffer many limitations. They can be expensive to build,
have a limited contact cycle life, take up a lot of room, and
switch slowly, compared to modern semiconductor devices.
These limitations are especially true for large power
contactor relays. To address these limitations, many relay
manufacturers offer "solid-state" relays, which use an SCR,
TRIAC, or transistor output instead of mechanical contacts to
switch the controlled power. The output device (SCR, TRIAC,
or transistor) is optically-coupled to an LED light source
inside the relay. The relay is turned on by energizing this
LED, usually with low-voltage DC power. This optical
isolation between input to output rivals the best that
electromechanical relays can offer.
Solid-state relay
Load
LED Opto-TRIAC
Being solid-state devices, there are no moving parts to wear
out, and they are able to switch on and off much faster than
any mechanical relay armature can move. There is no
sparking between contacts, and no problems with contact
corrosion. However, solid-state relays are still too expensive
to build in very high current ratings, and so
electromechanical contactors continue to dominate that
application in industry today.
One significant advantage of a solid-state SCR or TRIAC
relay over an electromechanical device is its natural
tendency to open the AC circuit only at a point of zero load
current. Because SCR's and TRIAC's are thyristors, their
inherent hysteresis maintains circuit continuity after the LED
is de-energized until the AC current falls below a threshold
value (the holding current). In practical terms what this
means is the circuit will never be interrupted in the middle
of a sine wave peak. Such untimely interruptions in a circuit
containing substantial inductance would normally produce
large voltage spikes due to the sudden magnetic field
collapse around the inductance. This will not happen ina
circuit broken by an SCR or TRIAC. This feature is called
zero-crossover switching.
One disadvantage of solid state relays is their tendency to
fail "shorted" on their outputs, while electromechanical relay
contacts tend to fail "open." In either case, it is possible fora
relay to fail in the other mode, but these are the most
common failures. Because a "fail-open" state is generally
considered safer than a "fail-closed" state,
electromechanical relays are still favored over their solid-
state counterparts in many applications.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
|| 4] l_—
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 6
LADDER LOGIC
"Ladder" diagrams
Digital logic functions
Permissive and interlock circuits
Motor control circuits
Fail-safe design
Programmable logic controllers
Contributors
"Ladder" diagrams
Ladder diagrams are specialized schematics commonly used
to document industrial control logic systems. They are called
"ladder" diagrams because they resemble a ladder, with two
vertical rails (supply power) and as many "rungs" (horizontal
lines) as there are control circuits to represent. If we wanted
to draw a simple ladder diagram showing a lamp that is
controlled by a hand switch, it would look like this:
L, Ls
Switch Lamp
The "L," and "L," designations refer to the two poles of a
120 VAC supply, unless otherwise noted. L; is the "hot"
conductor, and L, is the grounded ("neutral") conductor.
These designations have nothing to do with inductors, just
to make things confusing. The actual transformer or
generator supplying power to this circuit is omitted for
simplicity. In reality, the circuit looks something like this:
To 480 volt AC
power source (typical)
fuse fuse
te "control power"
transformer
Typically in industrial relay logic circuits, but not always, the
operating voltage for the switch contacts and relay coils will
be 120 volts AC. Lower voltage AC and even DC systems are
sometimes built and documented according to "ladder"
diagrams:
24 VDC
|
L; fuse
Switch
So long as the switch contacts and relay coils are all
adequately rated, it really doesn't matter what level of
voltage is chosen for the system to operate with.
Note the number "1" on the wire between the switch and the
lamp. In the real world, that wire would be labeled with that
number, using heat-shrink or adhesive tags, wherever it was
convenient to identify. Wires leading to the switch would be
labeled "L," and "1," respectively. Wires leading to the lamp
would be labeled "1" and "L;," respectively. These wire
numbers make assembly and maintenance very easy. Each
conductor has its own unique wire number for the control
system that its used in. Wire numbers do not change at any
junction or node, even if wire size, color, or length changes
going into or out of a connection point. Of course, it is
preferable to maintain consistent wire colors, but this is not
always practical. What matters is that any one, electrically
continuous point in a control circuit possesses the same wire
number. Take this circuit section, for example, with wire #25
as a Single, electrically continuous point threading to many
different devices:
25
In ladder diagrams, the load device (lamp, relay coil,
solenoid coil, etc.) is almost always drawn at the right-hand
side of the rung. While it doesn't matter electrically where
the relay coil is located within the rung, it does matter which
end of the ladder's power supply is grounded, for reliable
operation.
Take for instance this circuit:
Here, the lamp (load) is located on the right-hand side of the
rung, and so is the ground connection for the power source.
This is no accident or coincidence; rather, it is a purposeful
element of good design practice. Suppose that wire #1 were
to accidently come in contact with ground, the insulation of
that wire having been rubbed off so that the bare conductor
came in contact with grounded, metal conduit. Our circuit
would now function like this:
Fuse will blow
if switch is
closed! VW
Lamp cannot light!
accidental ground
With both sides of the lamp connected to ground, the lamp
will be "shorted out" and unable to receive power to light up.
If the switch were to close, there would be a short-circuit,
immediately blowing the fuse.
However, consider what would happen to the circuit with the
same fault (wire #1 coming in contact with ground), except
this time we'll swap the positions of switch and fuse (L> is
still grounded):
Switch has no
effect!
Lamp is energized!
accidental ground
This time the accidental grounding of wire #1 will force
power to the lamp while the switch will have no effect. It is
much safer to have a system that blows a fuse in the event
of a ground fault than to have a system that uncontrollably
energizes lamps, relays, or solenoids in the event of the
same fault. For this reason, the load(s) must always be
located nearest the grounded power conductor in the ladder
diagram.
e REVIEW:
e Ladder diagrams (sometimes called "ladder logic") area
type of electrical notation and symbology frequently
used to illustrate how electromechanical switches and
relays are interconnected.
e The two vertical lines are called "rails" and attach to
opposite poles of a power supply, usually 120 volts AC.
L, designates the "hot" AC wire and L, the "neutral"
(grounded) conductor.
Horizontal lines in a ladder diagram are called "rungs,"
each one representing a unique parallel circuit branch
between the poles of the power supply.
e Typically, wires in control systems are marked with
numbers and/or letters for identification. The rule is, all
permanently connected (electrically common) points
must bear the same label.
Digital logic functions
We can construct simply logic functions for our hypothetical
lamp circuit, using multiple contacts, and document these
circuits quite easily and understandably with additional
rungs to our original "ladder." If we use standard binary
notation for the status of the switches and lamp (0 for
unactuated or de-energized; 1 for actuated or energized), a
truth table can be made to show how the logic works:
L, L,
Now, the lamp will come on if either contact A or contact B is
actuated, because all it takes for the lamp to be energized is
to have at least one path for current from wire L, to wire 1.
What we have is a simple OR logic function, implemented
with nothing more than contacts and a lamp.
We can mimic the AND logic function by wiring the two
contacts in series instead of parallel:
Now, the lamp energizes only if contact A and contact B are
simultaneously actuated. A path exists for current from wire
L, to the lamp (wire 2) if and only if both switch contacts are
closed.
The logical inversion, or NOT, function can be performed on
a contact input simply by using a normally-closed contact
instead of a normally-open contact:
fof 1
Fe ec
Now, the lamp energizes if the contact is not actuated, and
de-energizes when the contact /s actuated.
If we take our OR function and invert each "input" through
the use of normally-closed contacts, we will end up with a
NAND function. In a special branch of mathematics known as
Boolean algebra, this effect of gate function identity
changing with the inversion of input signals is described by
DeMorgan's Theorem, a subject to be explored in more
detail in a later chapter.
or
>
B
The lamp will be energized if e/ther contact is unactuated. It
will go out only if both contacts are actuated simultaneously.
Likewise, if we take our AND function and invert each "input"
through the use of normally-closed contacts, we will end up
with a NOR function:
A pattern quickly reveals itself when ladder circuits are
compared with their logic gate counterparts:
e Parallel contacts are equivalent to an OR gate.
e Series contacts are equivalent to an AND gate.
e Normally-closed contacts are equivalent to a NOT gate
(inverter).
We can build combinational logic functions by grouping
contacts in series-parallel arrangements, as well. In the
following example, we have an Exclusive-OR function built
from a combination of AND, OR, and inverter (NOT) gates:
or
ee
The top rung (NC contact A in series with NO contact B) is
the equivalent of the top NOT/AND gate combination. The
bottom rung (NO contact A in series with NC contact B) is
the equivalent of the bottom NOT/AND gate combination.
The parallel connection between the two rungs at wire
number 2 forms the equivalent of the OR gate, in allowing
either rung 1 orrung 2 to energize the lamp.
To make the Exclusive-OR function, we had to use two
contacts per input: one for direct input and the other for
"inverted" input. The two "A" contacts are physically
actuated by the same mechanism, as are the two "B"
contacts. The common association between contacts is
denoted by the label of the contact. There is no limit to how
many contacts per switch can be represented in a ladder
diagram, as each new contact on any switch or relay (either
normally-open or normally-closed) used in the diagram is
simply marked with the same label.
Sometimes, multiple contacts on a single switch (or relay)
are designated by a compound labels, such as "A-1" and "A-
2" instead of two "A" labels. This may be especially useful if
you want to specifically designate which set of contacts on
each switch or relay is being used for which part of a circuit.
For simplicity's sake, I'll refrain from such elaborate labeling
in this lesson. If you see a common label for multiple
contacts, you know those contacts are all actuated by the
Same mechanism.
If we wish to invert the output of any switch-generated logic
function, we must use a relay with a normally-closed
contact. For instance, if we want to energize a load based on
the inverse, or NOT, of a normally-open contact, we could do
this:
Ly L
A CR1
We will call the relay, "control relay 1," or CR. When the coil
of CR, (symbolized with the pair of parentheses on the first
rung) is energized, the contact on the second rung opens,
thus de-energizing the lamp. From switch A to the coil of
CR, the logic function is noninverted. The normally-closed
contact actuated by relay coil CR; provides a logical inverter
function to drive the lamp opposite that of the switch's
actuation status.
Applying this inversion strategy to one of our inverted-input
functions created earlier, such as the OR-to-NAND, we can
invert the output with a relay to create a noninverted
function:
Li i
From the switches to the coil of CRj, the logical function is
that of a NAND gate. CR,'s normally-closed contact provides
one final inversion to turn the NAND function into an AND
function.
e REVIEW:
e Parallel contacts are logically equivalent to an OR gate.
e Series contacts are logically equivalent to an AND gate.
e Normally closed (N.C.) contacts are logically equivalent
to a NOT gate.
e Arelay must be used to invert the output of a logic gate
function, while simple normally-closed switch contacts
are sufficient to represent inverted gate /nputs.
Permissive and interlock circuits
A practical application of switch and relay logic is in control
systems where several process conditions have to be met
before a piece of equipment is allowed to start. A good
example of this is burner control for large combustion
furnaces. In order for the burners in a large furnace to be
started safely, the control system requests "permission" from
several process switches, including high and low fuel
pressure, air fan flow check, exhaust stack damper position,
access door position, etc. Each process condition is called a
permissive, and each permissive switch contact is wired in
series, so that if any one of them detects an unsafe
condition, the circuit will be opened:
low fuel high fuel minimum damper
pressure pressure — air flow open
——«
Green light = conditions met: safe to start
Red light = conditions not met: unsafe to start
If all permissive conditions are met, CR, will energize and
the green lamp will be lit. In real life, more than just a green
lamp would be energized: usually a control relay or fuel
valve solenoid would be placed in that rung of the circuit to
be energized when all the permissive contacts were "good:"
that is, all closed. If any one of the permissive conditions are
not met, the series string of switch contacts will be broken,
CR, will de-energize, and the red lamp will light.
Note that the high fuel pressure contact is normally-closed.
This is because we want the switch contact to open if the
fuel pressure gets too high. Since the "normal" condition of
any pressure switch is when zero (low) pressure is being
applied to it, and we want this switch to open with excessive
(high) pressure, we must choose a switch that is closed in its
normal state.
Another practical application of relay logic is in control
systems where we want to ensure two incompatible events
cannot occur at the same time. An example of this is in
reversible motor control, where two motor contactors are
wired to switch polarity (or phase sequence) to an electric
motor, and we don't want the forward and reverse
contactors energized simultaneously:
M1
A
3-phase p
AC
power 4
M1 = forward
M2 = reverse
M2
When contactor M, is energized, the 3 phases (A, B, and C)
are connected directly to terminals 1, 2, and 3 of the motor,
respectively. However, when contactor M> is energized,
phases A and B are reversed, A going to motor terminal 2
and B going to motor terminal 1. This reversal of phase wires
results in the motor spinning the opposite direction. Let's
examine the control circuit for these two contactors:
L, Lg
forward
reverse
a
Take note of the normally-closed "OL" contact, which is the
thermal overload contact activated by the "heater" elements
wired in series with each phase of the AC motor. If the
heaters get too hot, the contact will change from its normal
(closed) state to being open, which will prevent either
contactor from energizing.
This control system will work fine, so long as no one pushes
both buttons at the same time. If someone were to do that,
phases A and B would be short-circuited together by virtue
of the fact that contactor M, sends phases A and B straight
to the motor and contactor M> reverses them; phase A would
be shorted to phase B and vice versa. Obviously, this is a
bad control system design!
To prevent this occurrence from happening, we can design
the circuit so that the energization of one contactor prevents
the energization of the other. This is called interlocking, and
it is accomplished through the use of auxiliary contacts on
each contactor, as such:
Ly L
forward
reverse
lL. 5
Now, when M, is energized, the normally-closed auxiliary
contact on the second rung will be open, thus preventing Mp
from being energized, even if the "Reverse" pushbutton is
actuated. Likewise, M,'s energization is prevented when Mp
is energized. Note, as well, how additional wire numbers (4
and 5) were added to reflect the wiring changes.
It should be noted that this is not the only way to interlock
contactors to prevent a short-circuit condition. Some
contactors come equipped with the option of a mechanical
interlock: a lever joining the armatures of two contactors
together so that they are physically prevented from
simultaneous closure. For additional safety, electrical
interlocks may still be used, and due to the simplicity of the
circuit there is no good reason not to employ them in
addition to mechanical interlocks.
e REVIEW:
e Switch contacts installed in a rung of ladder logic
designed to interrupt a circuit if certain physical
conditions are not met are called permissive contacts,
because the system requires permission from these
inputs to activate.
Switch contacts designed to prevent a control system
from taking two incompatible actions at once (such as
powering an electric motor forward and backward
simultaneously) are called interlocks.
Motor control circuits
The interlock contacts installed in the previous section's
motor control circuit work fine, but the motor will run only as
long as each pushbutton switch is held down. If we wanted
to keep the motor running even after the operator takes his
or her hand off the control switch(es), we could change the
circuit in a couple of different ways: we could replace the
pushbutton switches with toggle switches, or we could add
some more relay logic to "latch" the control circuit with a
single, momentary actuation of either switch. Let's see how
the second approach is implemented, since it is commonly
used in industry:
reverse M1
1. 5
When the "Forward" pushbutton is actuated, M, will
energize, closing the normally-open auxiliary contact in
parallel with that switch. When the pushbutton is released,
the closed M, auxiliary contact will maintain current to the
coil of M,, thus latching the "Forward" circuit in the "on"
state. The same sort of thing will happen when the "Reverse"
pushbutton is pressed. These parallel auxiliary contacts are
sometimes referred to as sea/-in contacts, the word "seal"
meaning essentially the same thing as the word /atch.
However, this creates a new problem: how to stop the motor!
As the circuit exists right now, the motor will run either
forward or backward once the corresponding pushbutton
switch is pressed, and will continue to run as long as there is
power. To stop either circuit (forward or backward), we
require some means for the operator to interrupt power to
the motor contactors. We'll call this new switch, Stop:
Ly L
M1
reverse
Jl. §
Now, if either forward or reverse circuits are latched, they
may be "unlatched" by momentarily pressing the "Stop"
pushbutton, which will open either forward or reverse circuit,
de-energizing the energized contactor, and returning the
seal-in contact to its normal (open) state. The "Stop" switch,
having normally-closed contacts, will conduct power to
either forward or reverse circuits when released.
So far, so good. Let's consider another practical aspect of our
motor control scheme before we quit adding to it. If our
hypothetical motor turned a mechanical load with a lot of
momentum, such as a large air fan, the motor might
continue to coast for a substantial amount of time after the
stop button had been pressed. This could be problematic if
an operator were to try to reverse the motor direction
without waiting for the fan to stop turning. If the fan was still
coasting forward and the "Reverse" pushbutton was pressed,
the motor would struggle to overcome that inertia of the
large fan as it tried to begin turning in reverse, drawing
excessive current and potentially reducing the life of the
motor, drive mechanisms, and fan. What we might like to
have is some kind of a time-delay function in this motor
control system to prevent such a premature startup from
happening.
Let's begin by adding a couple of time-delay relay coils, one
in parallel with each motor contactor coil. If we use contacts
that delay returning to their normal state, these relays will
provide us a "memory" of which direction the motor was last
powered to turn. What we want each time-delay contact to
do is to open the starting-switch leg of the opposite rotation
circuit for several seconds, while the fan coasts to a halt.
L, i
stop _ forward
Bl 7 Wee pa
reverse
M1 M2
_| g TD1 5 2
If the motor has been running in the forward direction, both
M, and TD, will have been energized. This being the case,
the normally-closed, timed-closed contact of TD; between
wires 8 and 5 will have immediately opened the moment
TD, was energized. When the stop button is pressed, contact
TD, waits for the specified amount of time before returning
to its normally-closed state, thus holding the reverse
pushbutton circuit open for the duration so M, can't be
energized. When TD, times out, the contact will close and
the circuit will allow M, to be energized, if the reverse
pushbutton is pressed. In like manner, TD, will prevent the
"Forward" pushbutton from energizing M, until the
prescribed time delay after M, (and TDz) have been de-
energized.
The careful observer will notice that the time-interlocking
functions of TD; and TD, render the M, and Mz interlocking
contacts redundant. We can get rid of auxiliary contacts M,
and Mb, for interlocks and just use TD, and TD,'s contacts,
since they immediately open when their respective relay
coils are energized, thus "locking out" one contactor if the
other is energized. Each time delay relay will serve a dual
purpose: preventing the other contactor from energizing
while the motor is running, and preventing the same
contactor from energizing until a prescribed time after motor
shutdown. The resulting circuit has the advantage of being
simpler than the previous example:
Lj L
reverse M2
_| 5 =TOD1
M2
e REVIEW:
e Motor contactor (or "starter") coils are typically
designated by the letter "M" in ladder logic diagrams.
e Continuous motor operation with a momentary "start"
switch is possible if a normally-open "seal-in" contact
from the contactor is connected in parallel with the start
switch, so that once the contactor is energized it
maintains power to itself and keeps itself "latched" on.
e Time delay relays are commonly used in large motor
control circuits to prevent the motor from being started
(or reversed) until a certain amount of time has elapsed
from an event.
Fail-safe design
Logic circuits, whether comprised of electromechanical
relays or solid-state gates, can be built in many different
ways to perform the same functions. There is usually no one
"correct" way to design a complex logic circuit, but there are
usually ways that are better than others.
In control systems, safety is (or at least should be) an
important design priority. If there are multiple ways in which
a digital control circuit can be designed to perform a task,
and one of those ways happens to hold certain advantages
in safety over the others, then that design is the better one
to choose.
Let's take a look at a simple system and consider how it
might be implemented in relay logic. Suppose that a large
laboratory or industrial building is to be equipped with a fire
alarm system, activated by any one of several latching
switches installed throughout the facility. The system should
work so that the alarm siren will energize if any one of the
switches is actuated. At first glance it seems as though the
relay logic should be incredibly simple: just use normally-
open switch contacts and connect them all in parallel with
each other:
Lj E
switch 1 siren
switch 2
switch 3
switch 4
Essentially, this is the OR logic function implemented with
four switch inputs. We could expand this circuit to include
any number of switch inputs, each new switch being added
to the parallel network, but I'll limit it to four in this example
to keep things simple. At any rate, it is an elementary
system and there seems to be little possibility of trouble.
Except in the event of a wiring failure, that is. The nature of
electric circuits is such that "open" failures (open switch
contacts, broken wire connections, open relay coils, blown
fuses, etc.) are statistically more likely to occur than any
other type of failure. With that in mind, it makes sense to
engineer a circuit to be as tolerant as possible to sucha
failure. Let's suppose that a wire connection for Switch #2
were to fail open:
[; L
switch 1 siren
switch 2
. open wire connection!
switch 3 aa
switch 4
If this failure were to occur, the result would be that Switch
#2 would no longer energize the siren if actuated. This,
obviously, is not good in a fire alarm system. Unless the
system were regularly tested (a good idea anyway), no one
would know there was a problem until someone tried to use
that switch in an emergency.
What if the system were re-engineered so as to sound the
alarm in the event of an open failure? That way, a failure in
the wiring would result in a false alarm, a scenario much
more preferable than that of having a switch silently fail and
not function when needed. In order to achieve this design
goal, we would have to re-wire the switches so that an open
contact sounded the alarm, rather than a closed contact.
That being the case, the switches will have to be normally-
closed and in series with each other, powering a relay coil
which then activates a normally-closed contact for the siren:
switch 1 switch 3
switch 2 switch 4
siren
When all switches are unactuated (the regular operating
state of this system), relay CR, will be energized, thus
keeping contact CR, open, preventing the siren from being
powered. However, if any of the switches are actuated, relay
CR, will de-energize, closing contact CR; and sounding the
alarm. Also, if there is a break in the wiring anywhere in the
top rung of the circuit, the alarm will sound. When it is
discovered that the alarm is false, the workers in the facility
will know that something failed in the alarm system and that
it needs to be repaired.
Granted, the circuit is more complex than it was before the
addition of the control relay, and the system could still fail in
the "silent" mode with a broken connection in the bottom
rung, but its still a safer design than the original circuit, and
thus preferable from the standpoint of safety.
This design of circuit is referred to as fail-safe, due to its
intended design to default to the safest mode in the event of
a common failure such as a broken connection in the switch
wiring. Fail-safe design always starts with an assumption as
to the most likely kind of wiring or component failure, and
then tries to configure things so that such a failure will
cause the circuit to act in the safest way, the "safest way"
being determined by the physical characteristics of the
process.
Take for example an electrically-actuated (solenoid) valve for
turning on cooling water to a machine. Energizing the
solenoid coil will move an armature which then either opens
or closes the valve mechanism, depending on what kind of
valve we specify. A spring will return the valve to its
"normal" position when the solenoid is de-energized. We
already know that an open failure in the wiring or solenoid
coil is more likely than a short or any other type of failure, so
we should design this system to be in its safest mode with
the solenoid de-energized.
If its cooling water we're controlling with this valve, chances
are it is safer to have the cooling water turn on in the event
of a failure than to shut off, the consequences of a machine
running without coolant usually being severe. This means
we should specify a valve that turns on (opens up) when de-
energized and turns off (closes down) when energized. This
may seem "backwards" to have the valve set up this way,
but it will make for a safer system in the end.
One interesting application of fail-safe design is in the power
generation and distribution industry, where large circuit
breakers need to be opened and closed by electrical control
signals from protective relays. If a 50/51 relay
(instantaneous and time overcurrent) is going to command a
circuit breaker to trip (open) in the event of excessive
current, should we design it so that the relay closes a switch
contact to send a "trip" signal to the breaker, or opens a
switch contact to interrupt a regularly "on" signal to initiate
a breaker trip? We know that an open connection will be the
most likely to occur, but what is the safest state of the
system: breaker open or breaker closed?
At first, it would seem that it would be safer to have a large
circuit breaker trip (open up and shut off power) in the event
of an open fault in the protective relay control circuit, just
like we had the fire alarm system default to an alarm state
with any switch or wiring failure. However, things are not so
simple in the world of high power. To have a large circuit
breaker indiscriminately trip open is no small matter,
especially when customers are depending on the continued
supply of electric power to supply hospitals,
telecommunications systems, water treatment systems, and
other important infrastructures. For this reason, power
system engineers have generally agreed to design
protective relay circuits to output a closed contact signal
(power applied) to open large circuit breakers, meaning that
any open failure in the control wiring will go unnoticed,
simply leaving the breaker in the status quo position.
Is this an ideal situation? Of course not. If a protective relay
detects an overcurrent condition while the control wiring is
failed open, it will not be able to trip open the circuit
breaker. Like the first fire alarm system design, the "silent"
failure will be evident only when the system is needed.
However, to engineer the control circuitry the other way -- so
that any open failure would immediately shut the circuit
breaker off, potentially blacking out large potions of the
power grid -- really isn't a better alternative.
An entire book could be written on the principles and
practices of good fail-safe system design. At least here, you
know a couple of the fundamentals: that wiring tends to fail
open more often than shorted, and that an electrical control
system's (open) failure mode should be such that it
indicates and/or actuates the real-life process in the safest
alternative mode. These fundamental principles extend to
non-electrical systems as well: identify the most common
mode of failure, then engineer the system so that the
probable failure mode places the system in the safest
condition.
e REVIEW:
e The goal of fail-safe design is to make a control system
as tolerant as possible to likely wiring or component
failures.
e The most common type of wiring and component failure
isan "open" circuit, or broken connection. Therefore, a
fail-safe system should be designed to default to its
safest mode of operation in the case of an open circuit.
Programmable logic controllers
Before the advent of solid-state logic circuits, logical control
systems were designed and built exclusively around
electromechanical relays. Relays are far from obsolete in
modern design, but have been replaced in many of their
former roles as logic-level control devices, relegated most
often to those applications demanding high current and/or
high voltage switching.
Systems and processes requiring "on/off" control abound in
modern commerce and industry, but such control systems
are rarely built from either electromechanical relays or
discrete logic gates. Instead, digital computers fill the need,
which may be programmed to do a variety of logical
functions.
In the late 1960's an American company named Bedford
Associates released a computing device they called the
MODICON. As an acronym, it meant Modular Digital
Controller, and later became the name of a company
division devoted to the design, manufacture, and sale of
these special-purpose control computers. Other engineering
firms developed their own versions of this device, and it
eventually came to be known in non-proprietary terms as a
PLC, or Programmable Logic Controller. The purpose of a PLC
was to directly replace electromechanical relays as logic
elements, substituting instead a solid-state digital computer
with a stored program, able to emulate the interconnection
of many relays to perform certain logical tasks.
A PLC has many "input" terminals, through which it
interprets "high" and "low" logical states from sensors and
switches. It also has many output terminals, through which it
outputs "high" and "low" signals to power lights, solenoids,
contactors, small motors, and other devices lending
themselves to on/off control. In an effort to make PLCs easy
to program, their programming language was designed to
resemble ladder logic diagrams. Thus, an industrial
electrician or electrical engineer accustomed to reading
ladder logic schematics would feel comfortable
programming a PLC to perform the same control functions.
PLCs are industrial computers, and as such their input and
output signals are typically 120 volts AC, just like the
electromechanical control relays they were designed to
replace. Although some PLCs have the ability to input and
output low-level DC voltage signals of the magnitude used
in logic gate circuits, this is the exception and not the rule.
Signal connection and programming standards vary
somewhat between different models of PLC, but they are
similar enough to allow a "generic" introduction to PLC
programming here. The following illustration shows a simple
PLC, as it might appear from a front view. Two screw
terminals provide connection to 120 volts AC for powering
the PLC's internal circuitry, labeled L1 and L2. Six screw
terminals on the left-hand side provide connection to input
devices, each terminal representing a different input
"channel" with its own "X" label. The lower-left screw
terminal is a "Common" connection, which is generally
connected to L2 (neutral) of the 120 VAC power source.
@oxl
@Qox2
@ox3
@oxa
@Ooxs
@Qoxe
@ Common
Inside the PLC housing, connected between each input
terminal and the Common terminal, is an opto-isolator
device (Light-Emitting Diode) that provides an electrically
isolated "high" logic signal to the computer's circuitry (a
photo-transistor interprets the LED's light) when there is 120
VAC power applied between the respective input terminal
and the Common terminal. An indicating LED on the front
panel of the PLC gives visual indication of an "energized"
input:
Program ming
(Pet
Y10@
Y20@
Y30@
y40@
¥50@
y60@
Source@
Input X1 en pee
npu energize:
a "9 Y30@
PLC y40@
Y50@
y60@
c— Source@
port
Programming
~ Common
Output signals are generated by the PLC's computer
circuitry activating a switching device (transistor, TRIAC, or
even an electromechanical relay), connecting the "Source"
terminal to any of the "Y-" labeled output terminals. The
"Source" terminal, correspondingly, is usually connected to
the L1 side of the 120 VAC power source. As with each input,
an indicating LED on the front panel of the PLC gives visual
indication of an "energized" output:
@oxl
@ox2
@ox3
@oxa
@oxs
Output Y1 j ee
UTpu: energize
@Qoxe P "9 Y60
Program ming
@common
In this way, the PLC is able to interface with real-world
devices such as switches and solenoids.
The actual /ogic of the control system is established inside
the PLC by means of a computer program. This program
dictates which output gets energized under which input
conditions. Although the program itself appears to be a
ladder logic diagram, with switch and relay symbols, there
are no actual switch contacts or relay coils operating inside
the PLC to create the logical relationships between input
and output. These are imaginary contacts and coils, if you
will. The program is entered and viewed via a personal
computer connected to the PLC's programming port.
Consider the following circuit and PLC program:
L, L,
Programming
Personal cable
computer
display
When the pushbutton switch is unactuated (unpressed), no
power is sent to the X1 input of the PLC. Following the
program, which shows a normally-open X1 contact in series
with a Y1 coil, no "power" will be sent to the Y1 coil. Thus,
the PLC's Y1 output remains de-energized, and the indicator
lamp connected to it remains dark.
If the pushbutton switch is pressed, however, power will be
sent to the PLC's X1 input. Any and all X1 contacts
appearing in the program will assume the actuated (non-
normal) state, as though they were relay contacts actuated
by the energizing of a relay coil named "X1". In this case,
energizing the X1 input will cause the normally-open X1
contact will "close," sending "power" to the Y1 coil. When
the Y1 coil of the program "energizes," the real Y1 output
will become energized, lighting up the lamp connected to it:
L; 2
switch actuated
Programming
Personal cable
computer
display
It must be understood that the X1 contact, Y1 coil,
connecting wires, and "power" appearing in the personal
computer's display are all virtua/. They do not exist as real
electrical components. They exist as commands ina
computer program -- a piece of software only -- that just
happens to resemble a real relay schematic diagram.
Equally important to understand is that the personal
computer used to display and edit the PLC's program is not
necessary for the PLC's continued operation. Once a
program has been loaded to the PLC from the personal
computer, the personal computer may be unplugged from
the PLC, and the PLC will continue to follow the programmed
commands. | include the personal computer display in these
illustrations for your sake only, in aiding to understand the
relationship between real-life conditions (switch closure and
lamp status) and the program's status ("power" through
virtual contacts and virtual coils).
The true power and versatility of a PLC is revealed when we
want to alter the behavior of a control system. Since the PLC
iS a programmable device, we can alter its behavior by
changing the commands we give it, without having to
reconfigure the electrical components connected to it. For
example, suppose we wanted to make this switch-and-lamp
circuit function in an inverted fashion: push the button to
make the lamp turn off, and release it to make it turn on. The
"hardware" solution would require that a normally-closed
pushbutton switch be substituted for the normally-open
switch currently in place. The "software" solution is much
easier: just alter the program so that contact X1 is normally-
closed rather than normally-open.
In the following illustration, we have the altered system
shown in the state where the pushbutton is unactuated (not
being pressed):
L La
Programming
Personal cable
computer
display
In this next illustration, the switch is shown actuated
(pressed):
L La
switch actuated
Programming
Personal cable
computer
display
One of the advantages of implementing logical control in
software rather than in hardware is that input signals can be
re-used as many times in the program as is necessary. For
example, take the following circuit and program, designed to
energize the lamp if at least two of the three pushbutton
switches are simultaneously actuated:
To build an equivalent circuit using electromechanical
relays, three relays with two normally-open contacts each
would have to be used, to provide two contacts per input
switch. Using a PLC, however, we can program as many
contacts as we wish for each "X" input without adding
additional hardware, since each input and each output is
nothing more than a single bit in the PLC's digital memory
(either O or 1), and can be recalled as many times as
necessary.
Furthermore, since each output in the PLC is nothing more
than a bit in its memory as well, we can assign contacts in a
PLC program "actuated" by an output (Y) status. Take for
instance this next system, a motor start-stop control circuit:
L, L,
y20@| Motor
contactor
The pushbutton switch connected to input X1 serves as the
"Start" switch, while the switch connected to input X2 serves
as the "Stop." Another contact in the program, named Y1,
uses the output coil status as a seal-in contact, directly, so
that the motor contactor will continue to be energized after
the "Start" pushbutton switch is released. You can see the
normally-closed contact X2 appear in a colored block,
showing that it is in a closed ("electrically conducting")
state.
If we were to press the "Start" button, input X1 would
energize, thus "closing" the X1 contact in the program,
sending "power" to the Y1 "coil," energizing the Y1 output
and applying 120 volt AC power to the real motor contactor
coil. The parallel Y1 contact will also "close," thus latching
the "circuit" in an energized state:
L, Ls
Motp (actuated)
y20@| Motor
contactor
Now, if we release the "Start" pushbutton, the normally-open
X1 "contact" will return to its "open" state, but the motor will
continue to run because the Y1 seal-in "contact" continues
to provide "continuity" to "power" coil Y1, thus keeping the
Y1 output energized:
L, La
Molp ' (released)
y20@| Motor
contactor
To stop the motor, we must momentarily press the "Stop"
pushbutton, which will energize the X2 input and "open" the
normally-closed "contact," breaking continuity to the Y1
"coil:"
y20@| Motor
contactor
When the "Stop" pushbutton is released, input X2 will de-
energize, returning "contact" X2 to its normal, "closed"
state. The motor, however, will not start again until the
"Start" pushbutton is actuated, because the "seal-in" of Y1
has been lost:
y20@| Motor
contactor
Motor
stop
(released)
An important point to make here is that fai/-safe design is
just as important in PLC-controlled systems as it is in
electromechanical relay-controlled systems. One should
always consider the effects of failed (open) wiring on the
device or devices being controlled. In this motor control
circuit example, we have a problem: if the input wiring for
X2 (the "Stop" switch) were to fail open, there would be no
way to stop the motor!
The solution to this problem is a reversal of logic between
the X2 "contact" inside the PLC program and the actual
"Stop" pushbutton switch:
y20@| Motor
contactor
When the normally-closed "Stop" pushbutton switch is
unactuated (not pressed), the PLC's X2 input will be
energized, thus "closing" the X2 "contact" inside the
program. This allows the motor to be started when input X1
is energized, and allows it to continue to run when the
"Start" pushbutton is no longer pressed. When the "Stop"
pushbutton is actuated, input X2 will de-energize, thus
"opening" the X2 "contact" inside the PLC program and
shutting off the motor. So, we see there is no operational
difference between this new design and the previous design.
However, if the input wiring on input X2 were to fail open,
X2 input would de-energize in the same manner as when the
"Stop" pushbutton is pressed. The result, then, for a wiring
failure on the X2 input is that the motor will immediately
shut off. This is a safer design than the one previously
shown, where a "Stop" switch wiring failure would have
resulted in an inability to turn off the motor.
In addition to input (X) and output (Y) program elements,
PLCs provide "internal" coils and contacts with no intrinsic
connection to the outside world. These are used much the
same as "control relays" (CR1, CR2, etc.) are used in
standard relay circuits: to provide logic signal inversion
when necessary.
To demonstrate how one of these "internal" relays might be
used, consider the following example circuit and program,
designed to emulate the function of a three-input NAND
gate. Since PLC program elements are typically designed by
single letters, | will call the internal control relay "C1" rather
than "CR1" as would be customary in a relay control circuit:
In this circuit, the lamp will remain lit so long as any of the
pushbuttons remain unactuated (unpressed). To make the
lamp turn off, we will have to actuate (press) a//three
switches, like this:
All three switches actuated
This section on programmable logic controllers illustrates
just a small sample of their capabilities. As computers, PLCs
can perform timing functions (for the equivalent of time-
delay relays), drum sequencing, and other advanced
functions with far greater accuracy and reliability than what
is possible using electromechanical logic devices. Most PLCs
have the capacity for far more than six inputs and six
outputs. The following photograph shows several input and
output modules of a single Allen-Bradley PLC.
With each module having sixteen "points" of either input or
output, this PLC has the ability to monitor and control
dozens of devices. Fit into a control cabinet, a PLC takes up
little room, especially considering the equivalent space that
would be needed by electromechanical relays to perform the
Same functions:
pesereerussreys™***
a
r .
hewentratuwarsy:
One advantage of PLCs that simply cannot be duplicated by
electromechanical relays is remote monitoring and control
via digital computer networks. Because a PLC is nothing
more than a special-purpose digital computer, it has the
ability to communicate with other computers rather easily.
The following photograph shows a personal computer
displaying a graphic image of a real liquid-level process (a
pumping, or "lift," station for a municipal wastewater
treatment system) controlled by a PLC. The actual pumping
station is located miles away from the personal computer
display:
=r
oo -
=
es)
ee |
32
="s
a
a=)
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Roger Hollingsworth (May 2003): Suggested a way to
make the PLC motor control circuit fail-safe.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
Next
—
nts
E¢
—_
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 7
BOOLEAN ALGEBRA
Introduction
Boolean arithmetic
Boolean algebraic identities
Boolean algebraic properties
Boolean rules for simplification
Circuit simplification examples
The Exclusive-OR function
DeMorgan's Theorems
Converting truth tables into Boolean expressions
0+0=0
O+1=1
Le OS
LpPi=
Rules of addition for Boolean quantities
“Gee Toto, | don't think we're in Kansas anymore!"
Dorothy, in The Wizard of Oz
Introduction
Mathematical rules are based on the defining limits we place
on the particular numerical quantities dealt with. When we
say thatl + 1=2o0r3+4=7, weare implying the use of
integer quantities: the same types of numbers we all learned
to count in elementary education. What most people assume
to be self-evident rules of arithmetic -- valid at all times and
for all purposes -- actually depend on what we define a
number to be.
For instance, when calculating quantities in AC circuits, we
find that the "real" number quantities which served us so
well in DC circuit analysis are inadequate for the task of
representing AC quantities. We know that voltages add
when connected in series, but we also know that it is
possible to connect a 3-volt AC source in series with a 4-volt
AC source and end up with 5 volts total voltage (3 + 4 = 5)!
Does this mean the inviolable and self-evident rules of
arithmetic have been violated? No, it just means that the
rules of "real" numbers do not apply to the kinds of
quantities encountered in AC circuits, where every variable
has both a magnitude and a phase. Consequently, we must
use a different kind of numerical quantity, or object, for AC
circuits (complex numbers, rather than rea/numbers), and
along with this different system of numbers comes a
different set of rules telling us how they relate to one
another.
An expression such as "3 + 4 = 5" is nonsense within the
scope and definition of real numbers, but it fits nicely within
the scope and definition of complex numbers (think of a
right triangle with opposite and adjacent sides of 3 and 4,
with a hypotenuse of 5). Because complex numbers are two-
dimensional, they are able to "add" with one another
trigonometrically as single-dimension "real" numbers
cannot.
Logic is much like mathematics in this respect: the so-called
“Laws" of logic depend on how we define what a proposition
is. The Greek philosopher Aristotle founded a system of logic
based on only two types of propositions: true and false. His
bivalent (two-mode) definition of truth led to the four
foundational laws of logic: the Law of Identity (A is A); the
Law of Non-contradiction (A is not non-A); the Law of the
Excluded Middle (either A or non-A); and the Law of Rational
Inference. These so-called Laws function within the scope of
logic where a proposition is limited to one of two possible
values, but may not apply in cases where propositions can
hold values other than "true" or "false." In fact, much work
has been done and continues to be done on "multivalued,"
or fuzzy logic, where propositions may be true or false to a
limited degree. In such a system of logic, "Laws" such as the
Law of the Excluded Middle simply do not apply, because
they are founded on the assumption of bivalence. Likewise,
many premises which would violate the Law of Non-
contradiction in Aristotelian logic have validity in "fuzzy"
logic. Again, the defining limits of propositional values
determine the Laws describing their functions and relations.
The English mathematician George Boole (1815-1864)
sought to give symbolic form to Aristotle's system of logic.
Boole wrote a treatise on the subject in 1854, titled An
Investigation of the Laws of Thought, on Which Are Founded
the Mathematical Theories of Logic and Probabilities, which
codified several rules of relationship between mathematical
quantities limited to one of two possible values: true or
false, 1 or 0. His mathematical system became known as
Boolean algebra.
All arithmetic operations performed with Boolean quantities
have but one of two possible outcomes: either 1 or 0. There
is no such thing as "2" or "-L" or "1/2" in the Boolean world.
It is a world in which all other possibilities are invalid by fiat.
As one might guess, this is not the kind of math you want to
use when balancing a checkbook or calculating current
through a resistor. However, Claude Shannon of MIT fame
recognized how Boolean algebra could be applied to on-and-
off circuits, where all signals are characterized as either
"high" (1) or "low" (0). His 1938 thesis, titled A Symbolic
Analysis of Relay and Switching Circuits, put Boole's
theoretical work to use in a way Boole never could have
imagined, giving us a powerful mathematical tool for
designing and analyzing digital circuits.
In this chapter, you will find a lot of similarities between
Boolean algebra and "normal" algebra, the kind of algebra
involving so-called real numbers. Just bear in mind that the
system of numbers defining Boolean algebra is severely
limited in terms of scope, and that there can only be one of
two possible values for any Boolean variable: 1 or 0.
Consequently, the "Laws" of Boolean algebra often differ
from the "Laws" of real-number algebra, making possible
such statements as 1 + 1 = 1, which would normally be
considered absurd. Once you comprehend the premise of all
quantities in Boolean algebra being limited to the two
possibilities of 1 and 0, and the general philosophical
principle of Laws depending on quantitative definitions, the
"nonsense" of Boolean algebra disappears.
It should be clearly understood that Boolean numbers are
not the same as binary numbers. Whereas Boolean numbers
represent an entirely different system of mathematics from
real numbers, binary is nothing more than an alternative
notation for real numbers. The two are often confused
because both Boolean math and binary notation use the
Same two ciphers: 1 and O. The difference is that Boolean
quantities are restricted to a single bit (either 1 or 0),
whereas binary numbers may be composed of many bits
adding up in place-weighted form to a value of any finite
size. The binary number 10011, ("nineteen") has no more
place in the Boolean world than the decimal number 2,6
("two") or the octal number 32. ("twenty-six").
Boolean arithmetic
Let us begin our exploration of Boolean algebra by adding
numbers together:
0+ 0= 0
0O+1e= 1
1+0dQ0e-= 1
1+tle=ktl1
The first three sums make perfect sense to anyone familiar
with elementary addition. The last sum, though, is quite
possibly responsible for more confusion than any other
single statement in digital electronics, because it seems to
run contrary to the basic principles of mathematics. Well, it
does contradict principles of addition for real numbers, but
not for Boolean numbers. Remember that in the world of
Boolean algebra, there are only two possible values for any
quantity and for any arithmetic operation: 1 or 0. There is no
such thing as "2" within the scope of Boolean values. Since
the sum "1 + 1" certainly isn't 0, it must be 1 by process of
elimination.
It does not matter how many or few terms we add together,
either. Consider the following sums:
+4
co
= oOo fF &
oF
OF FF FB
+
+
oo
PrP FP
PrP FP FR
1
+]1e= 1
Take a close look at the two-term sums in the first set of
equations. Does that pattern look familiar to you? It should!
It is the same pattern of 1's and 0's as seen in the truth table
for an OR gate. In other words, Boolean addition corresponds
to the logical function of an "OR" gate, as well as to parallel
switch contacts:
+
oO
MI
oO
+
=
i]
=
+
Oo
I
=
There is no such thing as subtraction in the realm of Boolean
mathematics. Subtraction implies the existence of negative
numbers: 5 - 3 is the same thing as 5 + (-3), and in Boolean
algebra negative quantities are forbidden. There is no such
thing as division in Boolean mathematics, either, since
division is really nothing more than compounded
subtraction, in the same way that multiplication is
compounded addition.
Multiplication is valid in Boolean algebra, and thankfully it is
the same as in real-number algebra: anything multiplied by
0 is O, and anything multiplied by 1 remains unchanged:
0
0
i
1
x X X X
KF OF Oo
|
Fe o oOo Oo
This set of equations should also look familiar to you: it is
the same pattern found in the truth table for an AND gate. In
other words, Boolean multiplication corresponds to the
logical function of an "AND" gate, as well as to series switch
contacts:
oO
x
It
Oo
x
i]
=
Like "normal" algebra, Boolean algebra uses alphabetical
letters to denote variables. Unlike "normal" algebra, though,
Boolean variables are always CAPITAL letters, never lower-
case. Because they are allowed to possess only one of two
possible values, either 1 or 0, each and every variable has a
complement: the opposite of its value. For example, if
variable "A" has a value of 0, then the complement of A has
a value of 1. Boolean notation uses a bar above the variable
character to denote complementation, like this:
If:
0
Then: 1
A=
A=
If:
Then:
A=1
A=0
In written form, the complement of "A" denoted as "A-not" or
"A-bar". Sometimes a "prime" symbol is used to represent
complementation. For example, A' would be the complement
of A, much the same as using a prime symbol to denote
differentiation in calculus rather than the fractional notation
d/dt. Usually, though, the "bar" symbol finds more
widespread use than the "prime" symbol, for reasons that
will become more apparent later in this chapter.
Boolean complementation finds equivalency in the form of
the NOT gate, or a normally-closed switch or relay contact:
lf: A=0
Then: A=1 =
A A
A A ? a
lf: A=1
Then: A=0 =
A
zA a 1 0
1 Seo to}
The basic definition of Boolean quantities has led to the
simple rules of addition and multiplication, and has
excluded both subtraction and division as valid arithmetic
operations. We have a symbology for denoting Boolean
variables, and their complements. In the next section we will
proceed to develop Boolean identities.
e REVIEW:
e Boolean addition is equivalent to the OR logic function,
as well as paralle/ switch contacts.
e Boolean multiplication is equivalent to the AND logic
function, as well as series switch contacts.
e Boolean complementation is equivalent to the NOT logic
function, as well as normally-closed relay contacts.
Boolean algebraic identities
In mathematics, an /dentity is a statement true for all
possible values of its variable or variables. The algebraic
identity of x + 0 = x tells us that anything (x) added to zero
equals the original "anything," no matter what value that
"anything" (x) may be. Like ordinary algebra, Boolean
algebra has its own unique identities based on the bivalent
states of Boolean variables.
The first Boolean identity is that the sum of anything and
zero is the same as the original "anything." This identity is
no different from its real-number algebraic equivalent:
A+Oe-=A
. A
0 , ii
No matter what the value of A, the output will always be the
same: when A=1, the output will also be 1; when A=0O, the
output will also be O.
The next identity is most definitely different from any seen
in normal algebra. Here we discover that the sum of
anything and one is one:
=1
A+i1
A 1
A > 4
i
“fh fo ™*
1
No matter what the value of A, the sum of A and 1 will
always be 1. In a sense, the "L" signal overrides the effect of
A on the logic circuit, leaving the output fixed at a logic
level of 1.
Next, we examine the effect of adding A and A together,
which is the same as connecting both inputs of an OR gate
to each other and activating them with the same signal:
A+A=A
In real-number algebra, the sum of two identical variables is
twice the original variable's value (x + x = 2x), but
remember that there is no concept of "2" in the world of
Boolean math, only 1 and 0, so we cannot say thatA +A =
2A. Thus, when we add a Boolean quantity to itself, the sum
is equal to the original quantity:0 +0 =0,and1+1=1.
Introducing the uniquely Boolean concept of
complementation into an additive identity, we find an
interesting effect. Since there must be one "1" value
between any variable and its complement, and since the
sum of any Boolean quantity and 1 is 1, the sum of a
variable and its complement must be 1:
+A=l1
A
A A 1
A 7 4
1 ‘
z A
Just as there are four Boolean additive identities (A+0, A+1,
A+A, and A+A’'), so there are also four multiplicative
identities: AxO, Ax1, AxA, and AxA’. Of these, the first two
are no different from their equivalent expressions in regular
algebra:
0 A 0
0 meus
iD se ote es
1A =A
1 ; he ae
The third multiplicative identity expresses the result of a
Boolean quantity multiplied by itself. In normal algebra, the
product of a variable and itself is the square of that variable
(3 x 3 = 3? = 9). However, the concept of "square" implies a
quantity of 2, which has no meaning in Boolean algebra, so
we cannot say that A x A = A2. Instead, we find that the
product of a Boolean quantity and itself is the original
quantity, sinceO0 x0 =Oand1x1= tL:
A . my _ c :
The fourth multiplicative identity has no equivalent in
regular algebra because it uses the complement of a
variable, a concept unique to Boolean mathematics. Since
there must be one "0" value between any variable and its
complement, and since the product of any Boolean quantity
and 0 is O, the product of a variable and its complement
must be 0:
AA = 0
A A 0
A Va?
TD HAH
A
To summarize, then, we have four basic Boolean identities
for addition and four for multiplication:
Basic Boolean algebraic identities
Additive Multiplicative
A+0e=A OA = 0
A+l=l1 IA=A
A+A=A AA=A
A+A=1 AA = 0
Another identity having to do with complementation is that
of the double complement: a variable inverted twice.
Complementing a variable twice (or any even number of
times) results in the original Boolean value. This is
analogous to negating (multiplying by -1) in real-number
algebra: an even number of negations cancel to leave the
original value:
CR1 (samé) CR2
Boolean algebraic properties
Another type of mathematical identity, called a "property" or
a "law," describes how differing variables relate to each
other in a system of numbers. One of these properties is
known as the commutative property, and it applies equally
to addition and multiplication. In essence, the commutative
property tells us we can reverse the order of variables that
are either added together or multiplied together without
changing the truth of the expression:
Commutative property of addition
A+B=B+H+A,
ork }
(same) B
a
Commutative property of multiplication
AB = BA
; Sis
DY’
B A
Along with the commutative properties of addition and
multiplication, we have the associative property, again
applying equally well to addition and multiplication. This
property tells US we can associate groups of added or
multiplied variables together with parentheses without
altering the truth of the equations.
Associative property of addition
A+ (B+ C) = (A+B) +C
Associative property of multiplication
A(BC) = (AB)C
Lastly, we have the d/stributive property, illustrating how to
expand a Boolean expression formed by the product of a
sum, and in reverse shows us how terms may be factored out
of Boolean sums-of-products:
Distributive property
A(B + C) = AB + AC
To summarize, here are the three basic properties:
commutative, associative, and distributive.
Basic Boolean algebraic properties
Additive Multiplicative
A+B=B+H+A AB = BA
A+ (B+ C) = (A+B) +C A(BC) = (AB)C
A(B + C) = AB + AC
Boolean rules for simplification
Boolean algebra finds its most practical use in the
simplification of logic circuits. If we translate a logic circuit's
function into symbolic (Boolean) form, and apply certain
algebraic rules to the resulting equation to reduce the
number of terms and/or arithmetic operations, the simplified
equation may be translated back into circuit form for a logic
circuit performing the same function with fewer
components. If equivalent function may be achieved with
fewer components, the result will be increased reliability and
decreased cost of manufacture.
To this end, there are several rules of Boolean algebra
presented in this section for use in reducing expressions to
their simplest forms. The identities and properties already
reviewed in this chapter are very useful in Boolean
simplification, and for the most part bear similarity to many
identities and properties of "normal" algebra. However, the
rules shown in this section are all unique to Boolean
mathematics.
nee
>
+
He
to
AB
This rule may be proven symbolically by factoring an "A" out
of the two terms, then applying the rules of A+ 1=1 and
1A = A to achieve the final result:
A AB
Factoring A out of both terms
(1 B)
+
oo
mt Applying identiy A + 1 = 1
A
Applying identity 1A = A
Please note how the rule A + 1 = 1 was used to reduce the
(B + 1) term to 1. When a rule like "A + 1 = 1" is expressed
using the letter "A", it doesn't mean it only applies to
expressions containing "A". What the "A" stands for in a rule
like A+ 1=1 is any Boolean variable or collection of
variables. This is perhaps the most difficult concept for new
students to master in Boolean simplification: applying
standardized identities, properties, and rules to expressions
not in standard form.
For instance, the Boolean expression ABC + 1 also reduces
to 1 by means of the "A + 1 = 1" identity. In this case, we
recognize that the "A" term in the identity's standard form
can represent the entire "ABC" term in the original
expression.
The next rule looks similar to the first one shown in this
section, but is actually quite different and requires a more
clever proof:
A + AB
| Applying the previous rule to expand A term
A+ AB=A
A + AB + AB
Factoring B out of 2" and 3" terms
A + B(A + A)
Applying identity A + A = 1
Applying identity 1A = A
A + B(1)
A+B
Note how the last rule (A + AB = A) is used to "un-simplify"
the first "A" term in the expression, changing the "A" into an
"A + AB". While this may seem like a backward step, it
certainly helped to reduce the expression to something
simpler! Sometimes in mathematics we must take
"backward" steps to achieve the most elegant solution.
Knowing when to take such a step and when not to is part of
the art-form of algebra, just as a victory in a game of chess
almost always requires calculated sacrifices.
Another rule involves the simplification of a product-of-sums
expression:
(A + B) (A + C) = A+ BC
(A+B) (A+C) B
(same)
A + BC
And, the corresponding proof:
(A + B) (A + C)
| Distributing terms
BA + AC + AB + BC
| Applying identity AA = A
A + AC + AB + BC
| Applying rule A + AB
H
»
tothe A + Ac term
A + AB + BC
| Applying rule A + AB
to the A + AB term
i]
od
A+ BC
To summarize, here are the three new rules of Boolean
simplification expounded in this section:
Useful Boolean rules for simplification
A+ ABe=A
A+AB=A+B
(A + B)(A +C) =A + BC
Circuit simplification examples
Let's begin with a semiconductor gate circuit in need of
simplification. The "A," ""B," and "C" input signals are
assumed to be provided from switches, sensors, or perhaps
other gate circuits. Where these signals originate is of no
concern in the task of gate reduction.
Our first step in simplification must be to write a Boolean
expression for this circuit. This task is easily performed step
by step if we start by writing sub-expressions at the output
of each gate, corresponding to the respective input signals
for each gate. Remember that OR gates are equivalent to
Boolean addition, while AND gates are equivalent to Boolean
multiplication. For example, I'll write sub-expressions at the
outputs of the first three gates:
Finally, the output ("Q") is seen to be equal to the
expression AB + BC(B + C):
Q = AB + BC(B+C)
Now that we have a Boolean expression to work with, we
need to apply the rules of Boolean algebra to reduce the
expression to its simplest form (simplest defined as requiring
the fewest gates to implement):
AB + BC(B + C)
| Distributing terms
AB + BBC + BCC
Applying identity AA = A
to 2nd and 3rd terms
AB + BC + BC
Applying identtyA + A=A
| to 2nd and 3rd terms
AB + BC
| Factoring B out of terms
B(A + C)
The final expression, B(A + C), is much simpler than the
Original, yet performs the same function. If you would like to
verify this, you may generate a truth table for both
expressions and determine Q's status (the circuits’ output)
for all eight logic-state combinations of A, B, and C, for both
circuits. The two truth tables should be identical.
Now, we must generate a schematic diagram from this
Boolean expression. To do this, evaluate the expression,
following proper mathematical order of operations
(multiplication before addition, operations inside
parentheses before anything else), and draw gates for each
step. Remember again that OR gates are equivalent to
Boolean addition, while AND gates are equivalent to Boolean
multiplication. In this case, we would begin with the sub-
expression "A + C", which is an OR gate:
A ) >"
Cc
The next step in evaluating the expression "B(A + C)" is to
multiply (AND gate) the signal B by the output of the
previous gate (A + C):
A A+C
C QO = B(A+C)
B
Obviously, this circuit is much simpler than the original,
having only two logic gates instead of five. Such component
reduction results in higher operating speed (less delay time
from input signal transition to output signal transition), less
power consumption, less cost, and greater reliability.
Electromechanical relay circuits, typically being slower,
consuming more electrical power to operate, costing more,
and having a shorter average life than their semiconductor
counterparts, benefit dramatically from Boolean
simplification. Let's consider an example circuit:
As before, our first step in reducing this circuit to its simplest
form must be to develop a Boolean expression from the
schematic. The easiest way I've found to do this is to follow
the same steps I'd normally follow to reduce a series-parallel
resistor network to a single, total resistance. For example,
examine the following resistor network with its resistors
arranged in the same connection pattern as the relay
contacts in the former circuit, and corresponding total
resistance formula:
Re al
Ryotai = Ry // TUR//R,) -- R,] // (R; -- Rg)
Remember that parallel contacts are equivalent to Boolean
addition, while series contacts are equivalent to Boolean
multiplication. Write a Boolean expression for this relay
contact circuit, following the same order of precedence that
you would follow in reducing a series-parallel resistor
network to a total resistance. It may be helpful to write a
Boolean sub-expression to the left of each ladder "rung," to
help organize your expression-writing:
OQ = A+ B(A+C) + AC
Now that we have a Boolean expression to work with, we
need to apply the rules of Boolean algebra to reduce the
expression to its simplest form (simplest defined as requiring
the fewest relay contacts to implement):
A+ B(A + C) + AC
Distributing terms
A + AB + BC + AC
ApplyingruleA + AB =A
| to 1st and 2nd terms
A+ BC + AC
ApplyingruleA + AB =A
| to 1st and 3rd terms
A + BC
The more mathematically inclined should be able to see that
the two steps employing the rule "A + AB = A" may be
combined into a single step, the rule being expandable to:
"A8+AB+AC+AD+...=A"
A+ B(A + C) + AC
A
| Distributing terms
+ AB + BC + AC
Applying (expanded) ruleA + AB = A
| to 1st, 2nd, and 4th terms
A+ BC
As you can see, the reduced circuit is much simpler than the
original, yet performs the same logical function:
BC
REVIEW:
To convert a gate circuit to a Boolean expression, label
each gate output with a Boolean sub-expression
corresponding to the gates' input signals, until a final
expression is reached at the last gate.
To convert a Boolean expression to a gate circuit,
evaluate the expression using standard order of
operations: multiplication before addition, and
operations within parentheses before anything else.
To convert a ladder logic circuit to a Boolean expression,
label each rung with a Boolean sub-expression
corresponding to the contacts’ input signals, until a final
expression is reached at the last coil or light. To
determine proper order of evaluation, treat the contacts
as though they were resistors, and as if you were
determining total resistance of the series-parallel
network formed by them. In other words, look for
contacts that are either directly in series or directly in
parallel with each other first, then "collapse" them into
equivalent Boolean sub-expressions before proceeding
to other contacts.
e To convert a Boolean expression to a ladder logic circuit,
evaluate the expression using standard order of
operations: multiplication before addition, and
operations within parentheses before anything else.
The Exclusive-OR function
One element conspicuously missing from the set of Boolean
operations is that of Exclusive-OR. Whereas the OR function
is equivalent to Boolean addition, the AND function to
Boolean multiplication, and the NOT function (inverter) to
Boolean complementation, there is no direct Boolean
equivalent for Exclusive-OR. This hasn't stopped people from
developing a symbol to represent it, though:
Doe
This symbol is seldom used in Boolean expressions because
the identities, laws, and rules of simplification involving
addition, multiplication, and complementation do not apply
to it. However, there is a way to represent the Exclusive-OR
function in terms of OR and AND, as has been shown in
previous chapters: AB' + A'B
AB + AB
A ® B= AB + AB
As a Boolean equivalency, this rule may be helpful in
simplifying some Boolean expressions. Any expression
following the AB' + A'B form (two AND gates and an OR
gate) may be replaced by a single Exclusive-OR gate.
DeMorgan's Theorems
A mathematician named DeMorgan developed a pair of
important rules regarding group complementation in
Boolean algebra. By group complementation, I'm referring to
the complement of a group of terms, represented by a long
bar over more than one variable.
You should recall from the chapter on logic gates that
inverting all inputs to a gate reverses that gate's essential
function from AND to OR, or vice versa, and also inverts the
output. So, an OR gate with all inputs inverted (a Negative-
OR gate) behaves the same as a NAND gate, and an AND
gate with all inputs inverted (a Negative-AND gate) behaves
the same as a NOR gate. DeMorgan's theorems state the
Same equivalence in "backward" form: that inverting the
output of any gate results in the same function as the
opposite type of gate (AND vs. OR) with inverted inputs:
AB
i
w
. Is equivalent to...
A
AB =A+B
A long bar extending over the term AB acts as a grouping
symbol, and as such is entirely different from the product of
A and B independently inverted. In other words, (AB)' is not
equal to A'B'. Because the "prime" symbol (') cannot be
stretched over two variables like a bar can, we are forced to
use parentheses to make it apply to the whole term AB in
the previous sentence. A bar, however, acts as its own
grouping symbol when stretched over more than one
variable. This has profound impact on how Boolean
expressions are evaluated and reduced, as we shall see.
DeMorgan's theorem may be thought of in terms of breaking
a long bar symbol. When a long bar is broken, the operation
directly underneath the break changes from addition to
multiplication, or vice versa, and the broken bar pieces
remain over the individual variables. To illustrate:
DeMorgan’s Theorems
break! break!
W\ V7
NAND to Negative-OR NOR to Negative-AND
When multiple "layers" of bars exist in an expression, you
may only break one bar at a time, and it is generally easier
to begin simplification by breaking the longest (uppermost)
bar first. To illustrate, let's take the expression (A + (BC)')'
and reduce it using DeMorgan's Theorems:
A
A —
A+ BC
B BC
Cc
Following the advice of breaking the longest (uppermost)
bar first, I'll begin by breaking the bar covering the entire
expression as a first step:
C
a
| Breaking longest bar
(addition changes to multiplication)
A
Applying identity A =A
to BC
ABC
As a result, the original circuit is reduced to a three-input
AND gate with the A input inverted:
ABC
You should never break more than one bar in a single step,
as illustrated here:
A+ BC
| Breaking long bar between A and B;
/
Incorrect step: Breaking both bars between B and c
+c -
Applying identityA = A
| to B andc
wI|
A
Incorrect answer: AB + C
As tempting as it may be to conserve steps and break more
than one bar at atime, it often leads to an incorrect result,
so don't do it!
It is possible to properly reduce this expression by breaking
the short bar first, rather than the long bar first:
>
A
OQ
~<—g— +
Breaking shortest bar
(multiplication changes to addition)
+ ¢}
Applying associative property
to remove parentheses
|
A + + C
Breaking long bar in two places,
between 1st and 2nd terms;
between 2nd and 3rd terms
ABC
Applying identity A =A
to B andc
|
mw ~<—— wll aoe
QO
The end result is the same, but more steps are required
compared to using the first method, where the longest bar
was broken first. Note how in the third step we broke the
long bar in two places. This is a legitimate mathematical
operation, and not the same as breaking two bars in one
step! The prohibition against breaking more than one bar in
one step is nota prohibition against breaking a bar in more
than one place. Breaking in more than one place in a single
step is okay; breaking more than one barin a single step is
not.
You might be wondering why parentheses were placed
around the sub-expression B' + C', considering the fact that
| just removed them in the next step. | did this to emphasize
an important but easily neglected aspect of DeMorgan's
theorem. Since a long bar functions as a grouping symbol,
the variables formerly grouped by a broken bar must remain
grouped lest proper precedence (order of operation) be lost.
In this example, it really wouldn't matter if | forgot to put
parentheses in after breaking the short bar, but in other
cases it might. Consider this example, starting with a
different expression:
AB + CD
| Breaking bar in middle
Notice the grouping maintained
with parentheses ————+ (xB) (CD)
| Breaking both bars in middle
Correct answer: (A + B)(C + D)
AB + CD
| Breaking bar in middle
Parentheses omitted ———->» ZR CD
| Breaking both bars in middle
Incorrect answer: A+ BC +D
As you can see, maintaining the grouping implied by the
complementation bars for this expression is crucial to
obtaining the correct answer.
Let's apply the principles of DeMorgan's theorems to the
simplification of a gate circuit:
As always, our first step in simplifying this circuit must be to
generate an equivalent Boolean expression. We can do this
by placing a sub-expression label at the output of each gate,
as the inputs become known. Here's the first step in this
process:
Next, we can label the outputs of the first NOR gate and the
NAND gate. When dealing with inverted-output gates, | find
it easier to write an expression for the gate's output without
the final inversion, with an arrow pointing to just before the
inversion bubble. Then, at the wire leading out of the gate
(after the bubble), | write the full, complemented expression.
This helps ensure | don't forget a complementing bar in the
sub-expression, by forcing myself to split the expression-
writing task into two steps:
Finally, we write an expression (or pair of expressions) for
the last NOR gate:
Now, we reduce this expression using the identities,
properties, rules, and theorems (DeMorgan's) of Boolean
algebra:
A+ BC + AB
| Breaking longest bar
(A+ BC) (AB) -
Applying identity A = A
| witsrever double bars of
_ equal length are found
| Distributive property
to left term; applying identity
AA = OtoBandBinright
term
| Applying identity AA = A
a
| Applying identityA + O=A
The equivalent gate circuit for this much-simplified
expression is as follows:
e REVIEW
DeMorgan's Theorems describe the equivalence
between gates with inverted inputs and gates with
inverted outputs. Simply put, a NAND gate is equivalent
to a Negative-OR gate, and a NOR gate is equivalent to
a Negative-AND gate.
e When "breaking" a complementation bar in a Boolean
expression, the operation directly underneath the break
(addition or multiplication) reverses, and the broken bar
pieces remain over the respective terms.
e It is often easier to approach a problem by breaking the
longest (uppermost) bar before breaking any bars under
it. You must never attempt to break two bars in one step!
e Complementation bars function as grouping symbols.
Therefore, when a bar is broken, the terms underneath it
must remain grouped. Parentheses may be placed
around these grouped terms as a help to avoid changing
precedence.
Converting truth tables into Boolean
expressions
In designing digital circuits, the designer often begins with a
truth table describing what the circuit should do. The design
task is largely to determine what type of circuit will perform
the function described in the truth table. While some people
seem to have a natural ability to look at a truth table and
immediately envision the necessary logic gate or relay logic
circuitry for the task, there are procedural techniques
available for the rest of us. Here, Boolean algebra proves its
utility in a most dramatic way.
To illustrate this procedural method, we should begin with a
realistic design problem. Suppose we were given the task of
designing a flame detection circuit for a toxic waste
incinerator. The intense heat of the fire is intended to
neutralize the toxicity of the waste introduced into the
incinerator. Such combustion-based techniques are
commonly used to neutralize medical waste, which may be
infected with deadly viruses or bacteria:
Toxic waste
inlet
Toxic waste incinerator '
Fuel
~~ inlet
So long as a flame is maintained in the incinerator, it is safe
to inject waste into it to be neutralized. If the flame were to
be extinguished, however, it would be unsafe to continue to
inject waste into the combustion chamber, as it would exit
the exhaust un-neutralized, and pose a health threat to
anyone in close proximity to the exhaust. What we need in
this system is a sure way of detecting the presence of a
flame, and permitting waste to be injected only if a flame is
"proven" by the flame detection system.
Several different flame-detection technologies exist: optical
(detection of light), thermal (detection of high temperature),
and electrical conduction (detection of ionized particles in
the flame path), each one with its unique advantages and
disadvantages. Suppose that due to the high degree of
hazard involved with potentially passing un-neutralized
waste out the exhaust of this incinerator, it is decided that
the flame detection system be made redundant (multiple
sensors), so that failure of a single sensor does not lead to
an emission of toxins out the exhaust. Each sensor comes
equipped with a normally-open contact (open if no flame,
closed if flame detected) which we will use to activate the
inputs of a logic system:
Toxic waste
inlet
Toxic waste incinerator '
Waste shutoff
valve
Fuel
—~ inlet
ag
ogic system
ha seeeiaeel ee off as valve
if no flame detected)
Our task, now, is to design the circuitry of the logic system
to open the waste valve if and only if there is good flame
proven by the sensors. First, though, we must decide what
the logical behavior of this control system should be. Do we
want the valve to be opened if only one out of the three
sensors detects flame? Probably not, because this would
defeat the purpose of having multiple sensors. If any one of
the sensors were to fail in such a way as to falsely indicate
the presence of flame when there was none, a logic system
based on the principle of "any one out of three sensors
showing flame" would give the same output that a single-
sensor system would with the same failure. A far better
solution would be to design the system so that the valve is
commanded to open if and only if a// three sensors detect a
good flame. This way, any single, failed sensor falsely
showing flame could not keep the valve in the open
position; rather, it would require all three sensors to be
failed in the same manner -- a highly improbable scenario --
for this dangerous condition to occur.
Thus, our truth table would look like this:
sensor
inputs
A[B[C] Output
fofofol oO | ouput-o
Ojo;i] oOo | (close valve)
jo}ijo| o
o}ifif o
jtjojo| o
jHjo}i} oO
riftfof 0 | Output -1
Pifaj}il to] (open valve)
It does not require much insight to realize that this
functionality could be generated with a three-input AND
gate: the output of the circuit will be "high" if and only if
input A AND input B AND input C are all "high:"
Toxic waste
inlet
Toxic waste incinerator }
Waste shutoff
valve
Fuel
—~ inlet
sensor
B
If using relay circuitry, we could create this AND function by
wiring three relay contacts in series, or simply by wiring the
three sensor contacts in series, so that the only way
electrical power could be sent to open the waste valve is if
all three sensors indicate flame:
Toxic waste
inlet
Toxic waste incinerator '
Waste shutoff
valve
sensor || sensor |} sensor
A B C
L l
While this design strategy maximizes safety, it makes the
system very susceptible to sensor failures of the opposite
kind. Suppose that one of the three sensors were to fail in
such a way that it indicated no flame when there really was
a good flame in the incinerator's combustion chamber. That
single failure would shut off the waste valve unnecessarily,
resulting in lost production time and wasted fuel (feeding a
fire that wasn't being used to incinerate waste).
Fuel
~~ inlet
It would be nice to have a logic system that allowed for this
kind of failure without shutting the system down
unnecessarily, yet still provide sensor redundancy so as to
maintain safety in the event that any single sensor failed
"high" (showing flame at all times, whether or not there was
one to detect). A strategy that would meet both needs would
be a "two out of three" sensor logic, whereby the waste
valve is opened if at least two out of the three sensors show
good flame. The truth table for such a system would look like
this:
sensor
inputs
[ATE C] Oupar
fofofol o
Output = 0
ojo;i] oOo | (close valve)
jolijo| o
of upay ot
Here, it is not necessarily obvious what kind of logic circuit
would satisfy the truth table. However, a simple method for
designing such a circuit is found in a standard form of
Boolean expression called the Sum-Of-Products, or SOP,
form. As you might suspect, a Sum-Of-Products Boolean
expression is literally a set of Boolean terms added
(summed) together, each term being a multiplicative
(product) combination of Boolean variables. An example of
an SOP expression would be something like this: ABC + BC
+ DF, the sum of products "ABC," "BC," and "DF."
Sum-Of-Products expressions are easy to generate from truth
tables. All we have to do is examine the truth table for any
rows where the output is "high" (1), and write a Boolean
product term that would equal a value of 1 given those input
conditions. For instance, in the fourth row down in the truth
table for our two-out-of-three logic system, where A=O, B=1,
and C=1, the product term would be A'BC, since that term
would have a value of 1 if and only if A=0, B=1, and C=1:
sensor
inputs
Three other rows of the truth table have an output value of
1, so those rows also need Boolean product expressions to
represent them:
sensor
inputs
II
bh
sl
ao
Finally, we join these four Boolean product expressions
together by addition, to create a single Boolean expression
describing the truth table as a whole:
sensor
inputs
Output = ABC + ABC + ABC + ABC
Now that we have a Boolean Sum-Of-Products expression for
the truth table's function, we can easily design a logic gate
or relay logic circuit based on that expression:
Output = ABC + ABC + ABC + ABC
L, L,
A CR1
B CR2
Cc CR3
CR1i CR2 CR3 _ Output
ABC ‘oe:
4 x
CR1 CR2 CR3
ABC
CR1 CR2 CR3
— ABC
CR1 CR2 CR3
-——| | ABC
Unfortunately, both of these circuits are quite complex, and
could benefit from simplification. Using Boolean algebra
techniques, the expression may be significantly simplified:
ABC + ABC + ABC + ABC
| Factoring Bc out of 1®' and 4" terms
BC(A + A) + ABC + ABC
| Applying identityA + A = 1
BC(1) + ABC + ABC
Applying identity 1A = A
BC + ABC + ABC
| Factoring B out of 1“ and 3" terms
B(C + AC) + ABC
| Applying ruleA + AB = A + Bto
thec + Acterm
B(C + A) + ABC
| Distributing terms
BC + AB + ABC
Factoring A out of 2™ and 3" terms
BC + A(B + BC)
Applying ruleA + AB = A + Bto
theB + BCterm
BC + A(B + C)
| Distributing terms
BC + AB + AC
or Simplified result
AB + BC + AC
As a result of the simplification, we can now build much
simpler logic circuits performing the same function, in either
gate or relay form:
Output = AB + BC + AC
Output = AB + BC + AC
Either one of these circuits will adequately perform the task
of operating the incinerator waste valve based on a flame
verification from two out of the three flame sensors. At
minimum, this is what we need to have a Safe incinerator
system. We can, however, extend the functionality of the
system by adding to it logic circuitry designed to detect if
any one of the sensors does not agree with the other two.
If all three sensors are operating properly, they should
detect flame with equal accuracy. Thus, they should either
all register "low" (000: no flame) or all register "high" (111:
good flame). Any other output combination (001, 010, 011,
100, 101, or 110) constitutes a disagreement between
sensors, and may therefore serve as an indicator of a
potential sensor failure. If we added circuitry to detect any
one of the six "sensor disagreement" conditions, we could
use the output of that circuitry to activate an alarm.
Whoever is monitoring the incinerator would then exercise
judgment in either continuing to operate with a possible
failed sensor (inputs: 011, 101, or 110), or shut the
incinerator down to be absolutely safe. Also, if the
incinerator is shut down (no flame), and one or more of the
sensors still indicates flame (001, 010, 011, 100, 101, or
110) while the other(s) indicate(s) no flame, it will be known
that a definite sensor problem exists.
The first step in designing this "sensor disagreement"
detection circuit is to write a truth table describing its
behavior. Since we already have a truth table describing the
output of the "good flame" logic circuit, we can simply add
another output column to the table to represent the second
circuit, and make a table representing the entire logic
system:
Output = 0 Output = 0
(close valve) (sensors agree)
Output = 1 Output = 1
(open valve) (sensors disagree)
sensor
inputs Good Sensor
flame disagreement
[A[B]C] Ourpar | Ourpar |
fofof o | 0
While it is possible to generate a Sum-Of-Products
expression for this new truth table column, it would require
six terms, of three variables each! Such a Boolean
expression would require many steps to simplify, with a
large potential for making algebraic errors:
Output = 0 Output = 0
(close valve) (Sensors agree)
Output = 1 Output = 1
(open valve) (sensors disagree)
sensor ~
inputs Good Sensor
flame disagreement
ATE] C] Oupar | Ourpar |
ro fo
ABC
ABC
ABC
ABC
ABC
ABC
Output = ABC + ABC + ABC + ABC + ABC + ABC
An alternative to generating a Sum-Of-Products expression
to account for all the "high" (1) output conditions in the
truth table is to generate a Product-Of-Sums, or POS,
expression, to account for all the "low" (0) output conditions
instead. Being that there are much fewer instances of a
"low" output in the last truth table column, the resulting
Product-Of-Sums expression should contain fewer terms. As
its name suggests, a Product-Of-Sums expression is a set of
added terms (sums), which are multiplied (product)
together. An example of a POS expression would be (A + B)
(C + D), the product of the sums "A + B" and "C + D".
To begin, we identify which rows in the last truth table
column have "low" (0) outputs, and write a Boolean sum
term that would equal O for that row's input conditions. For
instance, in the first row of the truth table, where A=O, B=O,
and C=O, the sum term would be (A + B + C), since that
term would have a value of 0 if and only if A=0, B=0, and
C=0:
Output = 0 Output = 0
(close valve) (sensors agree)
Output = 1 Output = 1
(open valve) (sensors disagree)
sensor
inputs Good Sensor
flame disagreement
[A[B[C] Ourpar | Outpar
fof o | o |
(A + B + C}
Only one other row in the last truth table column has a "low"
(0) output, so all we need is one more sum term to complete
our Product-Of-Sums expression. This last sum term
represents a O output for an input condition of A=1, B=1
and C=1. Therefore, the term must be written as (A' + B'+
C'), because only the sum of the comp/emented input
variables would equal O for that condition only:
Output = 0 Output = 0
(close valve) (sensors agree)
Output = 1 Output = 1
(open valve) (sensors disagree)
sensor
inputs Good Sensor
_ flame disagreement
(A + B + C)
The completed Product-Of-Sums expression, of course, is the
multiplicative combination of these two sum terms:
Output = 0 Output = 0
(close valve) (Sensors agree)
Output = 1 Output = 1
(open valve) (sensors disagree)
sensor
inputs Good Sensor
flame disagreement
co
(A + B+ C)
Whereas a Sum-Of-Products expression could be
implemented in the form of a set of AND gates with their
outputs connecting to a single OR gate, a Product-Of-Sums
expression can be implemented as a set of OR gates feeding
into a single AND gate:
Output = A+B+C)
(
Correspondingly, whereas a Sum-Of-Products expression
could be implemented as a parallel collection of series-
connected relay contacts, a Product-Of-Sums expression can
be implemented as a series collection of parallel-connected
relay contacts:
Output = (A +B + C) (A + B + C)
L, L,
A CR1
B CR2
Cc CR3
CR1 CR1 Output
{)-
CR2
A+ Be OC) (A + BY C}
The previous two circuits represent different versions of the
“sensor disagreement" logic circuit only, not the "good
flame" detection circuit(s). The entire logic system would be
the combination of both "good flame" and "sensor
disagreement" circuits, shown on the same diagram.
Implemented in a Programmable Logic Controller (PLC), the
entire logic system might resemble something like this:
Sensor Waste valve
A solenoid
. Sensor
S disagreement
adel alarm lamp
Sensor
C
a
Programming
cable
Personal
computer
display
As you can see, both the Sum-Of-Products and Products-Of-
Sums standard Boolean forms are powerful tools when
applied to truth tables. They allow us to derive a Boolean
expression -- and ultimately, an actual logic circuit -- from
nothing but a truth table, which is a written specification for
what we want a logic circuit to do. To be able to go from a
written specification to an actual circuit using simple,
deterministic procedures means that it is possible to
automate the design process for a digital circuit. In other
words, a computer could be programmed to design a custom
logic circuit from a truth table specification! The steps to
take from a truth table to the final circuit are so
unambiguous and direct that it requires little, if any,
creativity or other original thought to execute them.
REVIEW:
Sum-Of-Products, or SOP, Boolean expressions may be
generated from truth tables quite easily, by determining
which rows of the table have an output of 1, writing one
product term for each row, and finally summing all the
product terms. This creates a Boolean expression
representing the truth table as a whole.
Sum-Of-Products expressions lend themselves well to
implementation as a set of AND gates (products) feeding
into a single OR gate (Sum).
Product-Of-Sums, or POS, Boolean expressions may also
be generated from truth tables quite easily, by
determining which rows of the table have an output of 0,
writing one sum term for each row, and finally
multiplying all the sum terms. This creates a Boolean
expression representing the truth table as a whole.
Product-Of-Sums expressions lend themselves well to
implementation as a set of OR gates (Sums) feeding into
a single AND gate (product).
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
Next
—
nts
E¢
—_
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 8
KARNAUGH MAPPING
Introduction
Venn diagrams and sets
Boolean Relationships on Venn Diagrams
Making a Venn diagram look like a Karnaugh map
Logic simplification with Karnaugh maps
Larger 4-variable Karnaugh maps
Minterm vs maxterm solution
Don't care cells in the Karnaugh map
Larger 5 & 6-variable Karnaugh maps
Original author: Dennis Crunkilton
Introduction
Why learn about Karnaugh maps? The Karnaugh map, like
Boolean algebra, is a simplification tool applicable to digital
logic. See the "Toxic waste incinerator" in the Boolean
algebra chapter for an example of Boolean simplification of
digital logic. The Karnaugh Map will simplify logic faster and
more easily in most cases.
Boolean simplification is actually faster than the Karnaugh
map for a task involving two or fewer Boolean variables. It is
still quite usable at three variables, but a bit slower. At four
input variables, Boolean algebra becomes tedious. Karnaugh
maps are both faster and easier. Karnaugh maps work well
for up to six input variables, are usable for up to eight
variables. For more than six to eight variables, simplification
should be by CAD (computer automated design).
Recommended logic simplification vs number of inputs
Boolean algebra computer automated
See eee
In theory any of the three methods will work. However, as a
practical matter, the above guidelines work well. We would
not normally resort to computer automation to simplify a
three input logic block. We could sooner solve the problem
with pencil and paper. However, if we had seven of these
problems to solve, say for a BCD (Binary Coded Decimal) to
seven segment decoder, we might want to automate the
process. A BCD to seven segment decoder generates the
logic signals to drive a seven segment LED (light emitting
diode) display.
Examples of computer automated design languages for
simplification of logic are PALASM, ABEL, CUPL, Verilog, and
VHDL. These programs accept a hardware descriptor
language input file which is based on Boolean equations and
produce an output file describing a reduced (or simplified)
Boolean solution. We will not require such tools in this
chapter. Let's move on to Venn diagrams as an introduction
to Karnaugh maps.
Venn diagrams and sets
Mathematicians use Venn diagrams to show the logical
relationships of sets (collections of objects) to one another.
Perhaps you have already seen Venn diagrams in your
algebra or other mathematics studies. If you have, you may
remember overlapping circles and the union and
intersection of sets. We will review the overlapping circles of
the Venn diagram. We will adopt the terms OR and AND
instead of union and intersection since that is the
terminology used in digital electronics.
The Venn diagram bridges the Boolean algebra from a
previous chapter to the Karnaugh Map. We will relate what
you already know about Boolean algebra to Venn diagrams,
then transition to Karnaugh maps.
A set is a collection of objects out of a universe as shown
below. The members of the set are the objects contained
within the set. The members of the set usually have
something in common; though, this is not a requirement.
Out of the universe of real numbers, for example, the set of
all positive integers {1,2,3...} is a set. The set {3,4,5} is an
example of a smaller set, or subset of the set of all positive
integers. Another example is the set of all males out of the
universe of college students. Can you think of some more
examples of sets?
Above left, we have a Venn diagram showing the set A in the
circle within the universe U, the rectangular area. If
everything inside the circle is A, then anything outside of
the circle is not A. Thus, above center, we label the
rectangular area outside of the circle A as A-not instead of U.
We show B and B-not in a similar manner.
What happens if both A and B are contained within the same
universe? We show four possibilities.
Let's take a closer look at each of the the four possibilities as
shown above.
The first example shows that set A and set B have nothing in
common according to the Venn diagram. There is no overlap
between the A and B circular hatched regions. For example,
suppose that sets A and B contain the following members:
set A = {1,2,3,4}
set B = {5,6,7,8}
None of the members of set A are contained within set B, nor
are any of the members of B contained within A. Thus, there
is no overlap of the circles.
In the second example in the above Venn diagram, Set A is
totally contained within set B How can we explain this
situation? Suppose that sets A and B contain the following
members:
set A = {1,2}
set B = {1,2,3,4,5,6,7,8}
All members of set A are also members of set B. Therefore,
set A is a subset of Set B. Since all members of set A are
members of set B, set A is drawn fully within the boundary of
set B.
There is a fifth case, not shown, with the four examples.
Hint: it is similar to the last (fourth) example. Draw a Venn
diagram for this fifth case.
The third example above shows perfect overlap between set
A and set B. It looks like both sets contain the same identical
members. Suppose that sets A and B contain the following:
set A = {1,2,3,4} set B = {1,2,3,4}
Therefore,
Set A = Set B
Sets And B are identically equal because they both have the
Same identical members. The A and B regions within the
corresponding Venn diagram above overlap completely. If
there is any doubt about what the above patterns represent,
refer to any figure above or below to be sure of what the
circular regions looked like before they were overlapped.
The fourth example above shows that there is something in
common between set A and set B in the overlapping region.
For example, we arbitrarily select the following sets to
illustrate our point:
set A = {1,2,3,4}
set B = {3,4,5,6}
Set A and Set B both have the elements 3 and 4 in common
These elements are the reason for the overlap in the center
common to A and B. We need to take a closer look at this
situation
Boolean Relationships on Venn
Diagrams
The fourth example has A partially overlapping B. Though,
we will first look at the whole of all hatched area below, then
later only the overlapping region. Let's assign some Boolean
expressions to the regions above as shown below. Below left
there is a red horizontal hatched area for A. There is a blue
vertical hatched area for B.
If we look at the whole area of both, regardless of the hatch
style, the sum total of all hatched areas, we get the
illustration above right which corresponds to the inclusive
OR function of A, B. The Boolean expression is A+ B. This is
shown by the 45° hatched area. Anything outside of the
hatched area corresponds to (A+ B)-not as shown above.
Let's move on to next part of the fourth example
The other way of looking at a Venn diagram with
overlapping circles is to look at just the part common to both
A and B, the double hatched area below left. The Boolean
expression for this common area corresponding to the AND
function is AB as shown below right. Note that everything
outside of double hatched AB is AB-not.
Note that some of the members of A, above, are members of
(AB)'. Some of the members of B are members of (AB)'. But,
none of the members of (AB)' are within the doubly hatched
area AB.
We have repeated the second example above left. Your fifth
example, which you previously sketched, is provided above
right for comparison. Later we will find the occasional
element, or group of elements, totally contained within
another group in a Karnaugh map.
Next, we show the development of a Boolean expression
involving a complemented variable below.
Example: (above)
Show a Venn diagram for A'B (A-not AND B).
Solution:
Starting above top left we have red horizontal shaded A’ (A-
not), then, top right, B. Next, lower left, we form the AND
function A'B by overlapping the two previous regions. Most
people would use this as the answer to the example posed.
However, only the double hatched A'B is shown far right for
clarity. The expression A’B is the region where both A’ and B
overlap. The clear region outside of A'B is (A'B)', which was
not part of the posed example.
Let's try something similar with the Boolean OR function.
Example:
Find B+A
(US
Solution:
Above right we start out with B which is complemented to
B'. Finally we overlay A on top of B’. Since we are interested
in forming the OR function, we will be looking for all hatched
area regardless of hatch style. Thus, A+ B' is all hatched area
above right. It is shown as a single hatch region below left
for clarity.
eMorgans theorem
ouble negation
Example:
Find (A+ B')'
Solution:
The green 45° A+ B' hatched area was the result of the
previous example. Moving on to a to,(A+B')' ,the present
example, above left, let us find the complement of A+B’,
which is the white clear area above left corresponding to
(A+ B')'. Note that we have repeated, at right, the AB'
double hatched result from a previous example for
comparison to our result. The regions corresponding to
(A+ B')' and AB’ above left and right respectively are
identical. This can be proven with DeMorgan's theorem and
double negation.
This brings up a point. Venn diagrams don't actually prove
anything. Boolean algebra is needed for formal proofs.
However, Venn diagrams can be used for verification and
visualization. We have verified and visualized DeMorgan's
theorem with a Venn diagram.
Example:
What does the Boolean expression A'+ B' look like on a Venn
Diagram?
A+B clear area
AB double hatch
Solution: above figure
Start out with red horizontal hatched A’ and blue vertical
hatched B' above. Superimpose the diagrams as shown. We
can still see the A’ red horizontal hatch superimposed on the
other hatch. It also fills in what used to be part of the B (B-
true) circle, but only that part of the B open circle not
common to the A open circle. If we only look at the B’ blue
vertical hatch, it fills that part of the open A circle not
common to B. Any region with any hatch at all, regardless of
type, corresponds to A'+B'. That is, everything but the open
white space in the center.
Example:
What does the Boolean expression (A'+ B')' look like on a
Venn Diagram?
Solution: above figure, lower left
Looking at the white open space in the center, it is
everything NOT in the previous solution of A'+B', which is
(A'+B')'.
Example:
Show that (A'+ B')' = AB
Solution: below figure, lower left
We previously showed on the above right diagram that the
white open region is (A'+B')'. On an earlier example we
showed a doubly hatched region at the intersection
(overlay) of AB. This is the left and middle figures repeated
here. Comparing the two Venn diagrams, we see that this
open region , (A'+B')', is the same as the doubly hatched
region AB (A AND B). We can also prove that (A'+ B')'= AB
by DeMorgan's theorem and double negation as shown
above.
Three variable Venn diagram
We show a three variable Venn diagram above with regions
A (red horizontal), B (blue vertical), and, C (green 45°). In
the very center note that all three regions overlap
representing Boolean expression ABC. There is also a larger
petal shaped region where A and B overlap corresponding to
Boolean expression AB. In a similar manner A and C overlap
producing Boolean expression AC. And B and C overlap
producing Boolean expression BC.
Looking at the size of regions described by AND expressions
above, we see that region size varies with the number of
variables in the associated AND expression.
A, 1-variable is a large circular region.
AB, 2-variable is a smaller petal shaped region.
ABC, 3-variable is the smallest region.
The more variables in the AND term, the smaller the
region.
Making a Venn diagram look like a
Karnaugh map
Starting with circle Ain a rectangular A’ universe in figure
(a) below, we morph a Venn diagram into almost a Karnaugh
map.
DI
a)
We expand circle A at (b) and (c), conform to the rectangular
A’ universe at (d), and change Ato a rectangle at (e).
Anything left outside of Ais A’ . We assign a rectangle to A’
at (f). Also, we do not use shading in Karnaugh maps. What
we have so far resembles a 1-variable Karnaugh map, but is
of little utility. We need multiple variables.
a)
ME]
tol
to
Figure (a) above is the same as the previous Venn diagram
showing A and A’ above except that the labels A and A’ are
above the diagram instead of inside the respective regions.
Imagine that we have go through a process similar to figures
(a-f) to get a "Square Venn diagram" for B and B' as we show
in middle figure (b). We will now superimpose the diagrams
in Figures (a) and (b) to get the result at (c), just like we
have been doing for Venn diagrams. The reason we do this is
so that we may observe that which may be common to two
overlapping regions-- say where A overlaps B. The lower
right cell in figure (c) corresponds to AB where A overlaps B.
B
B 0
B if
We don't waste time drawing a Karnaugh map like (c) above,
Sketching a simplified version as above left instead. The
column of two cells under A’ is understood to be associated
with A’, and the heading A is associated with the column of
cells under it. The row headed by B' is associated with the
cells to the right of it. In a similar manner B is associated
with the cells to the right of it. For the sake of simplicity, we
do not delineate the various regions as clearly as with Venn
diagrams.
The Karnaugh map above right is an alternate form used in
most texts. The names of the variables are listed next to the
diagonal line. The A above the diagonal indicates that the
variable A (and A’) is assigned to the columns. The O isa
substitute for A’, and the 1 substitutes for A. The B below
the diagonal is associated with the rows: O for B', and 1 for B
Example:
Mark the cell corresponding to the Boolean expression AB in
the Karnaugh map above with a 1
>|
a
|
a
B B
B B
Solution:
Shade or circle the region corresponding to A. Then, shade
or enclose the region corresponding to B. The overlap of the
two regions is AB. Place a 1 in this cell. We do not
necessarily enclose the A and B regions as at above left.
We develop a 3-variable Karnaugh map above, starting with
Venn diagram like regions. The universe (inside the black
rectangle) is split into two narrow narrow rectangular regions
for A’ and A. The variables B' and B divide the universe into
two square regions. C occupies a square region in the middle
of the rectangle, with C' split into two vertical rectangles on
each side of the C square.
In the final figure, we superimpose all three variables,
attempting to clearly label the various regions. The regions
are less obvious without color printing, more obvious when
compared to the other three figures. This 3-variable K-Map
(Karnaugh map) has 23 = 8 cells, the small squares within
the map. Each individual cell is uniquely identified by the
three Boolean Variables (A, B, C). For example, ABC’
uniquely selects the lower right most cell(*), A'B'C’ selects
the upper left most cell (x).
We don't normally label the Karnaugh map as shown above
left. Though this figure clearly shows map coverage by
single boolean variables of a 4-cell region. Karnaugh maps
are labeled like the illustration at right. Each cell is still
uniquely identified by a 3-variable product term, a Boolean
AND expression. Take, for example, ABC’ following the A row
across to the right and the BC’ column down, both
intersecting at the lower right cell ABC’. See (*) above
figure.
The above two different forms of a 3-variable Karnaugh map
are equivalent, and is the final form that it takes. The
version at right is a bit easier to use, since we do not have to
write down so many boolean alphabetic headers and
complement bars, just 1s and Os Use the form of map on the
right and look for the the one at left in some texts. The
column headers on the left B'C', B'C, BC, BC’ are
equivalent to 00, 01, 11, 10 on the right. The row headers
A, A’ are equivalent to 0, 1 on the right map.
Boolean expressions
Maurice Karnaugh, a telecommunications engineer,
developed the Karnaugh map at Bell Labs in 1953 while
designing digital logic based telephone switching circuits.
Now that we have developed the Karnaugh map with the aid
of Venn diagrams, let's put it to use. Karnaugh maps reduce
logic functions more quickly and easily compared to Boolean
algebra. By reduce we mean simplify, reducing the number
of gates and inputs. We like to simplify logic to a Jowest cost
form to save costs by elimination of components. We define
lowest cost as being the lowest number of gates with the
lowest number of inputs per gate.
Given a choice, most students do logic simplification with
Karnaugh maps rather than Boolean algebra once they learn
this tool.
ATS [ Ourpar_
fol a
ots
unspecified logic
Output = ABC + ABC + . . . ABC
We show five individual items above, which are just different
ways of representing the same thing: an arbitrary 2-input
digital logic function. First is relay ladder logic, then logic
gates, a truth table, a Karnaugh map, and a Boolean
equation. The point is that any of these are equivalent. Two
inputs A and B can take on values of either O or 1, high or
low, open or closed, True or False, as the case may be. There
are 22 = 4 combinations of inputs producing an output. This
iS applicable to all five examples.
These four outputs may be observed on a lamp in the relay
ladder logic, on a logic probe on the gate diagram. These
outputs may be recorded in the truth table, or in the
Karnaugh map. Look at the Karnaugh map as being a
rearranged truth table. The Output of the Boolean equation
may be computed by the laws of Boolean algebra and
transfered to the truth table or Karnaugh map. Which of the
five equivalent logic descriptions should we use? The one
which is most useful for the task to be accomplished.
The outputs of a truth table correspond on a one-to-one
basis to Karnaugh map entries. Starting at the top of the
truth table, the A=0, B=0 inputs produce an output a. Note
that this same output a is found in the Karnaugh map at the
A=0, B=0O cell address, upper left corner of K-map where the
A=0 row and B=0 column intersect. The other truth table
outputs B, x, 6 from inputs AB=01, 10, 11 are found at
corresponding K-map locations.
Below, we show the adjacent 2-cell regions in the 2-variable
K-map with the aid of previous rectangular Venn diagram
like Boolean regions.
Cells a and x are adjacent in the K-map as ellipses in the left
most K-map below. Referring to the previous truth table, this
is not the case. There is another truth table entry (B)
between them. Which brings us to the whole point of the
organizing the K-map into a square array, cells with any
Boolean variables in common need to be close to one
another so as to present a pattern that jumps out at us. For
cells a and x they have the Boolean variable B' in common.
We know this because B=0O (same as B') for the column
above cells a and x. Compare this to the square Venn
diagram above the kK-map.
A similar line of reasoning shows that B and 6 have Boolean
B (B=1) in common. Then, a and B have Boolean A’ (A=0) in
common. Finally, x and 5 have Boolean A (A=1) in common.
Compare the last two maps to the middle square Venn
diagram.
To summarize, we are looking for commonality of Boolean
variables among cells. The Karnaugh map is organized so
that we may see that commonality. Let's try some examples.
ATS [ Output :
fofol 0
ott.
ae a
Example:
Transfer the contents of the truth table to the Karnaugh map
above.
Solution:
The truth table contains two Ls. the K- map must have both
of them. locate the first 1 in the 2nd row of the truth table
above.
e note the truth table AB address
e locate the cell in the K-map having the same address
e place a 1 in that cell
Repeat the process for the 1 in the last line of the truth
table.
Example:
For the Karnaugh map in the above problem, write the
Boolean expression. Solution is below.
ve)
ve
nd
Solution:
Look for adjacent cells, that is, above or to the side of a cell.
Diagonal cells are not adjacent. Adjacent cells will have one
or more Boolean variables in common.
e Group (circle) the two Ls in the column
e Find the variable(s) top and/or side which are the same
for the group, Write this as the Boolean result. It is B in
Our case.
e Ignore variable(s) which are not the same for a cell
group. In our case A varies, is both 1 and 0, ignore
Boolean A.
e Ignore any variable not associated with cells containing
ls. B' has no ones under it. Ignore B'
e Result Out = B
This might be easier to see by comparing to the Venn
diagrams to the right, specifically the B column.
Example:
Write the Boolean expression for the Karnaugh map below.
B B
wo1i A
; A
Out =A
Solution: (above)
e Group (circle) the two l's in the row
e Find the variable(s) which are the same for the group,
Out = A’
Example:
For the Truth table below, transfer the outputs to the
Karnaugh, then write the Boolean expression for the result.
Output= A +B
Wrong Output= AB +B
Solution:
Transfer the 1s from the locations in the Truth table to the
corresponding locations in the K-map.
e Group (circle) the two L's in the column under B=1
Group (circle) the two 1's in the row right of A=1
Write product term for first group = B
Write product term for second group = A
Write Sum-Of-Products of above two terms Output =
A+B
The solution of the K-map in the middle is the simplest or
lowest cost solution. A less desirable solution is at far right.
After grouping the two Ls, we make the mistake of forming a
group of 1-cell. The reason that this is not desirable is that:
e The single cell has a product term of AB’
e The corresponding solution is Output = AB' + B
e This is not the simplest solution
The way to pick up this single 1 is to form a group of two
with the 1 to the right of it as shown in the lower line of the
middle K-map, even though this 1 has already been included
in the column group (B). We are allowed to re-use cells in
order to form larger groups. In fact, it is desirable because it
leads to a simpler result.
We need to point out that either of the above solutions,
Output or Wrong Output, are logically correct. Both circuits
yield the same output. It is a matter of the former circuit
being the lowest cost solution.
Example:
Fill in the Karnaugh map for the Boolean expression below,
then write the Boolean expression for the result.
Out= AB + AB + AB
O1 10 11
Solution: (above)
The Boolean expression has three product terms. There will
be a 1 entered for each product term. Though, in general,
the number of 1s per product term varies with the number of
variables in the product term compared to the size of the K-
map. The product term is the address of the cell where the 1
is entered. The first product term, A'B, corresponds to the O01
cell in the map. A 1 is entered in this cell. The other two P-
terms are entered for a total of three ls
Next, proceed with grouping and extracting the simplified
result as in the previous truth table problem.
Example:
Simplify the logic diagram below.
Out
Solution: (Figure below)
e Write the Boolean expression for the original logic
diagram as shown below
e Transfer the product terms to the Karnaugh map
e Form groups of cells as in previous examples
e Write Boolean expression for groups as in previous
examples
e Draw simplified logic diagram
Out= AB + AB + i
O1 10 13
Example:
Simplify the logic diagram below.
Out= AB + AB
O01 10
0
out = ae
B Exclusive-OR
Solution:
e Write the Boolean expression for the original logic
diagram shown above
e Transfer the product terms to the Karnaugh map.
e It is not possible to form groups.
e« No simplification is possible; leave it as it Is.
No logic simplification is possible for the above diagram.
This sometimes happens. Neither the methods of Karnaugh
maps nor Boolean algebra can simplify this logic further. We
show an Exclusive-OR schematic symbol above; however,
this is not a logical simplification. It just makes a schematic
diagram look nicer. Since it is not possible to simplify the
Exclusive-OR logic and it is widely used, it is provided by
manufacturers as a basic integrated circuit (7486).
Logic simplification with Karnaugh
maps
The logic simplification examples that we have done so
could have been performed with Boolean algebra about as
quickly. Real world logic simplification problems call for
larger Karnaugh maps so that we may do serious work. We
will work some contrived examples in this section, leaving
most of the real world applications for the Combinatorial
Logic chapter. By contrived, we mean examples which
illustrate techniques. This approach will develop the tools
we need to transition to the more complex applications in
the Combinatorial Logic chapter.
We show our previously developed Karnaugh map. We will
use the form on the right.
Note the sequence of numbers across the top of the map. It
is not in binary sequence which would be 00, 01, 10, 11. It
is OO, O01, 11 10, which is Gray code sequence. Gray code
sequence only changes one binary bit as we go from one
number to the next in the sequence, unlike binary. That
means that adjacent cells will only vary by one bit, or
Boolean variable. This is what we need to organize the
outputs of a logic function so that we may view
commonality. Moreover, the column and row headings must
be in Gray code order, or the map will not work as a
Karnaugh map. Cells sharing common Boolean variables
would no longer be adjacent, nor show visual patterns.
Adjacent cells vary by only one bit because a Gray code
sequence varies by only one bit.
If we sketch our own Karnaugh maps, we need to generate
Gray code for any size map that we may use. This is how we
generate Gray code of any size.
How to generate Gray code.
1. Write 0,1 ina column.
2. Draw a mirror under the column.
3. Reflect the numbers about the mirror.
4. Distinguish the numbers above the mirror with leading zeros.
5. Distinguish those below the mirror with
leading ones.
| | 6. Finished 2-bit Gray code.
'
0 0 0 00 00 00 OO 000 OOO
1 4) 0101 01 001 001
i L tt ak al O11 O11
0 0 1010 10 ) 010 010
7. Need 3-bit Gray code? Draw 1 0 1 0 1 1 0
Dubit cades reflect about 11 A, <a
mirror. 01 01 101
00 00 _100
8. Distinguish upper 4-numbers with leading zeros.
9. Distinguish lower 4-numbers with leading ones.
Note that the Gray code sequence, above right, only varies
by one bit as we go down the list, or bottom to top up the
list. This property of Gray code is often useful in digital
electronics in general. In particular, it is applicable to
Karnaugh maps.
Let us move on to some examples of simplification with 3-
variable Karnaugh maps. We show how to map the product
terms of the unsimplified logic to the K-map. We illustrate
how to identify groups of adjacent cells which leads to a
Sum-of-Products simplification of the digital logic.
Above we, place the 1's in the K-map for each of the product
terms, identify a group of two, then write a p-term (product
term) for the sole group as our simplified result.
Mapping the four product terms above yields a group of four
covered by Boolean A'
Mapping the four p-terms yields a group of four, which is
covered by one variable C.
After mapping the six p-terms above, identify the upper
group of four, pick up the lower two cells as a group of four
by sharing the two with two more from the other group.
Covering these two with a group of four gives a simpler
result. Since there are two groups, there will be two p-terms
in the Sum-of-Products result A'+ B
Out= ABC+ABC
C
A\OO 011110
0
1
Out= BC
The two product terms above form one group of two and
simplifies to BC
Mapping the four p-terms yields a single group of four, which
isB
Out= ABC+ABC+ABC+ABC
Mapping the four p-terms above yields a group of four.
Visualize the group of four by rolling up the ends of the map
to form a cylinder, then the cells are adjacent. We normally
mark the group of four as above left. Out of the variables A,
B, C, there is acommon variable: C'. C' is a O over all four
cells. Final result is C’.
Out= ABC+ABC+ABCH+ABC+ABC+ABC
The six cells above from the unsimplified equation can be
organized into two groups of four. These two groups should
give us two p-terms in our simplified result of A' + C’.
Below, we revisit the Toxic Waste Incinerator from the
Boolean algebra chapter. See Boolean algebra chapter for
details on this example. We will simplify the logic using a
Karnaugh map.
A\OO 011110
ota
Output = AB + BC + AC
The Boolean equation for the output has four product terms.
Map four 1's corresponding to the p-terms. Forming groups
of cells, we have three groups of two. There will be three p-
terms in the simplified result, one for each group. See "Toxic
Waste Incinerator", Boolean algebra chapter for a gate
diagram of the result, which is reproduced below.
Output = AB + BC + AC
Below we repeat the Boolean algebra simplification of Toxic
waste incinerator for comparison.
ABC + ABC + ABC + ABC
| Factoring Bc out of 1°‘ and 4" terms
BC(A + A) + ABC + ABC
| Applying identityA + A = 1
BC(1) + ABC + ABC
Applying identity 1A = A
BC + ABC + ABC
| Factoring B out of 1° and 3” terms
B(C + AC) + ABC
Applying ruleA + AB = A + Bto
thec + ACterm
B(C + A) + ABC
| Distributing terms
BC + AB + ABC
Factoring A out of 2™ and 3" terms
BC + A(B + BC)
Applying ruleA + AB = A + Bto
theB + BCterm
BC + A(B + C)
| Distributing terms
BC + AB + AC
or Simplified result
AB + BC + AC
Below we repeat the Toxic waste incinerator Karnaugh map
solution for comparison to the above Boolean algebra
simplification. This case illustrates why the Karnaugh map is
widely used for logic simplification.
C
A\OO 011110
Output = AB + BC + AC
The Karnaugh map method looks easier than the previous
page of boolean algebra.
Larger 4-variable Karnaugh maps
Knowing how to generate Gray code should allow us to build
larger maps. Actually, all we need to do is look at the left to
right sequence across the top of the 3-variable map, and
copy it down the left side of the 4-variable map. See below.
ral D . 4 4 .
“Ap\o0oO O1 11 10
00
Ol
11
10
The following four variable Karnaugh maps illustrate
reduction of Boolean expressions too tedious for Boolean
algebra. Reductions could be done with Boolean algebra.
However, the Karnaugh map is faster and easier, especially if
there are many logic reductions to do.
‘
Out= AB + CD
The above Boolean expression has seven product terms.
They are mapped top to bottom and left to right on the kK-
map above. For example, the first P-term A'B'CD is first row
3rd cell, corresponding to map location A=0O, B=O, C=1,
D=1. The other product terms are placed in a similar
manner. Encircling the largest groups possible, two groups of
four are shown above. The dashed horizontal group
corresponds the the simplified product term AB. The vertical
group corresponds to Boolean CD. Since there are two
groups, there will be two product terms in the Sum-Of-
Products result of Out= AB+ CD.
Fold up the corners of the map below like it is a napkin to
make the four cells physically adjacent.
Out= ABCD+ABCD+ABCD+ABCD
The four cells above are a group of four because they all
have the Boolean variables B’ and D' in common. In other
words, B=0 for the four cells, and D=0O for the four cells. The
other variables (A, B) are O in some cases, 1 in other cases
with respect to the four corner cells. Thus, these variables
(A, B) are not involved with this group of four. This single
group comes out of the map as one product term for the
simplified result: Out=B'C'
For the K-map below, roll the top and bottom edges into a
cylinder forming eight adjacent cells.
Out= ABCD + ABCD + ABCD + i
+ ABCD +ABCD + ABCD + ABCD
The above group of eight has one Boolean variable in
common: B=0. Therefore, the one group of eight is covered
by one p-term: B’. The original eight term Boolean
expression simplifies to Out=B'
The Boolean expression below has nine p-terms, three of
which have three Booleans instead of four. The difference is
that while four Boolean variable product terms cover one
cell, the three Boolean p-terms cover a pair of cells each.
The six product terms of four Boolean variables map in the
usual manner above as single cells. The three Boolean
variable terms (three each) map as cell pairs, which is shown
above. Note that we are mapping p-terms into the K-map,
not pulling them out at this point.
For the simplification, we form two groups of eight. Cells in
the corners are shared with both groups. This is fine. In fact,
this leads to a better solution than forming a group of eight
and a group of four without sharing any cells. Final Solution
is Out= B'+ D'
Below we map the unsimplified Boolean expression to the
Karnaugh map.
Above, three of the cells form into a groups of two cells. A
fourth cell cannot be combined with anything, which often
happens in "real world" problems. In this case, the Boolean
p-term ABCD is unchanged in the simplification process.
Result: Out= B'C'D'+ A'B'D'+ ABCD
Often times there is more than one minimum cost solution to
a simplification problem. Such is the case illustrated below.
Both results above have four product terms of three Boolean
variable each. Both are equally valid minimal cost solutions.
The difference in the final solution is due to how the cells are
grouped as shown above. A minimal cost solution is a valid
logic design with the minimum number of gates with the
minimum number of inputs.
Below we map the unsimplified Boolean equation as usual
and form a group of four as a first simplification step. It may
not be obvious how to pick up the remaining cells.
Out= ABCD + ABCD + ABCD
+ ABCD + ABCD + ABCD
+ ABCD + ABCD + ABCD
re CD wer
AR\OO 01 11 10
ooffafz\p |_|
AR 00 011110
Disa
AR 00 01 1110
emp |
Out= AC + AD + BC + BD
Pick up three more cells in a group of four, center above.
There are still two cells remaining. the minimal cost method
to pick up those is to group them with neighboring cells as
groups of four as at above right.
On a cautionary note, do not attempt to form groups of
three. Groupings must be powers of 2, that is, 1, 2, 4, 8 ...
Below we have another example of two possible minimal
cost solutions. Start by forming a couple of groups of four
after mapping the cells.
Out= ABCD+ABCD+ABCD+ABCD+ABCD
+ABCD + ABCD + ABCD + ABCD
Ap 00 O01 1110
oofY fy
AD 00 O1 1110
Zona
Out= CD + CD+ ABC
Out= CD + CD+ ABD
The two solutions depend on whether the single remaining
cell is grouped with the first or the second group of four asa
group of two cells. That cell either comes out as either ABC’
or ABD, your choice. Either way, this cell is covered by
either Boolean product term. Final results are shown above.
Below we have an example of a simplification using the
Karnaugh map at left or Boolean algebra at right. Plot C’ on
the map as the area of all cells covered by address C=O, the
8-cells on the left of the map. Then, plot the single ABCD
cell. That single cell forms a group of 2-cell as shown, which
simplifies to P-term ABD, for an end result of Out = C' +
ABD.
Out= C+ABCD Simplification by Boolean
r Algebra
Out= C+ABCD
Applying rule A + AB = A+B to
the C + ABCD term
Out= C + ABD
This (above) is a rare example of a four variable problem
that can be reduced with Boolean algebra without a lot of
work, assuming that you remember the theorems.
Minterm vs maxterm solution
So far we have been finding Sum-Of-Product (SOP) solutions
to logic reduction problems. For each of these SOP solutions,
there is also a Product-Of-Sums solution (POS), which could
be more useful, depending on the application. Before
working a Product-Of-Sums solution, we need to introduce
some new terminology. The procedure below for mapping
product terms is not new to this chapter. We just want to
establish a formal procedure for minterms for comparison to
the new procedure for maxterms.
Out= ABC
Out= ABC i oe
Minterm= ABC Minterm= ABC
Numeric= lll Numeric= 010
Out= ABC
A minterm is a Boolean expression resulting in 1 for the
output of a single cell, and Os for all other cells ina
Karnaugh map, or truth table. If a minterm has a single 1
and the remaining cells as Os, it would appear to cover a
minimum area of 1s. The illustration above left shows the
minterm ABC, a single product term, as a single 1 in a map
that is otherwise Os. We have not shown the Qs in our
Karnaugh maps up to this point, as it is customary to omit
them unless specifically needed. Another minterm A'BC' is
shown above right. The point to review is that the address of
the cell corresponds directly to the minterm being mapped.
That is, the cell 111 corresponds to the minterm ABC above
left. Above right we see that the minterm A' BC’ corresponds
directly to the cell 010. A Boolean expression or map may
have multiple minterms.
Referring to the above figure, Let's summarize the procedure
for placing a minterm in a K-map:
e Identify the minterm (product term) term to be mapped.
e Write the corresponding binary numeric value.
e Use binary value as an address to place a 1 in the K-map
e Repeat steps for other minterms (P-terms within a Sum-
Of-Products).
Numer.
Minter
A Boolean expression will more often than not consist of
multiple minterms corresponding to multiple cells ina
Karnaugh map as shown above. The multiple minterms in
this map are the individual minterms which we examined in
the previous figure above. The point we review for reference
is that the Ls come out of the K-map as a binary cell address
which converts directly to one or more product terms. By
directly we mean that a O corresponds to a complemented
variable, and a 1 corresponds to a true variable. Example:
010 converts directly to A'BC’. There was no reduction in
this example. Though, we do have a Sum-Of-Products result
from the minterms.
Referring to the above figure, Let's summarize the procedure
for writing the Sum-Of-Products reduced Boolean equation
from a K-map:
e Form largest groups of 1s possible covering all
minterms. Groups must be a power of 2.
e Write binary numeric value for groups.
e Convert binary value to a product term.
e Repeat steps for other groups. Each group yields a p-
terms within a Sum-Of-Products.
Nothing new so far, a formal procedure has been written
down for dealing with minterms. This serves as a pattern for
dealing with maxterms.
Next we attack the Boolean function which is O for a single
cell and Ls for all others.
Out = (A+B+C)
Maxterm = A+B+C
Numeric = 1 1 iit
Complement =
A maxterm is a Boolean expression resulting in a O for the
output of a single cell expression, and Is for all other cells in
the Karnaugh map, or truth table. The illustration above left
shows the maxterm (A+ B+ C), a single sum term, as a single
0 in a map that is otherwise Ls. If a maxterm has a single O
and the remaining cells as Ls, it would appear to cover a
maximum area of Ls.
There are some differences now that we are dealing with
something new, maxterms. The maxterm is a O, nota 1 in
the Karnaugh map. A maxterm is a sum term, (A+ B+ C) in
our example, not a product term.
It also looks strange that (A+ B+ C) is mapped into the cell
000. For the equation Out= (A+ B+ C)=0, all three variables
(A, B, C) must individually be equal to O. Only (0+ 0+ 0)=0
will equal O. Thus we place our sole O for minterm (A+ B+ C)
in cell A,B,C=000 in the K-map, where the inputs are allO .
This is the only case which will give us a O for our maxterm.
All other cells contain 1s because any input values other
than ((0,0,0) for (A+ B+ C) yields 1s upon evaluation.
Referring to the above figure, the procedure for placing a
maxterm in the K-map is:
Identify the Sum term to be mapped.
Write corresponding binary numeric value.
Form the complement
Use the complement as an address to place a O in the K-
map
e Repeat for other maxterms (Sum terms within Product-
of-Sums expression).
Out = (A+B+C)
Maxterm = A+B+C
Numeric = 0.60 60
Complement = a i. fk
Another maxterm A'+ B'+C' is shown above. Numeric 000
corresponds to A'+B'+C’. The complement is 111. Place a O
for maxterm (A'+ B'+ C’) in this cell (1,1,1) of the K-map as
shown above.
Why should (A'+ B'+C') cause a O to be in cell 111? When
A'+B'+C' is (1'+1'+1'), all Ls in, which is (0+ 0+ 0) after
taking complements, we have the only condition that will
give us aQ. All the Ls are complemented to all Os, which is O
when ORed.
Out = (A+B+C) (A+Bt+C)
Maxterm= (A+B+C) Maxterm= (A+B+C)
Numeric= he “ay ad Numeric= 1 1 0
Complement= O O O Complement= O O 1
A Boolean Product-Of-Sums expression or map may have
multiple maxterms as shown above. Maxterm (A+ B+ C)
yields numeric 111 which complements to 000, placing aO
in cell (0,0,0). Maxterm (A+ B+ C’) yields numeric 110
which complements to OO1, placing a O in cell (0,0,1).
Now that we have the k-map setup, what we are really
interested in is showing how to write a Product-Of-Sums
reduction. Form the Os into groups. That would be a group of
two below. Write the binary value corresponding to the sum-
term which is (0,0,X). Both A and B are O for the group. But,
C is both O and 1 so we write an X as a place holder for C.
Form the complement (1,1,X). Write the Sum-term (A+ B)
discarding the C and the X which held its' place. In general,
expect to have more sum-terms multiplied together in the
Product-Of-Sums result. Though, we have a simple example
here.
Out = (A+B+C)(A+B+C)
m =e
4 \00 011110
o Qf o)2 |: |
1
ABCz=00xX
Complement = 11 xX
Sum-term =(A+B)
Out =(A+B)
Let's summarize the procedure for writing the Product-Of-
Sums Boolean reduction for a K-map:
e Form largest groups of Os possible, covering all
maxterms. Groups must be a power of 2.
e Write binary numeric value for group.
e Complement binary numeric value for group.
e Convert complement value to a sum-term.
Repeat steps for other groups. Each group yields a sum-
term within a Product-Of-Sums result.
Example:
Simplify the Product-Of-Sums Boolean expression below,
providing a result in POS form.
Out= (A+B+C+D) (A+B+C+D) (A+B+C+D) (A+B+C+D)
(A+B+C+D) (A+B+C+D) (A+B+C+D)
Solution:
Transfer the seven maxterms to the map below as Os. Be
sure to complement the input variables in finding the proper
cell location.
Out= (A+B+C+D)(A+B+C+D) (A+B+C+D) (A+B+C+D)
(A+B+C+D) (A+B+C+D)(A+B+C+D)
We map the Os as they appear left to right top to bottom on
the map above. We locate the last three maxterms with
leader lines..
Once the cells are in place above, form groups of cells as
shown below. Larger groups will give a sum-term with fewer
inputs. Fewer groups will yield fewer sum-terms in the result.
input complement Sum-term
ABCD = X00l1 > X110 > (B+C+D)
ABCD = 0x01 > 1X10 > (A+ C+D )
ABCD = XXl0 > XxX0l1l > (C+D )
Out= (B+C+D) (A+C+D) (C+D)
We have three groups, so we expect to have three sum-
terms in our POS result above. The group of 4-cells yields a
2-variable sum-term. The two groups of 2-cells give us two 3-
variable sum-terms. Details are shown for how we arrived at
the Sum-terms above. For a group, write the binary group
input address, then complement it, converting that to the
Boolean sum-term. The final result is product of the three
sums.
Example:
Simplify the Product-Of-Sums Boolean expression below,
providing a result in SOP form.
Out= (A+B+C+D) (A+B+C+D) (A+B+C+D) (A+B+C+D)
(A+B+C+D) (A+B+C+D) (A+B+C+D)
Solution:
This looks like a repeat of the last problem. It is except that
we ask for a Sum-Of-Products Solution instead of the
Product-Of-Sums which we just finished. Map the maxterm
Os from the Product-Of-Sums given as in the previous
problem, below left.
Out= (A+B+C+D)(A+B+C+D) (A+B+C+D) (A+B+C+D)
(A+B+C+D) (A+B+C+D)(A+B+C+D)
Then fill in the implied 1s in the remaining cells of the map
above right.
Form groups of 1s to cover all 1s. Then write the Sum-Of-
Products simplified result as in the previous section of this
chapter. This is identical to a previous problem.
Out= (A+B+C+D)(A+B+C+D) (A+B+C+D) (A+B+C+D)
(A+B+C+D) (A+B+C+D) (A+B+C+D)
Out= CD + CD+ ABD
Out= (B+C+D) (A+C+D)(C+D)
Above we show both the Product-Of-Sums solution, from the
previous example, and the Sum-Of-Products solution from
the current problem for comparison. Which is the simpler
solution? The POS uses 3-OR gates and 1-AND gate, while
the SOP uses 3-AND gates and 1-OR gate. Both use four
gates each. Taking a closer look, we count the number of
gate inputs. The POS uses 8-inputs; the SOP uses 7-inputs.
By the definition of minimal cost solution, the SOP solution
is simpler. This is an example of a technically correct answer
that is of little use in the real world.
The better solution depends on complexity and the logic
family being used. The SOP solution is usually better if using
the TTL logic family, as NAND gates are the basic building
block, which works well with SOP implementations. On the
other hand, A POS solution would be acceptable when using
the CMOS logic family since all sizes of NOR gates are
available.
Out= (B+C+D) (A+C+D)(C+D) Out= CD + CD+ ABD
io]
D
The gate diagrams for both cases are shown above, Product-
Of-Sums left, and Sum-Of-Products right.
Below, we take a closer look at the Sum-Of-Products version
of our example logic, which is repeated at left.
Out=- CD + CD+ ABD
oO Pp
Out= CD + CD+ ABD
Out
Above all AND gates at left have been replaced by NAND
gates at right.. The OR gate at the output is replaced by a
NAND gate. To prove that AND-OR logic is equivalent to
NAND-NAND logic, move the inverter invert bubbles at the
output of the 3-NAND gates to the input of the final NAND as
shown in going from above right to below left.
on
@
x Out
¥
Z = 7
Out= 2Y2 DeMorgans
+¥+2 Double negation
Out= X
Out= X+Y+Z
Xx Out
=e
Z
Out= X+Y+Z
Above right we see that the output NAND gate with inverted
inputs is logically equivalent to an OR gate by DeMorgan's
theorem and double negation. This information is useful in
building digital logic in a laboratory setting where TTL logic
family NAND gates are more readily available in a wide
variety of configurations than other types.
The Procedure for constructing NAND-NAND logic, in place of
AND-OR logic is as follows:
Produce a reduced Sum-Of-Products logic design.
When drawing the wiring diagram of the SOP, replace all
gates (both AND and OR) with NAND gates.
Unused inputs should be tied to logic High.
In case of troubleshooting, internal nodes at the first
level of NAND gate outputs do NOT match AND-OR
diagram logic levels, but are inverted. Use the NAND-
NAND logic diagram. Inputs and final output are
identical, though.
Label any multiple packages U1, U2... etc.
Use data sheet to assign pin numbers to inputs and
outputs of all gates.
Example:
Let us revisit a previous problem involving an SOP
minimization. Produce a Product-Of-Sums solution. Compare
the POS solution to the previous SOP.
ABCD + ABCD + ABCD
+ ABCD + ABCD + ABCD
ABCD + ABCD + ABCD
Ap 00 O01 1110
oo] fay]
oy |
AB 00 011110
00/1 fi fi fos]
Solution:
Above left we have the original problem starting with a 9-
minterm Boolean unsimplified expression. Reviewing, we
formed four groups of 4-cells to yield a 4-product-term SOP
result, lower left.
In the middle figure, above, we fill in the empty spaces with
the implied Os. The Os form two groups of 4-cells. The solid
blue group is (A'+B), the dashed red group is (C'+D). This
yields two sum-terms in the Product-Of-Sums result, above
right Out = (A'+B)(C'+D)
Comparing the previous SOP simplification, left, to the POS
simplification, right, shows that the POS is the least cost
solution. The SOP uses 5-gates total, the POS uses only 3-
gates. This POS solution even looks attractive when using
TTL logic due to simplicity of the result. We can find AND
gates and an OR gate with 2-inputs.
Out
D
B B Out
Cc
D
Out= AC +AD +BC + BD Out= (A+B) (C+D)
The SOP and POS gate diagrams are shown above for our
comparison problem.
Given the pin-outs for the TTL logic family integrated circuit
gates below, label the maxterm diagram above right with
Circuit designators (Ul-a, U1-b, U2-a, etc), and pin numbers.
2
vec
A
B
c
D
7404
aan Out= (A+B) (C+D)
|
Each integrated circuit package that we use will receive a
circuit designator: U1, U2, U3. To distinguish between the
individual gates within the package, they are identified as a,
b, c, d, etc. The 7404 hex-inverter package is U1. The
individual inverters in it are are Ul-a, U1-b, U1-c, etc. U2 is
assigned to the 7432 quad OR gate. U3 is assigned to the
7408 quad AND gate. With reference to the pin numbers on
the package diagram above, we assign pin numbers to all
gate inputs and outputs on the schematic diagram below.
We can now build this circuit in a laboratory setting. Or, we
could design a printed circuit board for it. A printed circuit
board contains copper foil "wiring" backed by a non
conductive substrate of phenolic, or epoxy-fiberglass.
Printed circuit boards are used to mass produce electronic
circuits. Ground the inputs of unused gates.
Ul = 7404
U2 = 7432
Out= (A+B) (C+D) U3 = 7408
Label the previous POS solution diagram above left (third
figure back) with Circuit designators and pin numbers. This
will be similar to what we just did.
We can find 2-input AND gates, 7408 in the previous
example. However, we have trouble finding a 4-input OR
gate in our TTL catalog. The only kind of gate with 4-inputs
is the 7420 NAND gate shown above right.
We can make the 4-input NAND gate into a 4-input OR gate
by inverting the inputs to the NAND gate as shown below. So
we will use the 7420 4-input NAND gate as an OR gate by
inverting the inputs.
AB=A+B DeMorgan's
Y=
Y =A+B Double negation —j >} — >»
We will not use discrete inverters to invert the inputs to the
7420 4-input NAND gate, but will drive it with 2-input NAND
gates in place of the AND gates called for in the SOP,
minterm, solution. The inversion at the output of the 2-input
NAND gates supply the inversion for the 4-input OR gate.
7404
7400
7420
Out= (AC ) (AD) (BC) (BD) Boolean from diagram
Out= AC + AD + BC + BD DeMorgan’s
C
+ AD + BC + BD Double negation
The result is shown above. It is the only practical way to
actually build it with TTL gates by using NAND-NAND logic
replacing AND-OR logic.
For reference, this section introduces the terminology used
in some texts to describe the minterms and maxterms
assigned to a Karnaugh map. Otherwise, there is no new
material here.
2 (sigma) indicates sum and lower case "m" indicates
minterms. 2m indicates sum of minterms. The following
example is revisited to illustrate our point. Instead of a
Boolean equation description of unsimplified logic, we list
the minterms.
f(A,B,C,D) = 2 m(1, 2, 3, 4, 5,7, 8,9, 11, 12, 13, 15)
or
f(A,B,C,D) =
2(M1,Mz,M3,M4,M5,M7,Mg,Mg,M11,M12,M13,M 45)
The numbers indicate cell location, or address, within a
Karnaugh map as shown below right. This is certainly a
compact means of describing a list of minterms or cells ina
K-map.
BCD + ABCD + ABCD
+ ABCD + ABCD + ABCD
BCD + ABCD + ABCD
ANXOO 011110 4p 00 011110 ARA0O0 01 11 10
oof of [3 Ja | ota GRE
The Sum-Of-Products solution is not affected by the new
terminology. The minterms, 1s, in the map have been
grouped as usual and a Sum-OF-Products solution written.
Below, we show the terminology for describing a list of
maxterms. Product is indicated by the Greek NM (pi), and
upper case "M" indicates maxterms. MM indicates product of
maxterms. The same example illustrates our point. The
Boolean equation description of unsimplified logic, is
replaced by a list of maxterms.
f(A,B,C,D) = M(2, 6, 8, 9, 10, 11, 14)
or
f(A,B,C,D) = N(M,, Me, Ms, Mo, Mio, Mi. My.)
Once again, the numbers indicate K-map cell address
locations. For maxterms this is the location of Os, as shown
below. A Product-OF-Sums solution is completed in the usual
manner.
+D) (A +B+C+D)
f£(A,B,C,D)= IIM(2,6,8,9,10,11,14)
cD
AXOO 011110 AXO0 011110 ANoo 01 1110
00 ae oof iti fr fo} oof ifs fi fey
Don't care cells in the Karnaugh map
Up to this point we have considered logic reduction
problems where the input conditions were completely
specified. That is, a 3-variable truth table or Karnaugh map
had 2" = 23 or 8-entries, a full table or map. It is not always
necessary to fill in the complete truth table for some real-
world problems. We may have a choice to not fill in the
complete table.
For example, when dealing with BCD (Binary Coded
Decimal) numbers encoded as four bits, we may not care
about any codes above the BCD range of (0, 1, 2...9). The 4-
bit binary codes for the hexadecimal numbers (Ah, Bh, Ch,
Eh, Fh) are not valid BCD codes. Thus, we do not have to fill
in those codes at the end of a truth table, or K-map, if we do
not care to. We would not normally care to fill in those codes
because those codes (1010, 1011, 1100, 1101, 1110, 1111)
will never exist as long as we are dealing only with BCD
encoded numbers. These six invalid codes are don't cares as
far as we are concerned. That is, we do not care what output
our logic circuit produces for these don't cares.
Don't cares in a Karnaugh map, or truth table, may be either
ls or Os, as long as we don't care what the output is for an
input condition we never expect to see. We plot these cells
with an asterisk, *, among the normal 1s and Os. When
forming groups of cells, treat the don't care cell as eithera 1
or a0, or ignore the don't cares. This is helpful if it allows us
to form a larger group than would otherwise be possible
without the don't cares. There is no requirement to group all
or any of the don't cares. Only use them in a group if it
simplifies the logic.
Negece ace
Mi input comp- Sum
lement term
XX0 > XX1 > Cc
ie
oo
ie)
iow od
YO
0
pa
oY
°
Dy
Above is an example of a logic function where the desired
output is 1 for input ABC = 101 over the range from 000 to
101. We do not care what the output is for the other
possible inputs (110, 111). Map those two as don't cares.
We show two solutions. The solution on the right Out = AB'C
is the more complex solution since we did not use the don't
care cells. The solution in the middle, Out=AC, is less
complex because we grouped a don't care cell with the
single 1 to form a group of two. The third solution, a Product-
Of-Sums on the right, results from grouping a don't care with
three zeros forming a group of four Os. This is the same, less
complex, Out= AC. We have illustrated that the don't care
cells may be used as either 1s or Os, whichever is useful.
The electronics class of Lightning State College has been
asked to build the lamp logic for a stationary bicycle exhibit
at the local science museum. As a rider increases his
pedaling speed, lamps will light on a bar graph display. No
lamps will light for no motion. As speed increases, the lower
lamp, L1 lights, then L1 and L2, then, L1, L2, and L3, until all
lamps light at the highest speed. Once all the lamps
illuminate, no further increase in speed will have any effect
on the display.
A small DC generator coupled to the bicycle tire outputs a
voltage proportional to speed. It drives a tachometer board
which limits the voltage at the high end of speed where all
lamps light. No further increase in speed can increase the
voltage beyond this level. This is crucial because the
downstream A to D (Analog to Digital) converter puts out a
3-bit code, ABC, 2? or 8-codes, but we only have five lamps.
A is the most significant bit, C the least significant bit.
The lamp logic needs to respond to the six codes out of the
A to D. For ABC=000, no motion, no lamps light. For the five
codes (001 to 101) lamps L1, LL&L2, LL&L2&L3, up to all
lamps will light, as speed, voltage, and the A to D code
(ABC) increases. We do not care about the response to input
codes (110, 111) because these codes will never come out
of the A to D due to the limiting in the tachometer block. We
need to design five logic circuits to drive the five lamps.
Since, none of the lamps light for ABC= 000 out of the A to
D, enter a O in all K-maps for cell ABC= 000. Since we don't
care about the never to be encountered codes (110, 111),
enter asterisks into those two cells in all five K-maps.
Lamp L5 will only light for code ABC= 101. Enter a 1 in that
cell and five Os into the remaining empty cells of L5 K-map.
L4 will light initially for code ABC= 100, and will remain
illuminated for any code greater, ABC= 101, because all
lamps below L5 will light when L5 lights. Enter 1s into cells
100 and 101 of the L4 map so that it will light for those
codes. Four O's fill the remaining L4 cells
L3 will initially light for code ABC=0O11. It will also light
whenever L5 and L4 illuminate. Enter three 1s into cells
011, 100, 101 for L3 map. Fill three Os into the remaining
L3 cells.
L2 lights for ABC=010 and codes greater. Fill Ls into cells
010, 011, 100, 101, and two Os in the remaining cells.
The only time L1 is not lighted is for no motion. There is
already a O in cell ABC=000. All the other five cells receive
Ls.
Group the 1's as shown above, using don't cares whenever a
larger group results. The L1 map shows three product terms,
corresponding to three groups of 4-cells. We used both don't
cares in two of the groups and one don't care on the third
group. The don't cares allowed us to form groups of four.
In a similar manner, the L2 and L4 maps both produce
groups of 4-cells with the aid of the don't care cells. The L4
reduction is striking in that the L4 lamp is controlled by the
most significant bit from the A to D converter, L5S= A. No
logic gates are required for lamp L4. In the L3 and L5 maps,
single cells form groups of two with don't care cells. In all
five maps, the reduced Boolean equation is less complex
than without the don't cares.
The gate diagram for the circuit is above. The outputs of the
five K-map equations drive inverters. Note that the Ll OR
gate is not a 3-input gate but a 2-input gate having inputs
(A+B), C, outputting A+ B+C The open collector inverters,
7406, are desirable for driving LEDs, though, not part of the
K-map logic design. The output of an open collecter gate or
inverter is open circuited at the collector internal to the
integrated circuit package so that all collector current may
flow through an external load. An active high into any of the
inverters pulls the output low, drawing current through the
LED and the current limiting resistor. The LEDs would likely
be part of a solid state relay driving 120VAC lamps fora
museum exhibit, not shown here.
Larger 5 & 6-variable Karnaugh maps
Larger Karnaugh maps reduce larger logic designs. How
large is large enough? That depends on the number of
inputs, fan-ins, to the logic circuit under consideration. One
of the large programmable logic companies has an answer.
Altera's own data, extracted from its library of customer
designs, supports the value of heterogeneity. By
examining logic cones, mapping them onto LUT-based
nodes and sorting them by the number of inputs that
would be best at each node, Altera found that the
distribution of fan-ins was nearly flat between two and
six inputs, with a nice peak at five.
The answer is no more than six inputs for most all designs,
and five inputs for the average logic design. The five
variable Karnaugh map follows.
7 000 O01 O11 010 110 111 101 100
ARB
5- variable Karnaugh map (Gray code)
The older version of the five variable K-map, a Gray Code
map or reflection map, is shown above. The top (and side for
a 6-variable map) of the map is numbered in full Gray code.
The Gray code reflects about the middle of the code. This
style map is found in older texts. The newer preferred style
is below.
B 000 O01 O11 010 100 101 111 110
5- variable Karnaugh map (overlay)
The overlay version of the Karnaugh map, shown above, is
simply two (four for a 6-variable map) identical maps except
for the most significant bit of the 3-bit address across the
top. If we look at the top of the map, we will see that the
numbering is different from the previous Gray code map. If
we ignore the most significant digit of the 3-digit numbers,
the sequence 00, O1, 11, 10 is at the heading of both sub
maps of the overlay map. The sequence of eight 3-digit
numbers is not Gray code. Though the sequence of four of
the least significant two bits is.
Let's put our 5-variable Karnaugh Map to use. Design a
circuit which has a 5-bit binary input (A, B, C, D, E), with A
being the MSB (Most Significant Bit). It must produce an
output logic High for any prime number detected in the
input data.
eas
Br <
ius gage
EVN Pe
eT ) |
eae ABCE
5- variable Karnaugh map (Gray code)
We show the solution above on the older Gray code
(reflection) map for reference. The prime numbers are
(1,2,3,5,7,11,13,17,19,23,29,31). Plota Lin each
corresponding cell. Then, proceed with grouping of the cells.
Finish by writing the simplified result. Note that 4-cell group
A'B'E consists of two pairs of cell on both sides of the mirror
line. The same is true of the 2-cell group AB'DE. It is a group
of 2-cells by being reflected about the mirror line. When
using this version of the K-map look for mirror images in the
other half of the map.
Out = A'B'E + B'C'E + A'C'DE + A'CD'E + ABCE + AB'DE +
A'B'C'D
Below we show the more common version of the 5-variable
map, the overlay map.
5- variable Karnaugh map (overlay)
If we compare the patterns in the two maps, some of the
cells in the right half of the map are moved around since the
addressing across the top of the map is different. We also
need to take a different approach at spotting commonality
between the two halves of the map. Overlay one half of the
map atop the other half. Any overlap from the top map to
the lower map is a potential group. The figure below shows
that group AB'DE is composed of two stacked cells. Group
A'B'E consists of two stacked pairs of cells.
For the A'B'E group of 4-cells ABCDE = OOxx1 for the
group. That is A,B,E are the same 001 respectively for the
group. And, CD=xx that is it varies, no commonality in
CD= xx for the group of 4-cells. Since ABCDE = OOxxl1, the
group of 4-cells is covered by A'B'XXE = A'B'E.
The above 5-variable overlay map is shown stacked.
An example of a six variable Karnaugh map follows. We have
mentally stacked the four sub maps to see the group of 4-
cells corresponding to Out = C’'F'
A magnitude comparator (used to illustrate a 6-variable kK
map) compares two binary numbers, indicating if they are
equal, greater than, or less than each other on three
respective outputs. A three bit magnitude comparator has
two inputs A5A,Ap and B>B,Bo An integrated circuit
magnitude comparator (7485) would actually have four
inputs, But, the Karnaugh map below needs to be kept to a
reasonable size. We will only solve for the A>B output.
Below, a 6-variable Karnaugh map aids simplification of the
logic for a 3-bit magnitude comparator. This is an overlay
type of map. The binary address code across the top and
down the left side of the map is not a full 3-bit Gray code.
Though the 2-bit address codes of the four sub maps is Gray
code. Find redundant expressions by stacking the four sub
maps atop one another (Shown above). There could be cells
common to all four maps, though not in the example below.
It does have cells common to pairs of sub maps.
”A Magnitude 4<5
Comparator a-_—p
B
A>B
The A>B output above is ABC>XYZ on the map below.
Z
000 O01 O11 010 100 101 111 110
Out = AX+ABY+BXY+ABCZ+ACYZ+BCXZ+CXYZ
6- variable Karnaugh map (overlay)
Where ever ABC is greater than XYZ, a 1 is plotted. In the
first line ABC=000 cannot be greater than any of the values
of XYZ. No Is in this line. In the second line, ABC=001, only
the first cell ABCXYZ= 001000 is ABC greater than XYZ. A
single 1 is entered in the first cell of the second line. The
fourth line, ABC=010, has a pair of 1s. The third line,
ABC= 011 has three 1s. Thus, the map is filled with 1s in
any cells where ABC is greater than XXZ.
In grouping cells, form groups with adjacent sub maps if
possible. All but one group of 16-cells involves cells from
pairs of the sub maps. Look for the following groups:
e 1 group of 16-cells
e 2 groups of 8-cells
e 4 groups of 4-cells
The group of 16-cells, AX’ occupies all of the lower right sub
map; though, we don't circle it on the figure above.
One group of 8-cells is composed of a group of 4-cells in the
upper sub map overlaying a similar group in the lower left
map. The second group of 8-cells is composed of a similar
group of 4-cells in the right sub map overlaying the same
group of 4-cells in the lower left map.
The four groups of 4-cells are shown on the Karnaugh map
above with the associated product terms. Along with the
product terms for the two groups of 8-cells and the group of
16-cells, the final Sum-Of-Products reduction is shown, all
seven terms. Counting the 1s in the map, there is a total of
16+6+6=28 ones. Before the K-map logic reduction there
would have been 28 product terms in our SOP output, each
with 6-inputs. The Karnaugh map yielded seven product
terms of four or less inputs. This is really what Karnaugh
maps are all about!
The wiring diagram is not shown. However, here is the parts
list for the 3-bit magnitude comparator for ABC>XYZ using 4
TTL logic family parts:
e 1 ea 7410 triple 3-input NAND gate AX’, ABY', BX'yY'
e 2 ea 7420 dual 4-input NAND gate ABCZ’', ACY'Z',
BCX'Z', CX'Y'Z'
e 1 ea 7430 8-input NAND gate for output of 7 -P-terms
¢ REVIEW:
Boolean algebra, Karnaugh maps, and CAD (Computer
Aided Design) are methods of logic simplification. The
goal of logic simplification is a minimal cost solution.
A minimal cost solution is a valid logic reduction with
the minimum number of gates with the minimum
number of inputs.
Venn diagrams allow us to visualize Boolean
expressions, easing the transition to Karnaugh maps.
e Karnaugh map cells are organized in Gray code order so
that we may visualize redundancy in Boolean
expressions which results in simplification.
e The more common Sum-Of-Products (Sum of Minters)
expressions are implemented as AND gates (products)
feeding a single OR gate (sum).
e Sum-Of-Products expressions (AND-OR logic) are
equivalent to a NAND-NAND implementation. All AND
gates and OR gates are replaced by NAND gates.
e Less often used, Product-Of-Sums expressions are
implemented as OR gates (Sums) feeding into a single
AND gate (product). Product-Of-Sums expressions are
based on the Os, maxterms, in a Karnaugh map.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—||+4]l—
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 9
COMBINATIONAL LOGIC
FUNCTIONS
Introduction
A Half-Adder
A Full-Adder
Decoder
Encoder
Demultiplexers
Multiplexers
Using_multiple combinational circuits
Original author: David Zitzelsberger
Introduction
The term "combinational" comes to us from mathematics. In
mathematics a combination is an unordered set, which is a
formal way to say that nobody cares which order the items
came in. Most games work this way, if you rolled dice one at
a time and get a 2 followed by a 3 it is the same as if you
had rolled a 3 followed by a 2. With combinational logic, the
circuit produces the same output regardless of the order the
inputs are changed.
There are circuits which depend on the when the inputs
change, these circuits are called sequential logic. Even
though you will not find the term "sequential logic" in the
chapter titles, the next several chapters will discuss
sequential logic.
Practical circuits will have a mix of combinational and
sequential logic, with sequential logic making sure
everything happens in order and combinational logic
performing functions like arithmetic, logic, or conversion.
You have already used combinational circuits. Each logic
gate discussed previously is a combinational logic function.
Let's follow how two NAND gate works if we provide them
inputs in different orders.
We begin with both inputs being O.
00
We then set one input high.
10
We then set the other input high.
11
So NAND gates do not care about the order of the inputs,
and you will find the same true of all the other gates covered
up to this point (AND, XOR, OR, NOR, XNOR, and NOT).
A Half-Adder
As a first example of useful combinational logic, let's build a
device that can add two binary digits together. We can
quickly calculate what the answers should be:
0+0=0 Q0+t1e=1 1+0O0e= 1 1 +
1 = 105
So we well need two inputs (a and b) and two outputs. The
low order output will be called 2 because it represents the
sum, and the high order output will be called C,,,, because it
represents the carry out.
The truth table is
Simplifying boolean equations or making some Karnaugh
map will produce the same circuit shown below, but start by
looking at the results. The 2 column is our familiar XOR gate,
while the Co; column is the AND gate. This device is called a
half-adder for reasons that will make sense in the next
section.
stb
py
B
or in ladder logic
A Full-Adder
The half-adder is extremely useful until you want to add
more that one binary digit quantities. The slow way to
develop a two binary digit adders would be to make a truth
table and reduce it. Then when you decide to make a three
binary digit adder, do it again. Then when you decide to
make a four digit adder, do it again. Then when ... The
circuits would be fast, but development time would be slow.
Looking at a two binary digit sum shows what we need to
extend addition to multiple binary digits.
Ii
11
11
110
Look at how many inputs the middle column uses. Our adder
needs three inputs; a, b, and the carry from the previous
sum, and we can use our two-input adder to build a three
input adder.
2 is the easy part. Normal arithmetic tells us that if 2 =a+b
+ C,, and 2, =a+b,then z= 2, + Cp.
What do we do with C, and C,? Let's look at three input
sums and quickly calculate:
Ca_ ee oD St
0+0+0=0 06+0+12= 1 0+1+02= 1
0 +1+41= 10
1+0+02=1 1+0+1= 10 1+1+0= 10
1+1+t1e= 4141
If you have any concern about the low order bit, please
confirm that the circuit and ladder calculate it correctly.
In order to calculate the high order bit, notice that it is 1 in
both cases when a + b produces a Cj. Also, the high order
bit is 1 when a + b produces a 2, and C,, is a1. So We will
have a carry when C, OR (2, AND C,,). Our complete three
input adder is:
For some designs, being able to eliminate one or more types
of gates can be important, and you can replace the final OR
gate with an XOR gate without changing the results.
We can now connect two adders to add 2 bit quantities.
Cour
L, L,
Ao is the low order bit of A, Aj is the high order bit of A, Bg is
the low order bit of B, B, is the high order bit of B, Zpis the
low order bit of the sum, 2, is the high order bit of the sum,
and Coy is the Carry.
A two binary digit adder would never be made this way.
Instead the lowest order bits would also go through a full
adder.
L,
There are several reasons for this, one being that we can
then allow a circuit to determine whether the lowest order
carry should be included in the sum. This allows for the
chaining of even larger sums. Consider two different ways to
look at a four bit sum.
111 l<-+ l11<+-
0110 | O01 | 10
1011 | 10 | 141
eters am ul saeaeer 1 Geet
10001 1 +-100 +-101
If we allow the program to add a two bit number and
remember the carry for later, then use that carry in the next
sum the program can add any number of bits the user wants
even though we have only provided a two-bit adder. Small
PLCs can also be chained together for larger numbers.
These full adders can also can be expanded to any number
of bits space allows. As an example, here's how to do an 8
bit adder.
This is the same result as using the two 2-bit adders to make
a 4-bit adder and then using two 4-bit adders to make an 8-
bit adder or re-duplicating ladder logic and updating the
numbers.
3
bt
FA
Fa
Ao Bo A,B, A, Bo A,B, Ay By As Bs Ag Be A; By
Each "2+" is a 2-bit adder and made of two full adders. Each
"4+" is a 4-bit adder and made of two 2-bit adders. And the
result of two 4-bit adders is the same 8-bit adder we used
full adders to build.
For any large combinational circuit there are generally two
approaches to design: you can take simpler circuits and
replicate them; or you can design the complex circuit as a
complete device.
Using simpler circuits to build complex circuits allows a you
to spend less time designing but then requires more time for
signals to propagate through the transistors. The 8-bit adder
design above has to wait for all the C, 5, signals to move
from Ag + Bo up to the inputs of 27.
If a designer builds an 8-bit adder as a complete device
simplified to a sum of products, then each signal just travels
through one NOT gate, one AND gate and one OR gate. A
seventeen input device has a truth table with 131,072
entries, and reducing 131,072 entries to a sum of products
will take some time.
When designing for systems that have a maximum allowed
response time to provide the final result, you can begin by
using simpler circuits and then attempt to replace portions
of the circuit that are too slow. That way you spend most of
your time on the portions of a circuit that matter.
Decoder
A decoder is a circuit that changes a code into a set of
signals. It is called a decoder because it does the reverse of
encoding, but we will begin our study of encoders and
decoders with decoders because they are simpler to design.
A common type of decoder is the line decoder which takes
an n-digit binary number and decodes it into 2" data lines.
The simplest is the 1-to-2 line decoder. The truth table is
A is the address and D is the dataline. Dp is NOT A and Dj is
A. The circuit looks like
A Do
D,
Only slightly more complex is the 2-to-4 line decoder. The
truth table is
Developed into a circuit it looks like
>
5
>
>
6
O
=)
4
‘
‘
4
-
>
fs)
O
4
‘
‘
4
o
>
6
O
N
4
‘
‘
4
>
>
ra
O
| |
Larger line decoders can be designed in a similar fashion,
but just like with the binary adder there is a way to make
larger decoders by combining smaller decoders. An alternate
circuit for the 2-to-4 line decoder is
Replacing the 1-to-2 Decoders with their circuits will show
that both circuits are equivalent. In a similar fashion a 3-to-8
line decoder can be made from a 1-to-2 line decoder and a
2-to-4 line decoder, and a 4-to-16 line decoder can be made
from two 2-to-4 line decoders.
You might also consider making a 2-to-4 decoder ladder from
1-to-2 decoder ladders. If you do it might look something
like this:
For some logic it may be required to build up logic like this.
For an eight-bit adder we only know how to sum eight bits
by summing one bit at a time. Usually it is easier to design
ladder logic from boolean equations or truth tables rather
than design logic gates and then "translate" that into ladder
logic.
A typical application of a line decoder circuit is to select
among multiple devices. A circuit needing to select among
sixteen devices could have sixteen control lines to select
which device should "listen". With a decoder only four
control lines are needed.
Encoder
An encoder is a circuit that changes a set of signals into a
code. Let's begin making a 2-to-1 line encoder truth table by
reversing the 1-to-2 decoder truth table.
One question we need to answer is what to do with those
other inputs? Do we ignore them? Do we have them
generate an additional error output? In many circuits this
problem is solved by adding sequential logic in order to
know not just what input is active but also which order the
inputs became active.
A more useful application of combinational encoder design
is a binary to 7-segment encoder. The seven segments are
given according
Our truth table is:
Is [lp [hy [lo [De] Ds] D.|D5|D2|D;| Do:
fo fo fo fo ft [a fa fo fa fa fa
0 fo fo [1 [of o fs Jo fo fa Jo
fo fo ft fo ft {ofa fa fa fo fr
fo fo tt ft ft [ofa fa fo fa fr
oft fo fo fof a fa fa fo fy Jo
fo 4 fo fa fa [a fo [a fo ft [a |
oft | Jo fa | fo fa fa fa fa
ofa fa fa [a [0 |i [o fo [t fo |
1 jo fo jo fa fia fa [a fa ft [a |
jt fo fo fa fa fia fa fa fo ft fa |
Deciding what to do with the remaining six entries of the
truth table is easier with this circuit. This circuit should not
be expected to encode an undefined combination of inputs,
SO we can leave them as "don't care" when we design the
circuit. The equations were simplified with karnaugh maps.
D,=1I,+ I, + I,Ip + IpIo
The collection of equations is Summarised here:
Do= I; a I, Ts +I,Iy + T, 1p sf To ly Ip
Di= I3 + I, +1, + Ip
D» =I,I, + Tag
D3= 13; +1,1, + 1,1) +
K4
214
Dy= Ty 15+ Tyg + 14 kp
De= I, + In 1, + 1, 1p + Iolo
De= I, + I, + IsIp t+ Iolo
The circuit is:
see,
scalp, Do= L4+L1,4+Lh
L, +1, 1p thlky
And the corresponding ladder diagram:
L, L,
Do=134 1,4 blot
Ds=1 +b) +1 lp+bly
D,=1,+1,+1,)+1
D=1,+ Lh +h p+,
D,=1,+1,4], +p
Do= 4h], 4+bLhthl +bh
Demultiplexers
A demultiplexer, sometimes abbreviated dmux, is a circuit
that has one input and more than one output. It is used
when a circuit wishes to send a signal to one of many
devices. This description sounds similar to the description
given for a decoder, but a decoder is used to select among
many devices while a demultiplexer is used to send a signal
among many devices.
A demultiplexer is used often enough that it has its own
schematic symbol
| Do
D,
A
The truth table for a 1-to-2 demultiplexer is
Using our 1-to-2 decoder as part of the circuit, we can
express this circuit easily
1-to-2 line
decoder
This circuit can be expanded two different ways. You can
increase the number of signals that get transmitted, or you
can increase the number of inputs that get passed through.
To increase the number of inputs that get passed through
just requires a larger line decoder. Increasing the number of
signals that get transmitted is even easier.
As an example, a device that passes one set of two signals
among four signals is a "two-bit 1-to-2 demultiplexer". Its
circuit is
lo | Do
shows that it could be two one-bit 1-to-2 demultiplexers
without changing its expected behavior.
A 1-to-4 demultiplexer can easily be built from 1-to-2
demultiplexers as follows.
Multiplexers
A multiplexer, abbreviated mux, is a device that has
multiple inputs and one output.
The schematic symbol for multiplexers is
lo
, v
A
The truth table for a 2-to-1 multiplexer is
Using a 1-to-2 decoder as part of the circuit, we can express
this circuit easily.
1-to-2 line
decoder
Multiplexers can also be expanded with the same naming
conventions as demultiplexers. A 4-to-1 multiplexer circuit is
That is the formal definition of a multiplexer. Informally,
there is a lot of confusion. Both demultiplexers and
multiplexers have similar names, abbreviations, schematic
symbols and circuits, so confusion is easy. The term
multiplexer, and the abbreviation mux, are often used to
also mean a demultiplexer, or a multiplexer and a
demultiplexer working together. So when you hear about a
multiplexer, it may mean something quite different.
Using multiple combinational circuits
As an example of using several circuits together, we are
going to make a device that will have 16 inputs,
representing a four digit number, to a four digit 7-segment
display but using just one binary-to-7-segment encoder.
First, the overall architecture of our circuit provides what
looks like our the description provided.
7-segment
encoder
Follow this circuit through and you can confirm that it
matches the description given above. There are 16 primary
inputs. There are two more inputs used to select which digit
will be displayed. There are 28 outputs to control the four
digit 7-segment display. Only four of the primary inputs are
encoded at a time. You may have noticed a potential
question though.
When one of the digits are selected, what do the other three
digits display? Review the circuit for the demultiplexers and
notice that any line not selected by the A input is zero. So
the other three digits are blank. We don't have a problem,
only one digit displays at a time.
Let's get a perspective on just how complex this circuit is by
looking at the equivalent ladder logic.
Ly
A, Aq Dao
A, Ag Dio
A, Ay Dag
A, Ay Dag
A, Ay Da,
A, Ag Diy
A, Ay Da,
hh Ay Da,
A, Ay Doo
A, Ag Diz
A; Ag Daz
A, Ay Daa
A, Ay Dos
A, Ay Dy
A; Aq Dag
A, Ay Daa
ly
@.5 @ 6 O28 ar
=
e ;
=
=
O13 gO 20 218.28 20,18 Oe
D,
&
Ly
A, Ag Dy
A, Ag D,
A, Ay D,
A, Ag D,
A, Ay Dz
A, Ag De
A, Ag Dz
A, Ay Ds
A, Ay Ds
A, Aq Ds
A, Ag Ds
A, Ag D,
A, Ag D,
A Ag D,
A, Ag Dg
A, Ay Ds
A, Ay Ds
A, Ay Ds
A, Ag Ds
A, Ag Dg
A, Ay Dg
A, Ay Dg
A, Ay Dg
oO [?)
O » = on
Lie SAP Se ha 0 oe
bw
etpetwoelge
-
OP Oe
Notice how quickly this large circuit was developed from
smaller parts. This is true of most complex circuits: they are
composed of smaller parts allowing a designer to abstract
away some complexity and understand the circuit asa
whole. Sometimes a designer can even take components
that others have designed and remove the detail design
work.
In addition to the added quantity of gates, this design
suffers from one additional weakness. You can only see one
display one digit at a time. If there was some way to rotate
through the four digits quickly, you could have the
appearance of all four digits being displayed at the same
time. That is a job for a sequential circuit, which is the
subject of the next several chapters.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—| | 4]
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 10
MULTIVIBRATORS
Digital logic with feedback
The S-R latch
The gated S-R latch
The D latch
Edge-triggered latches: Flip-Flops
The J-K flip-flop
Monostable multivibrators
Digital logic with feedback
With simple gate and combinational logic circuits, there is a
definite output state for any given input state. Take the truth
table of an OR gate, for instance:
5
L.
A
tae Output
B
For each of the four possible combinations of input states (0-
0, 0-1, 1-0, and 1-1), there is one, definite, unambiguous
output state. Whether we're dealing with a multitude of
cascaded gates or a single gate, that output state is
determined by the truth table(s) for the gate(s) in the
circuit, and nothing else.
L
However, if we alter this gate circuit so as to give signal
feedback from the output to one of the inputs, strange
things begin to happen:
A
A CR1
Output
CR1
We know that if Ais 1, the output must be 1, as well. Such is
the nature of an OR gate: any "high" (1) input forces the
output "high" (1). lf Ais "low" (0), however, we cannot
guarantee the logic level or state of the output in our truth
table. Since the output feeds back to one of the OR gate's
inputs, and we know that any 1 input to an OR gates makes
the output 1, this circuit will "latch" in the 1 output state
after any time that A is 1. When A is O, the output could be
either 0 or 1, depending on the circuit's prior state! The
proper way to complete the above truth table would be to
insert the word /atch in place of the question mark, showing
that the output maintains its last state when A is 0.
Any digital circuit employing feedback is called a
multivibrator. The example we just explored with the OR
gate was a very simple example of what is called a bistable
multivibrator. It is called "bistable" because it can hold
stable in one of two possible output states, either O or 1.
There are also monostab/e multivibrators, which have only
one stable output state (that other state being momentary),
which we'll explore later; and astab/e multivibrators, which
have no stable state (oscillating back and forth between an
output of O and 1).
A very simple astable multivibrator is an inverter with the
output fed directly back to the input:
Inverter with feedback
eat
CR1 CR1
Output
When the input is 0, the output switches to 1. That 1 output
gets fed back to the input as a 1. When the input is 1, the
output switches to 0. That O output gets fed back to the
input as a O, and the cycle repeats itself. The result is a high
frequency (several megahertz) oscillator, if implemented
with a solid-state (semiconductor) inverter gate:
If implemented with relay logic, the resulting oscillator will
be considerably slower, cycling at a frequency well within
the audio range. The buzzer or vibrator circuit thus formed
was used extensively in early radio circuitry, as a way to
convert steady, low-voltage DC power into pulsating DC
power which could then be stepped up in voltage through a
transformer to produce the high voltage necessary for
operating the vacuum tube amplifiers. Henry Ford's
engineers also employed the buzzer/transformer circuit to
create continuous high voltage for operating the spark plugs
on Model T automobile engines:
"Model T” high-voltage
ignition coil
= ian
\
<
>
Borrowing terminology from the old mechanical buzzer
(vibrator) circuits, solid-state circuit engineers referred to
any circuit with two or more vibrators linked together as a
multivibrator. The astable multivibrator mentioned
previously, with only one "vibrator," is more commonly
implemented with multiple gates, as we'll see later.
The most interesting and widely used multivibrators are of
the bistable variety, so we'll explore them in detail now.
The S-R latch
A bistable multivibrator has two stable states, as indicated
by the prefix b/in its name. Typically, one state is referred to
as set and the other as reset. The simplest bistable device,
therefore, is known as a set-reset, or S-R, latch.
To create an S-R latch, we can wire two NOR gates in such a
way that the output of one feeds back to the input of
another, and vice versa, like this:
R
S
The Q and not-Q outputs are supposed to be in opposite
states. | say "Supposed to" because making both the S and R
inputs equal to 1 results in both Q and not-Q being O. For
this reason, having both S and R equal to 1 is called an
invalid or illegal state for the S-R multivibrator. Otherwise,
making S=1 and R=O "sets" the multivibrator so that Q=1
and not-Q=0. Conversely, making R=1 and S=O "resets" the
multivibrator in the opposite state. When S and R are both
equal to 0, the multivibrator's outputs "latch" in their prior
states. Note how the same multivibrator function can be
implemented in ladder logic, with the same results:
Re [o]
By definition, a condition of Q=1 and not-Q=0 is set. A
condition of Q=0 and not-Q=1 is reset. These terms are
universal in describing the output states of any
multivibrator circuit.
The astute observer will note that the initial power-up
condition of either the gate or ladder variety of S-R latch is
such that both gates (coils) start in the de-energized mode.
As such, one would expect that the circuit will start up in an
invalid condition, with both Q and not-Q outputs being in
the same state. Actually, this is true! However, the invalid
condition is unstable with both S and R inputs inactive, and
the circuit will quickly stabilize in either the set or reset
condition because one gate (or relay) is bound to react a
little faster than the other. If both gates (or coils) were
precisely identical, they would oscillate between high and
low like an astable multivibrator upon power-up without ever
reaching a point of stability! Fortunately for cases like this,
such a precise match of components is a rare possibility.
It must be noted that although an astable (continually
oscillating) condition would be extremely rare, there will
most likely be a cycle or two of oscillation in the above
circuit, and the final state of the circuit (set or reset) after
power-up would be unpredictable. The root of the problem is
a race condition between the two relays CR; and CR>.
A race condition occurs when two mutually-exclusive events
are simultaneously initiated through different circuit
elements by a single cause. In this case, the circuit elements
are relays CR, and CR3, and their de-energized states are
mutually exclusive due to the normally-closed interlocking
contacts. If one relay coil is de-energized, its normally-closed
contact will keep the other coil energized, thus maintaining
the circuit in one of two states (set or reset). Interlocking
prevents both relays from latching. However, if both relay
coils start in their de-energized states (such as after the
whole circuit has been de-energized and is then powered
up) both relays will "race" to become latched on as they
receive power (the "single cause") through the normally-
closed contact of the other relay. One of those relays will
inevitably reach that condition before the other, thus
opening its normally-closed interlocking contact and de-
energizing the other relay coil. Which relay "wins" this race
is dependent on the physical characteristics of the relays
and not the circuit design, so the designer cannot ensure
which state the circuit will fall into after power-up.
Race conditions should be avoided in circuit design
primarily for the unpredictability that will be created. One
way to avoid such a condition is to insert a time-delay relay
into the circuit to disable one of the competing relays for a
short time, giving the other one a clear advantage. In other
words, by purposely slowing down the de-energization of
one relay, we ensure that the other relay will always "win"
and the race results will always be predictable. Here is an
example of how a time-delay relay might be applied to the
above circuit to avoid the race condition:
L, L,
lL second
When the circuit powers up, time-delay relay contact TD, in
the fifth rung down will delay closing for 1 second. Having
that contact open for 1 second prevents relay CR» from
energizing through contact CR, in its normally-closed state
after power-up. Therefore, relay CR, will be allowed to
energize first (with a 1-second head start), thus opening the
normally-closed CR, contact in the fifth rung, preventing
CR> from being energized without the S input going active.
The end result is that the circuit powers up cleanly and
predictably in the reset state with S=O and R=0O.
It should be mentioned that race conditions are not
restricted to relay circuits. Solid-state logic gate circuits may
also suffer from the ill effects of race conditions if improperly
designed. Complex computer programs, for that matter, may
also incur race problems if improperly designed. Race
problems are a possibility for any sequential system, and
may not be discovered until some time after initial testing of
the system. They can be very difficult problems to detect
and eliminate.
A practical application of an S-R latch circuit might be for
starting and stopping a motor, using normally-open,
momentary pushbutton switch contacts for both start (S)
and stop (R) switches, then energizing a motor contactor
with either a CR; or CR> contact (or using a contactor in
place of CR, or CR>). Normally, a much simpler ladder logic
circuit is employed, such as this:
L, L,
Motor "on"
In the above motor start/stop circuit, the CR, contact in
parallel with the start switch contact is referred to as a "Seal-
in" contact, because it "seals" or latches control relay CR, in
the energized state after the start switch has been released.
To break the "seal," or to "unlatch" or "reset" the circuit, the
stop pushbutton is pressed, which de-energizes CR, and
restores the seal-in contact to its normally open status.
Notice, however, that this circuit performs much the same
function as the S-R latch. Also note that this circuit has no
inherent instability problem (if even a remote possibility) as
does the double-relay S-R latch design.
In semiconductor form, S-R latches come in prepackaged
units so that you don't have to build them from individual
gates. They are symbolized as such:
5 Q
R Q
e REVIEW:
e A bistable multivibrator is one with two stable output
states.
In a bistable multivibrator, the condition of Q=1 and
not-Q=0 is defined as set. A condition of Q=0O and not-
Q=1 is conversely defined as reset. If Q and not-Q
happen to be forced to the same state (both O or both
1), that state is referred to as invalid.
e In an S-R latch, activation of the S input sets the circuit,
while activation of the R input resets the circuit. If both
S and R inputs are activated simultaneously, the circuit
will be in an invalid condition.
e A race condition is a state in a sequential system where
two mutually-exclusive events are simultaneously
initiated by a single cause.
The gated S-R latch
It is sometimes useful in logic circuits to have a multivibrator
which changes state only when certain conditions are met,
regardless of its S and R input states. The conditional input
is called the enable, and is symbolized by the letter E. Study
the following example to see how this works:
Esky o | 0 |
; fo [oo | taich | Tatch_|
a [ofofi[ratch | Tatcn |
ri [o| tatoh | latch
foo] taich | lath
opto [7
aC a
opfhto [0 _]
When the E=0, the outputs of the two AND gates are forced
to O, regardless of the states of either S or R. Consequently,
the circuit behaves as though S and R were both 0, latching
the Q and not-Q outputs in their last states. Only when the
enable input is activated (1) will the latch respond to the S
and R inputs. Note the identical function in ladder logic:
0 |
0 |
E[s/R} Q | Q |
foo atch | Tatch
foi | atch | atch
latch latch
E |
cx
0 |
oft jo
ofi fit latch | tatch_|
Se eee
jo] + | 0 |
ES a
foo Tatch [latch
fof
a
A practical application of this might be the same motor
control circuit (with two normally-open pushbutton switches
for start and stop), except with the addition of a master
lockout input (E) that disables both pushbuttons from
having control over the motor when its low (0).
Once again, these multivibrator circuits are available as
prepackaged semiconductor devices, and are symbolized as
such:
S Q
E
R Q
It is also common to see the enable input designated by the
letters "EN" instead of just "E."
e REVIEW:
e The enable input on a multivibrator must be activated
for either S or R inputs to have any effect on the output
state.
e This enable input is sometimes labeled "E", and other
times as "EN",
The D latch
Since the enable input on a gated S-R latch provides a way
to latch the Q and not-Q outputs without regard to the
status of S or R, we can eliminate one of those inputs to
create a multivibrator latch circuit with no "illegal" input
states. Such a circuit is called a D latch, and its internal logic
looks like this:
D|
D
Note that the R input has been replaced with the
complement (inversion) of the old S input, and the S input
has been renamed to D. As with the gated S-R latch, the D
latch will not respond to a signal input if the enable input is
0 -- it simply stays latched in its last state. When the enable
input is 1, however, the Q output follows the D input.
Since the R input of the S-R circuitry has been done away
with, this latch has no "invalid" or "illegal" state. Q and not-
Q are always opposite of one another. If the above diagram
is confusing at all, the next diagram should make the
concept simpler:
Like both the S-R and gated S-R latches, the D latch circuit
may be found as its own prepackaged circuit, complete with
a standard symbol:
D Q
E
Q
The D latch is nothing more than a gated S-R latch with an
inverter added to make R the complement (inverse) of S.
Let's explore the ladder logic equivalent of a D latch,
modified from the basic ladder diagram of an S-R latch:
ED. e@ [| o_
fo [ Tatch | atch
Fa Je
haste 4
|
0 |
Ey
An application for the D latch is a 1-bit memory circuit. You
can "write" (store) a O or 1 bit in this latch circuit by making
the enable input high (1) and setting D to whatever you
want the stored bit to be. When the enable input is made
low (0), the latch ignores the status of the D input and
merrily holds the stored bit value, outputting at the stored
value at Q, and its inverse on output not-Q.
e REVIEW:
e A D latch is like an S-R latch with only one input: the "D"
input. Activating the D input sets the circuit, and de-
activating the D input resets the circuit. Of course, this
is only if the enable input (E) is activated as well.
Otherwise, the output(s) will be latched, unresponsive to
the state of the D input.
e D latches can be used as 1-bit memory circuits, storing
either a "high" or a "low" state when disabled, and
"reading" new data from the D input when enabled.
Edge-triggered latches: Flip-Flops
So far, we've studied both S-R and D latch circuits with
enable inputs. The latch responds to the data inputs (S-R or
D) only when the enable input is activated. In many digital
applications, however, it is desirable to limit the
responsiveness of a latch circuit to a very short period of
time instead of the entire duration that the enabling input is
activated. One method of enabling a multivibrator circuit is
called edge triggering, where the circuit's data inputs have
control only during the time that the enable input is
transitioning from one state to another. Let's compare timing
diagrams for a normal D latch versus one that is edge-
triggered:
Regular D-latch response
Outputs respond to input (D)
during these time periods
Positive edge-triggered D-latch response
D_J WIJ LSJ LS LE
Be - sar ee
a ,
a cr
Outputs respond to input (D)
only when enable signal transitions
from low to high
In the first timing diagram, the outputs respond to input D
whenever the enable (E) input is high, for however long it
remains high. When the enable signal falls back to a low
state, the circuit remains latched. In the second timing
diagram, we note a distinctly different response in the circuit
output(s): it only responds to the D input during that brief
moment of time when the enable signal changes, or
transitions, from low to high. This is known as positive edge-
triggering.
There is such a thing as negative edge triggering as well,
and it produces the following response to the same input
signals:
Negative edge-triggered D-latch response
D ao ee
rs ee Cae | emma) EE mea KR
es er sans oe
a i
Outputs respond to input (D)
only when enable signal transitions
from high to low
Whenever we enable a multivibrator circuit on the
transitional edge of a square-wave enable signal, we call it a
flip-flop instead of a /Jatch. Consequently, and edge-triggered
S-R circuit is more properly Known as an S-R flip-flop, and an
edge-triggered D circuit as a D flip-flop. The enable signal is
renamed to be the clock signal. Also, we refer to the data
inputs (S, R, and D, respectively) of these flip-flops as
synchronous inputs, because they have effect only at the
time of the clock pulse edge (transition), thereby
synchronizing any output changes with that clock pulse,
rather than at the whim of the data inputs.
But, how do we actually accomplish this edge-triggering? To
create a "gated" S-R latch from a regular S-R latch is easy
enough with a couple of AND gates, but how do we
implement logic that only pays attention to the rising or
falling edge of a changing digital signal? What we need isa
digital circuit that outputs a brief pulse whenever the input
is activated for an arbitrary period of time, and we can use
the output of this circuit to briefly enable the latch. We're
getting a little ahead of ourselves here, but this is actually a
kind of monostable multivibrator, which for now we'll call a
pulse detector.
Input | Pulse detector
circuit \
| aa
Output
Input _ J LJ LJ LJ LJ Le
Output _J| =f ™$5|J J J
The duration of each output pulse is set by components in
the pulse circuit itself. In ladder logic, this can be
accomplished quite easily through the use of a time-delay
relay with a very short delay time:
L, L,
Input
TD1
Implementing this timing function with semiconductor
components is actually quite easy, as it exploits the inherent
time delay within every logic gate (Known as propagation
delay). What we do is take an input signal and split it up two
ways, then place a gate or a series of gates in one of those
signal paths just to delay it a bit, then have both the original
signal and its delayed counterpart enter into a two-input
gate that outputs a high signal for the brief moment of time
that the delayed signal has not yet caught up to the low-to-
high change in the non-delayed signal. An example circuit
for producing a clock pulse on a low-to-high input signal
transition is shown here:
Input
Delayed input
Input Lo
Delayedinput™ —__|_ sf. .©63@§hT['—
Output f] ————
—+» ~— Brief period of time when
both inputs of the AND gate
are high
This circuit may be converted into a negative-edge pulse
detector circuit with only a change of the final gate from
AND to NOR:
Input
Delayed input
Input EES
Delayed input™ —__|_==sf ...©@8©©
Output as
ae
Brief period of time when
both inputs of the NOR gate
are low
Now that we know how a pulse detector can be made, we
can show it attached to the enable input of a latch to turn it
into a flip-flop. In this case, the circuit is a S-R flip-flop:
[ci{s|R]_ Q | Q |
Ffo}o] latch | latch |
jofi} o |
ah
fifo
a
jo [1 |
[x{i fo] latch | latch _|
Only when the clock signal (C) is transitioning from low to
high is the circuit responsive to the S and R inputs. For any
other condition of the clock signal ("x") the circuit will be
latched.
A ladder logic version of the S-R flip-flop is shown here:
is is
cad
ro [7
i a
oo]
—
C]S|R|
S| 0} 0 |
Fjo} i
{| 0 |
x 10 [0 |
xo}
x {i [Oo
x fifi
Relay contact CR3 in the ladder diagram takes the place of
the old E contact in the S-R latch circuit, and is closed only
during the short time that both C is closed and time-delay
contact TR, is closed. In either case (gate or ladder circuit),
we see that the inputs S and R have no effect unless C is
transitioning from a low (0) to a high (1) state. Otherwise,
the flip-flop's outputs latch in their previous states.
It is important to note that the invalid state for the S-R flip-
flop is maintained only for the short period of time that the
pulse detector circuit allows the latch to be enabled. After
that brief time period has elapsed, the outputs will latch into
either the set or the reset state. Once again, the problem of
a race condition manifests itself. With no enable signal, an
invalid output state cannot be maintained. However, the
valid "latched" states of the multivibrator -- set and reset --
are mutually exclusive to one another. Therefore, the two
gates of the multivibrator circuit will "race" each other for
supremacy, and whichever one attains a high output state
first will "win."
The block symbols for flip-flops are slightly different from
that of their respective latch counterparts:
S Q D Q
C C
R Q Q
The triangle symbol next to the clock inputs tells us that
these are edge-triggered devices, and consequently that
these are flip-flops rather than latches. The symbols above
are positive edge-triggered: that is, they "clock" on the
rising edge (low-to-high transition) of the clock signal.
Negative edge-triggered devices are symbolized with a
bubble on the clock input line:
Both of the above flip-flops will "clock" on the falling edge
(high-to-low transition) of the clock signal.
¢ REVIEW:
e A flip-flop is a latch circuit with a "pulse detector" circuit
connected to the enable (E) input, so that it is enabled
only for a brief moment on either the rising or falling
edge of a clock pulse.
e Pulse detector circuits may be made from time-delay
relays for ladder logic applications, or from
semiconductor gates (exploiting the phenomenon of
propagation delay).
The J-K flip-flop
Another variation on a theme of bistable multivibrators is
the J-K flip-flop. Essentially, this is a modified version of an
S-R flip-flop with no "invalid" or "illegal" output state. Look
closely at the following diagram to see how this is
accomplished:
[r[o]o| Taich | Tatch
latch
pfo [i] latch latch
[xi fo] latch | latch
What used to be the S and R inputs are now called the J and
K inputs, respectively. The old two-input AND gates have
been replaced with 3-input AND gates, and the third input of
each gate receives feedback from the Q and not-Q outputs.
What this does for us is permit the J input to have effect only
when the circuit is reset, and permit the K input to have
effect only when the circuit is set. In other words, the two
inputs are interlocked, to use a relay logic term, so that they
cannot both be activated simultaneously. If the circuit is
"set," the J input is inhibited by the O status of not-Q through
the lower AND gate; if the circuit is "reset," the K input is
inhibited by the O status of Q through the upper AND gate.
When both J and K inputs are 1, however, something unique
happens. Because of the selective inhibiting action of those
3-input AND gates, a "set" state inhibits input J so that the
flip-flop acts as if J=O while K=1 when in fact both are 1. On
the next clock pulse, the outputs will switch ("toggle") from
set (Q=1 and not-Q=0) to reset (Q=0 and not-Q=1).
Conversely, a "reset" state inhibits input K so that the flip-
flop acts as if J=1 and K=O when in fact both are 1. The next
clock pulse toggles the circuit again from reset to set.
See if you can follow this logical sequence with the ladder
logic equivalent of the J-K flip-flop:
[ is
nw
po | dt
a ae
J CR3 CR2
EE SH Kd Ec fet Ma ed)
Fes et ad =) Gs as =a ES
aK i Fe eG EE)
The end result is that the S-R flip-flop's "invalid" state is
eliminated (along with the race condition it engendered)
and we get a useful feature as a bonus: the ability to toggle
between the two (bistable) output states with every
transition of the clock input signal.
There is no such thing as a J-K latch, only J-K flip-flops.
Without the edge-triggering of the clock input, the circuit
would continuously toggle between its two output states
when both J and K were held high (1), making it an astable
device instead of a bistable device in that circumstance. If
we want to preserve bistable operation for all combinations
of input states, we must use edge-triggering so that it
toggles only when we tell it to, one step (clock pulse) ata
time.
The block symbol for a J-K flip-flop is a whole lot less
frightening than its internal circuitry, and just like the S-R
and D flip-flops, J-K flip-flops come in two clock varieties
(negative and positive edge-triggered):
¢ REVIEW:
e A J-K flip-flop is nothing more than an S-R flip-flop with
an added layer of feedback. This feedback selectively
enables one of the two set/reset inputs so that they
cannot both carry an active signal to the multivibrator
circuit, thus eliminating the invalid condition.
When both J and K inputs are activated, and the clock
input is pulsed, the outputs (Q and not-Q) will swap
states. That is, the circuit will togg/e from a set state toa
reset state, or vice versa.
The normal data inputs to a flip flop (D, S and R, or J and K)
are referred to as synchronous inputs because they have
effect on the outputs (Q and not-Q) only in step, or in sync,
with the clock signal transitions. These extra inputs that |
now bring to your attention are called asynchronous
because they can set or reset the flip-flop regardless of the
status of the clock signal. Typically, they're called preset and
Clear.
PRE PRE PRE
CLR CLR CLR
When the preset input is activated, the flip-flop will be set
(Q=1, not-Q=0) regardless of any of the synchronous inputs
or the clock. When the clear input is activated, the flip-flop
will be reset (Q=0, not-Q=1), regardless of any of the
synchronous inputs or the clock. So, what happens if both
preset and clear inputs are activated? Surprise, surprise: we
get an invalid state on the output, where Q and not-Q go to
the same state, the same as our old friend, the S-R latch!
Preset and clear inputs find use when multiple flip-flops are
ganged together to perform a function on a multi-bit binary
word, and a single line is needed to set or reset them all at
once.
Asynchronous inputs, just like synchronous inputs, can be
engineered to be active-high or active-low. If they're active-
low, there will be an inverting bubble at that input lead on
the block symbol, just like the negative edge-trigger clock
inputs.
PRE PRE PRE
S Q D Q J Q
C C C
R Q Q K Q
CLR CLR CLR
Sometimes the designations "PRE" and "CLR" will be shown
with inversion bars above them, to further denote the
negative logic of these inputs:
REVIEW:
Asynchronous inputs on a flip-flop have control over the
outputs (Q and not-Q) regardless of clock input status.
e These inputs are called the preset (PRE) and clear (CLR).
The preset input drives the flip-flop to a set state while
the clear input drives it to a reset state.
It is possible to drive the outputs of a J-K flip-flop to an
invalid condition using the asynchronous inputs,
because all feedback within the multivibrator circuit is
overridden.
Monostable multivibrators
We've already seen one example of a monostable
multivibrator in use: the pulse detector used within the
circuitry of flip-flops, to enable the latch portion for a brief
time when the clock input signal transitions from either low
to high or high to low. The pulse detector is classified as a
monostable multivibrator because it has only one stable
state. By stable, | mean a state of output where the device is
able to latch or hold to forever, without external prodding. A
latch or flip-flop, being a bistable device, can hold in either
the "set" or "reset" state for an indefinite period of time.
Once its set or reset, it will continue to latch in that state
unless prompted to change by an external input. A
monostable device, on the other hand, is only able to hold in
one particular state indefinitely. Its other state can only be
held momentarily when triggered by an external input.
A mechanical analogy of a monostable device would be a
momentary contact pushbutton switch, which spring-returns
to its normal (stable) position when pressure is removed
from its button actuator. Likewise, a standard wall (toggle)
switch, such as the type used to turn lights on and off ina
house, is a bistable device. It can latch in one of two modes:
on or Off.
All monostable multivibrators are timed devices. That is,
their unstable output state will hold only for a certain
minimum amount of time before returning to its stable state.
With semiconductor monostable circuits, this timing
function is typically accomplished through the use of
resistors and capacitors, making use of the exponential
charging rates of RC circuits. A comparator is often used to
compare the voltage across the charging (or discharging)
Capacitor with a steady reference voltage, and the on/off
output of the comparator used for a logic signal. With ladder
logic, time delays are accomplished with time-delay relays,
which can be constructed with semiconductor/RC circuits
like that just mentioned, or mechanical delay devices which
impede the immediate motion of the relay's armature. Note
the design and operation of the pulse detector circuit in
ladder logic:
L, L,
Input
1 second
TD1
Output
Input — J OL LOS LL
Output — JL LC CL
—> |}«— 1 second
No matter how long the input signal stays high (1), the
output remains high for just 1 second of time, then returns
to its normal (stable) low state.
For some applications, it is necessary to have a monostable
device that outputs a longer pulse than the input pulse
which triggers it. Consider the following ladder logic circuit:
Input TD1
10 seconds
Output
Input —_ JL =EurndS-d o©LLL_JD LLL
Output S) = LI L
—<—— = — <—— —> —
7 seconds 10 seconds 10 seconds
When the input contact closes, TD, contact immediately
closes, and stays closed for 10 seconds after the input
contact opens. No matter how short the input pulse is, the
output stays high (1) for exactly 10 seconds after the input
drops low again. This kind of monostable multivibrator is
called a one-shot. More specifically, it is a retriggerable one-
shot, because the timing begins after the input drops toa
low state, meaning that multiple input pulses within 10
seconds of each other will maintain a continuous high
output:
"Retriggering” action
Input — J LI LJ LSS
Output _- SOL
_—_ =
10 seconds
One application for a retriggerable one-shot is that of a
single mechanical contact debouncer. As you can see from
the above timing diagram, the output will remain high
despite "bouncing" of the input signal from a mechanical
switch. Of course, in a real-life switch debouncer circuit,
you'd probably want to use a time delay of much shorter
duration than 10 seconds, as you only need to "debounce"
pulses that are in the millisecond range.
Switch .
momentarily
actuate
"Dirty" signal I
"Clean" signal
What if we only wanted a 10 second timed pulse output from
a relay logic circuit, regardless of how many input pulses we
received or how long-lived they may be? In that case, we'd
have to couple a pulse-detector circuit to the retriggerable
one-shot time delay circuit, like this:
0.5 second
TD1 Input TD2 TD2
10 seconds
Output
Input IU] =< Des
Output | L__| |__| Le
~ <_ o —<o ~
10 sec. 10 sec. 10 sec.
Time delay relay TD, provides an "on" pulse to time delay
relay coil TD for an arbitrarily short moment (in this circuit,
for at least 0.5 second each time the input contact is
actuated). As soon as TD, is energized, the normally-closed,
timed-closed TD, contact in series with it prevents coil TD,
from being re-energized as long as its timing out (10
seconds). This effectively makes it unresponsive to any more
actuations of the input switch during that 10 second period.
Only after TD, times out does the normally-closed, timed-
closed TD> contact in series with it allow coil TD to be
energized again. This type of one-shot is called a
nonretriggerable one-shot.
One-shot multivibrators of both the retriggerable and
nonretriggerable variety find wide application in industry for
siren actuation and machine sequencing, where an
intermittent input signal produces an output signal of a set
time.
e REVIEW:
e A monostable multivibrator has only one stable output
state. The other output state can only be maintained
temporarily.
Monostable multivibrators, sometimes called one-shots,
come in two basic varieties: retriggerable and
nonretriggerable.
e One-shot circuits with very short time settings may be
used to debounce the "dirty" signals created by
mechanical switch contacts.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4]
—| | +4/l—
Lessons In Electric Circuits
-- Volume IV
Chapter 11
SEQUENTIAL CIRCUITS
Binary count sequence
Asynchronous counters
Synchronous counters
Counter modulus
Finite State Machines
Contributors
Bibliography
«& INCOMPLETE ***
Binary count sequence
If we examine a four-bit binary count sequence from 0000 to
1111, a definite pattern will be evident in the "oscillations" of
the bits between 0 and 1:
0000
0001
0010
O011
0100
O101
0110
Ot
1000
1001
1010
yA ge
1100
1101
a Uae Fag
oe De
Note how the least significant bit (LSB) toggles between 0
and 1 for every step in the count sequence, while each
succeeding bit toggles at one-half the frequency of the one
before it. The most significant bit (MSB) only toggles once
during the entire sixteen-step count sequence: at the
transition between 7 (0111) and 8 (1000).
If we wanted to design a digital circuit to "count" in four-bit
binary, all we would have to do is design a series of
frequency divider circuits, each circuit dividing the frequency
of a square-wave pulse by a factor of 2:
csp) 0 J1 lofi lofi lofi lof lof lofi lofr |
J-K flip-flops are ideally suited for this task, because they
have the ability to "toggle" their output state at the
command of a clock pulse when both J and K inputs are made
"high" (1):
signal B
signal A
If we consider the two signals (A and B) in this circuit to
represent two bits of a binary number, signal A being the LSB
and signal B being the MSB, we see that the count sequence
iS backward: from 11 to 10 to O1 to 00 and back again to 11.
Although it might not be counting in the direction we might
have assumed, at least it counts!
The following sections explore different types of counter
circuits, all made with J-K flip-flops, and all based on the
exploitation of that flip-flop's toggle mode of operation.
e REVIEW:
e Binary count sequences follow a pattern of octave
frequency division: the frequency of oscillation for each
bit, from LSB to MSB, follows a divide-by-two pattern. In
other words, the LSB will oscillate at the highest
frequency, followed by the next bit at one-half the LSB's
frequency, and the next bit at one-half the frequency of
the bit before it, etc.
e Circuits may be built that "count" in a binary sequence,
using J-K flip-flops set up in the "toggle" mode.
Asynchronous counters
In the previous section, we saw a circuit using one J-K flip-flop
that counted backward in a two-bit binary sequence, from 11
to 10 to O1 to OO. Since it would be desirable to have a circuit
that could count forward and not just backward, it would be
worthwhile to examine a forward count sequence again and
look for more patterns that might indicate how to build such
a circuit.
Since we know that binary count sequences follow a pattern
of octave (factor of 2) frequency division, and that J-K flip-
flop multivibrators set up for the "toggle" mode are capable
of performing this type of frequency division, we can envision
a circuit made up of several J-K flip-flops, cascaded to
produce four bits of output. The main problem facing us is to
determine how to connect these flip-flops together so that
they toggle at the right times to produce the proper binary
sequence. Examine the following binary count sequence,
paying attention to patterns preceding the "toggling" of a bit
between 0 and 1:
0000
0001
0010
O011
0100
O101
0110
2 Ka
1000
1001
POO
ye ge Oe
1100
i ie Ee 0 a
1.6
yi a ln
Note that each bit in this four-bit sequence toggles when the
bit before it (the bit having a lesser significance, or place-
weight), toggles in a particular direction: from 1 to 0. Small
arrows indicate those points in the sequence where a bit
toggles, the head of the arrow pointing to the previous bit
transitioning from a "high" (1) state to a "low" (0) state:
0000
0001
—
0010
0011
0100
0101
Sard
0110
O1l1l
+S
1000
1001
>
1010
yo i a
>
1100
L201
=>
1110
yg i ie
Starting with four J-K flip-flops connected in such a way to
always be in the "toggle" mode, we need to determine how to
connect the clock inputs in such a way so that each
succeeding bit toggles when the bit before it transitions from
1 to 0. The Q outputs of each flip-flop will serve as the
respective binary bits of the final, four-bit count:
If we used flip-flops with negative-edge triggering (bubble
symbols on the clock inputs), we could simply connect the
clock input of each flip-flop to the Q output of the flip-flop
before it, so that when the bit before it changes fromaltoa
0, the "falling edge" of that signal would "clock" the next flip-
flop to toggle the next bit:
A four-bit “up” counter
This circuit would yield the following output waveforms,
when "clocked" by a repetitive source of pulses from an
oscillator:
Clock
ba oe a oh od ed Ld
On O;LlIO;L{IO};]LI OL LOF;LTIo;LIToajy1. {ail
L L L L L L L
Q, 0 Of;1l 1L1}/;0 O71 17/0 Of;1 1/0 O71 1
Oo; 0.0 U Ut 2 £ LE 8 Us? 2st 4
QO, 8.0 8-0 ee et tt tei db 2 ld
The first flip-flop (the one with the Qj) output), has a positive-
edge triggered clock input, so it toggles with each rising
edge of the clock signal. Notice how the clock signal in this
example has a duty cycle less than 50%. I've shown the
signal in this manner for the purpose of demonstrating how
the clock signal need not be symmetrical to obtain reliable,
"clean" output bits in our four-bit binary sequence. In the
very first flip-flop circuit shown in this chapter, | used the
clock signal itself as one of the output bits. This is a bad
practice in counter design, though, because it necessitates
the use of a Square wave signal with a 50% duty cycle
(“high" time = "low" time) in order to obtain a count
sequence where each and every step pauses for the same
amount of time. Using one J-K flip-flop for each output bit,
however, relieves us of the necessity of having a symmetrical
clock signal, allowing the use of practically any variety of
high/low waveform to increment the count sequence.
As indicated by all the other arrows in the pulse diagram,
each succeeding output bit is toggled by the action of the
preceding bit transitioning from "high" (1) to "low" (0). This is
the pattern necessary to generate an "up" count sequence.
A less obvious solution for generating an "up" sequence
using positive-edge triggered flip-flops is to "clock" each flip-
flop using the Q' output of the preceding flip-flop rather than
the Q output. Since the Q' output will always be the exact
opposite state of the Q output on a J-K flip-flop (no invalid
states with this type of flip-flop), a high-to-low transition on
the Q output will be accompanied by a low-to-high transition
on the Q' output. In other words, each time the Q output of a
flip-flop transitions from 1 to 0, the Q' output of the same
flip-flop will transition from O to 1, providing the positive-
going clock pulse we would need to toggle a positive-edge
triggered flip-flop at the right moment:
A different way of making a four-bit “up” counter
One way we could expand the capabilities of either of these
two counter circuits is to regard the Q' outputs as another set
of four binary bits. If we examine the pulse diagram for such
a circuit, we see that the Q' outputs generate a down-
counting sequence, while the Q outputs generate an up-
counting sequence:
A simultaneous “up” and “down” counter
"Up" count sequence
65.0 0: 0. 0 OO 8 ee Ee
"Down" count sequence
Oh Oe a eT eB
ol Pr Pl
= —
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lo
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Ke
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Unfortunately, all of the counter circuits shown thusfar share
a common problem: the ripple effect. This effect is seen in
certain types of binary adder and data conversion circuits,
and is due to accumulative propagation delays between
cascaded gates. When the Q output of a flip-flop transitions
from 1 to O, it commands the next flip-flop to toggle. If the
next flip-flop toggle is a transition from 1 to O, it will
command the flip-flop after it to toggle as well, and so on.
However, since there is always some small amount of
propagation delay between the command to toggle (the
clock pulse) and the actual toggle response (Q and Q'
outputs changing states), any subsequent flip-flops to be
toggled will toggle some time after the first flip-flop has
toggled. Thus, when multiple bits toggle in a binary count
sequence, they will not all toggle at exactly the same time:
Pulse diagram showing (exaggerated) propagation delays
As you can see, the more bits that toggle with a given clock
pulse, the more severe the accumulated delay time from LSB
to MSB. When a clock pulse occurs at such a transition point
(say, on the transition from 0111 to 1000), the output bits
will "ripple" in sequence from LSB to MSB, as each
succeeding bit toggles and commands the next bit to toggle
as well, with a small amount of propagation delay between
each bit toggle. If we take a close-up look at this effect
during the transition from 0111 to 1000, we can see that
there will be fa/se output counts generated in the brief time
period that the "ripple" effect takes place:
Count False Count
7
counts
, 4d t4
Qo al 0 0 0 0
QO; a 1};0 0 QO
Qo 1 1 1;0 0
oi: fe) 0 0 0]1
Instead of cleanly transitioning from a "0111" output to a
"1000" output, the counter circuit will very quickly ripple
from 0111 to 0110 to 0100 to 0000 to 1000, or from 7 to 6to
4to Oand then to 8. This behavior earns the counter circuit
the name of ripple counter, or asynchronous counter.
In many applications, this effect is tolerable, since the ripple
happens very, very quickly (the width of the delays has been
exaggerated here as an aid to understanding the effects). If
all we wanted to do was drive a Set of light-emitting diodes
(LEDs) with the counter's outputs, for example, this brief
ripple would be of no consequence at all. However, if we
wished to use this counter to drive the "select" inputs of a
multiplexer, index a memory pointer in a microprocessor
(computer) circuit, or perform some other task where false
outputs could cause spurious errors, it would not be
acceptable. There is a way to use this type of counter circuit
In applications sensitive to false, ripple-generated outputs,
and it involves a principle known as strobing.
Most decoder and multiplexer circuits are equipped with at
least one input called the "enable." The output(s) of such a
circuit will be active only when the enable input is made
active. We can use this enable input to strobe the circuit
receiving the ripple counter's output so that it is disabled
(and thus not responding to the counter output) during the
brief period of time in which the counter outputs might be
rippling, and enabled only when sufficient time has passed
since the last clock pulse that all rippling will have ceased. In
most cases, the strobing signal can be the same clock pulse
that drives the counter circuit:
Receiving circuit
EN
Clock signal!
Binary
count
input
Counter circuit
With an active-low Enable input, the receiving circuit will
respond to the binary count of the four-bit counter circuit
only when the clock signal is "low." As soon as the clock
pulse goes "high," the receiving circuit stops responding to
the counter circuit's output. Since the counter circuit is
positive-edge triggered (as determined by the first flip-flop
clock input), all the counting action takes place on the low-
to-high transition of the clock signal, meaning that the
receiving circuit will become disabled just before any
toggling occurs on the counter circuit's four output bits. The
receiving circuit will not become enabled until the clock
signal returns to a low state, which should be a long enough
time after all rippling has ceased to be "safe" to allow the
new count to have effect on the receiving circuit. The crucial
parameter here is the clock signal's "high" time: it must be at
least as long as the maximum expected ripple period of the
counter circuit. If not, the clock signal will prematurely
enable the receiving circuit, while some rippling is still taking
place.
Another disadvantage of the asynchronous, or ripple, counter
circuit is limited speed. While all gate circuits are limited in
terms of maximum signal frequency, the design of
asynchronous counter circuits compounds this problem by
making propagation delays additive. Thus, even if strobing is
used in the receiving circuit, an asynchronous counter circuit
cannot be clocked at any frequency higher than that which
allows the greatest possible accumulated propagation delay
to elapse well before the next pulse.
The solution to this problem is a counter circuit that avoids
ripple altogether. Such a counter circuit would eliminate the
need to design a "strobing" feature into whatever digital
circuits use the counter output as an input, and would also
enjoy a much greater operating speed than its asynchronous
equivalent. This design of counter circuit is the subject of the
next section.
e REVIEW:
e An "up" counter may be made by connecting the clock
inputs of positive-edge triggered J-K flip-flops to the Q'
outputs of the preceding flip-flops. Another way is to use
negative-edge triggered flip-flops, connecting the clock
inputs to the Q outputs of the preceding flip-flops. In
either case, the J and K inputs of all flip-flops are
connected to V,, or Vgg so as to always be "high."
e Counter circuits made from cascaded J-K flip-flops where
each clock input receives its pulses from the output of
the previous flip-flop invariably exhibit a ripple effect,
where false output counts are generated between some
steps of the count sequence. These types of counter
circuits are called asynchronous counters, or ripple
counters.
e Strobing is a technique applied to circuits receiving the
output of an asynchronous (ripple) counter, so that the
false counts generated during the ripple time will have
no ill effect. Essentially, the enab/e input of such a circuit
is connected to the counter's clock pulse in such a way
that it is enabled only when the counter outputs are not
changing, and will be disabled during those periods of
changing counter outputs where ripple occurs.
Synchronous counters
A synchronous counter, in contrast to an asynchronous
counter, is one whose output bits change state
simultaneously, with no ripple. The only way we can build
such a counter circuit from J-K flip-flops is to connect all the
clock inputs together, so that each and every flip-flop
receives the exact same clock pulse at the exact same time:
Now, the question is, what do we do with the J and K inputs?
We know that we still have to maintain the same divide-by-
two frequency pattern in order to count in a binary sequence,
and that this pattern is best achieved utilizing the "toggle"
mode of the flip-flop, so the fact that the J and K inputs must
both be (at times) "high" is clear. However, if we simply
connect all the J and K inputs to the positive rail of the power
supply as we did in the asynchronous circuit, this would
clearly not work because all the flip-flops would toggle at the
same time: with each and every clock pulse!
This circuit will not function as a counter!
Let's examine the four-bit binary counting sequence again,
and see if there are any other patterns that predict the
toggling of a bit. Asynchronous counter circuit design is
based on the fact that each bit toggle happens at the same
time that the preceding bit toggles from a "high" to a "low"
(from 1 to 0). Since we cannot clock the toggling of a bit
based on the toggling of a previous bit in a synchronous
counter circuit (to do so would create a ripple effect) we must
find some other pattern in the counting sequence that can be
used to trigger a bit toggle:
Examining the four-bit binary count sequence, another
predictive pattern can be seen. Notice that just before a bit
toggles, all preceding bits are "high:"
0000
0001
0010
001
0100
0101
0110
O11)
1000
1001
1010
tou
1100
1101
1110
1111
This pattern is also something we can exploit in designing a
counter circuit. lf we enable each J-K flip-flop to toggle based
on whether or not all preceding flip-flop outputs (Q) are
"high," we can obtain the same counting sequence as the
asynchronous circuit without the ripple effect, since each
flip-flop in this circuit will be clocked at exactly the same
time:
A four-bit synchronous “up” counter
This flip-flop This flip-flop This flip-flop This flip-flop
toggles on every toggles only if toggles only if toggles only if
clock pulse Q, is “high” Q, AND Q, Q, AND Q, AND Q:
are “high” are “high”
The result is a four-bit synchronous "up" counter. Each of the
higher-order flip-flops are made ready to toggle (both J and K
inputs "high") if the Q outputs of all previous flip-flops are
"high." Otherwise, the J and K inputs for that flip-flop will
both be "low," placing it into the "latch" mode where it will
maintain its present output state at the next clock pulse.
Since the first (LSB) flip-flop needs to toggle at every clock
pulse, its J and K inputs are connected to V,, or Vgg, where
they will be "high" all the time. The next flip-flop need only
"recognize" that the first flip-flop's Q output is high to be
made ready to toggle, so no AND gate is needed. However,
the remaining flip-flops should be made ready to toggle only
when a//lower-order output bits are "high," thus the need for
AND gates.
To make a synchronous "down" counter, we need to build the
circuit to recognize the appropriate bit patterns predicting
each toggle state while counting down. Not surprisingly,
when we examine the four-bit binary count sequence, we see
that all preceding bits are "low" prior to a toggle (following
the sequence from bottom to top):
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
oa Dt eb
Since each J-K flip-flop comes equipped with a Q' output as
well as a Q output, we can use the Q' outputs to enable the
toggle mode on each succeeding flip-flop, being that each Q'
will be "high" every time that the respective Q is "low:"
A four-bit synchronous "down" counter
Qo Qu Q2 Q3
This flip-flop This flip-flop This flip-flop This flip-flop
toggles on every toggles only if toggles only if toggles only if
clock pulse Q, is “high” Q, AND Q, Q, AND Q, AND Q-
are “high” are “high
Taking this idea one step further, we can build a counter
circuit with selectable between "up" and "down" count
modes by having dual lines of AND gates detecting the
appropriate bit conditions for an "up" and a "down" counting
sequence, respectively, then use OR gates to combine the
AND gate outputs to the J and K inputs of each succeeding
flip-flop:
A four-bit synchronous “updown" counter
al Q: Qs
Up/Down —t ees a et
-__/ ] -L__/
J} 1) | J |Q| | - J Q
i —! Cl >— ra
a mt E 2 Pa —> | my Hh
“ | LK Oo j —~ ILK Oo ~ K O
p— —I— oo > Pp
> a | , 1)
1__/ —1_/
+ +
This circuit isn't as complex as it might first appear. The
Up/Down control input line simply enables either the upper
string or lower string of AND gates to pass the Q/Q' outputs
to the succeeding stages of flip-flops. If the Up/Down control
line is "high," the top AND gates become enabled, and the
circuit functions exactly the same as the first ("up")
synchronous counter circuit shown in this section. If the
Up/Down control line is made "low," the bottom AND gates
become enabled, and the circuit functions identically to the
second ("down" counter) circuit shown in this section.
To illustrate, here is a diagram showing the circuit in the "up"
counting mode (all disabled circuitry shown in grey rather
than black):
Counter in "up" counting mode
Qs
[ede |
|
lel
3
.
a:
fo
Here, shown in the "down" counting mode, with the same
grey coloring representing disabled circuitry:
Counter in “down” counting mode
Up/down counter circuits are very useful devices. A common
application is in machine motion control, where devices
called rotary shaft encoders convert mechanical rotation into
a series of electrical pulses, these pulses "clocking" a counter
circuit to track total motion:
Light sensor 2 a & @
(phototransistor)
Counter
Rotary shaftencoder =
As the machine moves, it turns the encoder shaft, making
and breaking the light beam between LED and
phototransistor, thereby generating clock pulses to
increment the counter circuit. Thus, the counter integrates,
or accumulates, total motion of the shaft, serving as an
electronic indication of how far the machine has moved. If all
we care about is tracking total motion, and do not care to
account for changes in the direction of motion, this
arrangement will suffice. However, if we wish the counter to
increment with one direction of motion and decrement with
the reverse direction of motion, we must use an up/down
counter, and an encoder/decoding circuit having the ability
to discriminate between different directions.
If we re-design the encoder to have two sets of
LED/phototransistor pairs, those pairs aligned such that their
square-wave output signals are 90° out of phase with each
other, we have what is Known as a quadrature output
encoder (the word "quadrature" simply refers to a 90°
angular separation). A phase detection circuit may be made
from a D-type flip-flop, to distinguish a clockwise pulse
sequence from a counter-clockwise pulse sequence:
Up/Down
SS Counter
Rotary shaft encoder
(quadrature output)
When the encoder rotates clockwise, the "D" input signal
square-wave will lead the "C" input square-wave, meaning
that the "D" input will already be "high" when the "C"
transitions from "low" to "high," thus setting the D-type flip-
flop (making the Q output "high") with every clock pulse. A
"high" Q output places the counter into the "Up" count mode,
and any clock pulses received by the clock from the encoder
(from either LED) will increment it. Conversely, when the
encoder reverses rotation, the "D" input will lag behind the
"C" input waveform, meaning that it will be "low" when the
"C" waveform transitions from "low" to "high," forcing the D-
type flip-flop into the reset state (making the Q output "low")
with every clock pulse. This "low" signal commands the
counter circuit to decrement with every clock pulse from the
encoder.
This circuit, or something very much like it, is at the heart of
every position-measuring circuit based on a pulse encoder
sensor. Such applications are very common in robotics, CNC
machine tool control, and other applications involving the
measurement of reversible, mechanical motion.
Counter modulus
INCOMPLETE
Finite State Machines
Up to now, every circuit that was presented was a
combinatorial circuit. That means that its output is
dependent only by its current inputs. Previous inputs for that
type of circuits have no effect on the output.
However, there are many applications where there is a need
for our circuits to have "memory"; to remember previous
inputs and calculate their outputs according to them. A
circuit whose output depends not only on the present input
but also on the history of the input is called a sequential
circult.
In this section we will learn how to design and build such
sequential circuits. In order to see how this procedure works,
we will use an example, on which we will study our topic.
So let's suppose we have a digital quiz game that works ona
clock and reads an input from a manual button. However, we
want the switch to transmit only one HIGH pulse to the
circuit. lf we hook the button directly on the game circuit it
will transmit HIGH for as few clock cycles as our finger can
achieve. On a common clock frequency our finger can never
be fast enough.
The desing procedure has specific steps that must be
followed in order to get the work done:
Step 1
The first step of the design procedure is to define with simple
but clear words what we want our circuit to do:
“Our mission is to design a secondary circuit that will
transmit a HIGH pulse with duration of only one cycle when
the manual button Is pressed, and won't transmit another
pulse until the button is depressed and pressed again."
Step 2
The next step is to design a State Diagram. This is a diagram
that is made from circles and arrows and describes visually
the operation of our circuit. In mathematic terms, this
diagram that describes the operation of our sequential circuit
is a Finite State Machine.
Make a note that this is a Moore Finite State Machine. Its
output is a function of only its current state, not its input.
That Is in contrast with the Mealy Finite State Machine, where
input affects the output. In this tutorial, only the Moore Finite
State Machine will be examined.
The State Diagram of our circuit is the following: (Figure
below)
Activate Pu
se
rst_n
1
Wait Loop
0
A State Diagram
Every circle represents a "state", a well-defined condition
that our machine can be found at.
In the upper half of the circle we describe that condition. The
description helps us remember what our circuit is supposed
to do at that condition.
e The first circle is the "stand-by" condition. This is where
our circuit starts from and where it waits for another
button press.
e The second circle is the condition where the button has
just been just pressed and our circuit needs to transmit a
HIGH pulse.
e The third circle is the condition where our circuit waits for
the button to be released before it returns to the "stand-
by" condition.
In the lower part of the circle is the output of our circuit. If we
want our circuit to transmit a HIGH on a specific state, we put
a 1 on that state. Otherwise we put a 0.
Every arrow represents a "transition" from one state to
another. A transition happens once every clock cycle.
Depending on the current Input, we may go to a different
state each time. Notice the number in the middle of every
arrow. This is the current Input.
For example, when we are in the "Initial-Stand by" state and
we "read" a 1, the diagram tells us that we have to go to the
"Activate Pulse" state. If we read a O we must stay on the
"Initial-Stand by" state.
So, what does our "Machine" do exactly? It starts from the
"Initial - Stand by" state and waits until a 1 is read at the
Input. Then it goes to the "Activate Pulse" state and
transmits a HIGH pulse on its output. If the button keeps
being pressed, the circuit goes to the third state, the "Wait
Loop". There it waits until the button is released (Input goes
0) while transmitting a LOW on the output. Then it's all over
again!
This is possibly the most difficult part of the design
procedure, because it cannot be described by simple steps. It
takes exprerience and a bit of sharp thinking in order to set
up a State Diagram, but the rest is just a set of
predetermined steps.
Step 3
Next, we replace the words that describe the different states
of the diagram with binary numbers. We start the
enumeration from 0 which is assigned on the initial state. We
then continue the enumeration with any state we like, until
all states have their number.
The result looks something like this: (Figure below)
0
rst_n
A State Diagram with Coded States
Step 4
Afterwards, we fill the State Table. This table has a very
specific form. | will give the table of our example and use it to
explain how to fill it in. (Figure below)
Current State Next State
| B Anext | Bnext
A
oe
Wt .
0
=i Ooo
ro °
—i — Oo
ooey
oo
—_ aa)
— Oo
A State Table
The first columns are as many as the bits of the highest
number we assigned the State Diagram. If we had 5 states,
we would have used up to the number 100, which means we
would use 3 columns. For our example, we used up to the
number 10, so only 2 columns will be needed. These columns
describe the Current State of our circuit.
To the right of the Current State columns we write the /nput
Columns. These will be as many as our Input variables. Our
example has only one Input.
Next, we write the Next State Columns. These are as many as
the Current State columns.
Finally, we write the Outputs Columns. These are aS many as
our outputs. Our example has only one output. Since we
have built a More Finite State Machine, the output is
dependent on only the current input states. This is the
reason the outputs column has two 1: to result in an output
Boolean function that is independant of input |. Keep on
reading for further details.
The Current State and Input columns are the Inputs of our
table. We fill them in with all the binary numbers from 0 to
9(Number of Current State columns + Number of Input columns) _4
It is simpler than it sounds fortunately. Usually there will be
more rows than the actual States we have created in the
State Diagram, but that's ok.
Each row of the Next State columns is filled as follows: We fill
it in with the state that we reach when, in the State Diagram,
from the Current State of the same row we follow the Input of
the same row. If have to fill in a row whose Current State
number doesn't correspond to any actual State in the State
Diagram we fill it with Don't Care terms (X). After all, we don't
care where we can go from a State that doesn't exist. We
wouldn't be there in the first place! Again it is simpler than it
sounds.
The outputs column is filled by the output of the
corresponding Current State in the State Diagram.
The State Table is complete! It describes the behaviour of our
circuit as fully as the State Diagram does.
Step 5a
The next step is to take that theoretical "Machine" and
implement it in a circuit. Most often than not, this
implementation involves Flip Flops. This guide is dedicated
to this kind of implementation and will describe the
procedure for both D - Flip Flops as well as JK - Flip Flops. T -
Flip Flops will not be included as they are too similar to the
two previous cases.
The selection of the Flip Flop to use is arbitrary and usually is
determined by cost factors. The best choice is to perform
both analysis and decide which type of Flip Flop results in
minimum number of logic gates and lesser cost.
First we will examine how we implement our "Machine" with
D-Flip Flops.
We will need as many D - Flip Flops as the State columns, 2
in our example. For every Flip Flop we will add one more
column in our State table (Figure below) with the name of the
Flip Flop's input, "D" for this case. The column that
corresponds to each Flip Flop describes what input we
must give the Flip Flop in order to go from the
Current State to the Next State. For the D- Flip Flop this
IS easy: The necessary input is equal to the Next State. In the
rows that contain X's we fill X's in this column as well.
Current State Next State Outputs Flip Flop Inputs
A | B Anext | Bnext Y Da | Des
0
fad Lead Land Gol bel od od
ad Lasd Kal Kal laa Gal acl
Hr CoH FH O°
A State Table with D - Flip Flop Excitations
Step 5b
We can do the same steps with JK - Flip Flops. There are some
differences however. A JK - Flip Flop has two inputs, therefore
we need to add two columns for each Flip Flop. The content
of each cell is dictated by the JK's excitation table: (Figure
below)
JK - Flip Flop Excitation Table
This table says that if we want to go from State Q to State
Qnext) We need to use the specific input for each terminal. For
example, to go from 0 to 1, we need to feed J with 1 and we
don't care which input we feed to terminal K.
Current State Next State Outputs Flip Flop Inputs
Anet | Bnext | Ka | Bb |
A State Table with JK - Flip Flop Excitations
Step 6
We are in the final stage of our procedure. What remains, is
to determine the Boolean functions that produce the inputs
of our Flip Flops and the Output. We will extract one Boolean
funtion for each Flip Flop input we have. This can be done
with a Karnaugh Map. The input variables of this map are the
Current State variables as well as the Inputs.
That said, the input functions for our D - Flip Flops are the
following: (Figure below)
Karnaugh Maps for the D - Flip Flop Inputs
D,ag=A-I+B-I =(A+B)-I
Dp=ABI
If we chose to use JK - Flip Flops our functions would be the
following: (Figure below)
OT
Karnaugh Map for the JK - Flip Flop Input
JA=BI
a= t
Jp=Al
A p=
A Karnaugh Map will be used to determine the function of the
Output as well: (Figure below)
Karnaugh Map for the Output variable Y
Y=A-B
Step 7
We design our circuit. We place the Flip Flops and use logic
gates to form the Boolean functions that we calculated. The
gates take input from the output of the Flip Flops and the
Input of the circuit. Don't forget to connect the clock to the
Flip Flops!
The D - Flip Flop version: (Figure below)
The completed D - Flip Flop Sequential Circuit
The JK - Flip Flop version: (Figure below)
The completed JK - Flip Flop Sequential Circuit
This is it! We have successfully designed and constructed a
Sequential Circuit. At first it might seem a daunting task, but
after practice and repetition the procedure will become
trivial. Sequential Circuits can come in handy as control parts
of bigger circuits and can perform any sequential logic task
that we can think of. The sky is the limit! (or the circuit
board, at least)
e REVIEW:
e A Sequential Logic function has a "memory" feature and
takes into account past inputs in order to decide on the
output.
e The Finite State Machine is an abstract mathematical
model of a sequential logic function. It has finite inputs,
outputs and number of states.
e FSMs are implemented in real-life circuits through the
use of Flip Flops
e The implementation procedure needs a specific order of
steps (algorithm), in order to be carried out.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See Appendix
2 (Contributor List) for dates and contact information.
George Zogopoulos Papaliakos (November 2010): Author
of Finite State Machines section.
Bibliography
1. [CLL] C. L. Liu, Elements of Discrete Mathematics, 2nd
Edition
. [MMM] M. Morris Mano, Digital Design, 3rd Edition
. [SLW] “Sequential logic” at
http://en.wikipedia.org/wiki/Sequential%5Fcircuit
4. [JKF] “JK flip-flop”, Flip-flop (electronics) at
http://en.wikipedia.org/wiki/JKY%5Fflip%5Fflop%23)|/K%5F
flip-flop
WN
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—||+4/]l—
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 12
SHIFT REGISTERS
Introduction
Serial-in/serial-out shift register
o Serial-in/serial-out devices
e Parallel-in, serial-out shift register
o Parallel-in/serial-out devices
o Practical applications
Serial-in, parallel-out shift register
o Serial-in/ parallel-out devices
o Practical applications
o Parallel-in/ parallel-out and universal devices
o Practical applications
e Ring counters
o Johnson counters
» Johnson counter devices
» Practical applications
e references
Original author: Dennis Crunkilton
Introduction
Shift registers, like counters, are a form of sequential logic.
Sequential logic, unlike combinational logic is not only
affected by the present inputs, but also, by the prior history.
In other words, sequential logic remembers past events.
Shift registers produce a discrete delay of a digital signal or
waveform. A waveform synchronized to a clock, a repeating
square wave, is delayed by "n" discrete clock times, where
"n" is the number of shift register stages. Thus, a four stage
shift register delays "data in" by four clocks to "data out".
The stages in a shift register are delay stages, typically type
"D" Flip-Flops or type "JK" Flip-flops.
Formerly, very long (several hundred stages) shift registers
served as digital memory. This obsolete application is
reminiscent of the acoustic mercury delay lines used as
early computer memory.
Serial data transmission, over a distance of meters to
kilometers, uses shift registers to convert parallel data to
serial form. Serial data communications replaces many slow
parallel data wires with a single serial high speed circuit.
Serial data over shorter distances of tens of centimeters,
uses shift registers to get data into and out of
microprocessors. Numerous peripherals, including analog to
digital converters, digital to analog converters, display
drivers, and memory, use shift registers to reduce the
amount of wiring in circuit boards.
Some specialized counter circuits actually use shift registers
to generate repeating waveforms. Longer shift registers, with
the help of feedback generate patterns so long that they
look like random noise, pseudo-noise.
Basic shift registers are classified by structure according to
the following types:
e Serial-in/serial-out
e Parallel-in/serial-out
e Serial-in/parallel-out
e Universal parallel-in/parallel-out
e Ring counter
data in __,} |, data out
clock __,)
stage A stage B stage C stage D
Serial-in, serial-out shitt register with 4-stages
Above we show a block diagram of a serial-in/serial-out shift
register, which is 4-stages long. Data at the input will be
delayed by four clock periods from the input to the output of
the shift register.
Data at "data in", above, will be present at the Stage A
output after the first clock pulse. After the second pulse
stage A data is transfered to stage B output, and "data in" is
transfered to stage A output. After the third clock, stage C is
replaced by stage B; stage B is replaced by stage A; and
stage A is replaced by "data in". After the fourth clock, the
data originally present at "data in" is at stage D, "output".
The "first in" data is "first out" as it is shifted from "data in"
to "data out".
Dy Dz De Dp
data in __,} , data out
clock _,}
stage A stage B stage C stage D
Parallel-in, serial-out shift register with 4-stages
Data is loaded into all stages at once of a parallel-in/serial-
out shift register. The data is then shifted out via "data out"
by clock pulses. Since a 4- stage shift register is shown
above, four clock pulses are required to shift out all of the
data. In the diagram above, stage D data will be present at
the "data out" up until the first clock pulse; stage C data will
be present at "data out" between the first clock and the
second clock pulse; stage B data will be present between
the second clock and the third clock; and stage A data will
be present between the third and the fourth clock. After the
fourth clock pulse and thereafter, successive bits of "data in"
should appear at "data out" of the shift register after a delay
of four clock pulses.
If four switches were connected to Dy, through Dp, the status
could be read into a microprocessor using only one data pin
and a clock pin. Since adding more switches would require
no additional pins, this approach looks attractive for many
inputs.
data in __,} |_, data out
clock _,}
stage A stage B stage C stage D
!
Qs Qs Qe Qp
Serial-in, parallel-out shift register with 4-stages
Above, four data bits will be shifted in from "data in" by four
clock pulses and be available at Q, through Qp for driving
external circuitry such as LEDs, lamps, relay drivers, and
horns.
After the first clock, the data at "data in" appears at Qa. After
the second clock, The old Q, data appears at Qz; Qa receives
next data from "data in". After the third clock, Qp data is at
Qc. After the fourth clock, Qc data is at Qp. This stage
contains the data first present at "data in". The shift register
should now contain four data bits.
Da Dz f Dp
data in __,} |, data out
clock _,}
mode »!
stage A stage B stage C stage D
|! ! !
Qs Qs Q- Q5
Parallel-in, parallel-out shift register with 4-stages
A parallel-in/parallel-out shift register combines the function
of the parallel-in, serial-out shift register with the function of
the serial-in, parallel-out shift register to yield the universal
shift register. The "do anything" shifter comes at a price- the
increased number of I/O (Input/Output) pins may reduce the
number of stages which can be packaged.
Data presented at D, through Dp is parallel loaded into the
registers. This data at Q, through Qpy may be shifted by the
number of pulses presented at the clock input. The shifted
data is available at Q, through Qp. The "mode" input, which
may be more than one input, controls parallel loading of
data from Dy, through Dp, shifting of data, and the direction
of shifting. There are shift registers which will shift data
either left or right.
data out
data in
clock _,} =
Qb
stage A stage B stage C stage D
Ring Counter, shift register output fed back to input
If the serial output of a shift register is connected to the
serial input, data can be perpetually shifted around the ring
as long as clock pulses are present. If the output is inverted
before being fed back as shown above, we do not have to
worry about loading the initial data into the "ring counter".
Serial-in/serial-out shift register
Serial-in, serial-out shift registers delay data by one clock
time for each stage. They will store a bit of data for each
register. A serial-in, serial-out shift register may be one to 64
bits in length, longer if registers or packages are cascaded.
Below is a single stage shift register receiving data which is
not synchronized to the register clock. The "data in" at the D
pin of the type D FF (Flip-Flop) does not change levels when
the clock changes for low to high. We may want to
synchronize the data to a system wide clock in a circuit
board to improve the reliability of a digital logic circuit.
ty ts & ty
data in D Q
clock
c L} Lf
clock data in | |
: _f Le
Qs
Data present at clock time is transfered from D to Q.
The obvious point (as compared to the figure below)
illustrated above is that whatever "data in" is present at the
D pin of a type D FF is transfered from D to output Q at clock
time. Since our example shift register uses positive edge
sensitive storage elements, the output Q follows the D input
when the clock transitions from low to high as shown by the
up arrows on the diagram above. There is no doubt what
logic level is present at clock time because the data is stable
well before and after the clock edge. This is seldom the case
in multi-stage shift registers. But, this was an easy example
to start with. We are only concerned with the positive, low to
high, clock edge. The falling edge can be ignored. It is very
easy to see Q follow D at clock time above. Compare this to
the diagram below where the "data in" appears to change
with the positive clock edge.
data in
clock data in
Q ?
Qw ?
Does the clock t, see a 0 or a1 at data in at D?_ Which output is correct,
Qe or Qu?
Since "data in" appears to changes at clock time t, above,
what does the type D FF see at clock time? The short over
simplified answer is that it sees the data that was present at
D prior to the clock. That is what is transfered to Q at clock
time t,;. The correct waveform is Qc. At t; Q goes to a zero if
it is not already zero. The D register does not see a one until
time t5, at which time Q goes high.
t ts t
data in D Q 7 : "
clock
clock data in
Q
Qs
‘ | |-— delay of 1 clock
period
Data present ty before clock time at Dis transfered toQ.
Since data, above, present at D is clocked to Q at clock time,
and Q cannot change until the next clock time, the D FF
delays data by one clock period, provided that the data is
already synchronized to the clock. The Q, waveform is the
Same as "data in" with a one clock period delay.
A more detailed look at what the input of the type D Flip-
Flop sees at clock time follows. Refer to the figure below.
Since "data in" appears to changes at clock time (above), we
need further information to determine what the D FF sees. If
the "data in" is from another shift register stage, another
same type D FF, we can draw some conclusions based on
data sheet information. Manufacturers of digital logic make
available information about their parts in data sheets,
formerly only available in a collection called a data book.
Data books are still available; though, the manufacturer's
web site is the modern source.
clock |
dara in D
. ls i
obey ——
Data must be present (t,) before the clock and after(t,,) the clock. Data is
delayed from D to Q by propagation delay (tp)
The following data was extracted from the CD4006b data
sheet for operation at 5Vpc, which serves as an example to
illustrate timing.
foal
e t;=100ns
e ty=60ns
¢ tp=200-400ns typ/max
ts is the setup time, the time data must be present before
clock time. In this case data must be present at D 100ns
prior to the clock. Furthermore, the data must be held for
hold time ty=60ns after clock time. These two conditions
must be met to reliably clock data from D to Q of the Flip-
Flop.
There is no problem meeting the setup time of 60ns as the
data at D has been there for the whole previous clock period
if it comes from another shift register stage. For example, at
a clock frequency of 1 Mhz, the clock period is 1000 us,
plenty of time. Data will actually be present for 1000us prior
to the clock, which is much greater than the minimum
required t, of 60ns.
The hold time ty=60ns is met because D connected to Q of
another stage cannot change any faster than the
propagation delay of the previous stage tp=200ns. Hold time
is met as long as the propagation delay of the previous D FF
is greater than the hold time. Data at D driven by another
stage Q will not change any faster than 200ns for the
CD4006b.
To summarize, output Q follows input D at nearly clock time
if Flip-Flops are cascaded into a multi-stage shift register.
Q, Qs Qe
data in Q data out
clock
Serial-in, serial-out shift register using type "D" storage elements
Three type D Flip-Flops are cascaded Q to D and the clocks
paralleled to form a three stage shift register above.
Qs Qe Qe
data in Q data out
Serial-in, serial-out shift register using type "JK” storage elements
Type JK FFs cascaded Q to J, Q' to K with clocks in parallel to
yield an alternate form of the shift register above.
A serial-in/serial-out shift register has a clock input, a data
input, and a data output from the last stage. In general, the
other stage outputs are not available Otherwise, it would be
a serial-in, parallel-out shift register.
The waveforms below are applicable to either one of the
preceding two versions of the serial-in, serial-out shift
register. The three pairs of arrows show that a three stage
shift register temporarily stores 3-bits of data and delays it
by three clock periods from input to output.
clock : Li Lil
data ———
o KE
—
At clock time t; a "data in" of O is clocked from D to Q of all
three stages. In particular, D of stage A sees a logic 0, which
is clocked to Qa, where it remains until time tp.
At clock time t, a "data in" of 1 is clocked from D to Q,j. At
stages B and C, a O, fed from preceding stages is clocked to
Qp and Qe.
At clock time t3 a "data in" of O is clocked from D to Qa. Qa
goes low and stays low for the remaining clocks due to "data
in" being O. Qg goes high at t3 due to a 1 from the previous
stage. Qc is still low after tz due to a low from the previous
stage.
Qc finally goes high at clock ty due to the high fed to D from
the previous stage Qg,. All earlier stages have Os shifted into
them. And, after the next clock pulse at ts, all logic 1s will
have been shifted out, replaced by Os
Serial-in/serial-out devices
We will take a closer look at the following parts available as
integrated circuits, courtesy of Texas Instruments. For
complete device data sheets follow the links.
e CD4006b 18-bit serial-in/ serial-out shift register
[*]
e CD4031b 64-bit serial-in/ serial-out shift register
hal
e CD4517b dual 64-bit serial-in/ serial-out shift register
bal
The following serial-in/ serial-out shift registers are 4000
series CMOS (Complementary Metal Oxide Semiconductor)
family parts. As such, They will accept a Vpp, positive power
supply of 3-Volts to 15-Volts. The Vcc pin is grounded. The
maximum frequency of the shift clock, which varies with
Vpp, is a few megahertz. See the full data sheet for details.
Lo {So Vg ( pin 7) = Gnd, Vpp (pin 14) = 43 to +18 Voc
clock CL a
CL and CL to all 18-stages & latch.
CD4006b Serial-in/ serial-out shift register
The 18-bit CD4006b consists of two stages of 4-bits and two
more stages of 5-bits with a an output tap at 4-bits. Thus,
the 5-bit stages could be used as 4-bit shift registers. To get
a full 18-bit shift register the output of one shift register
must be cascaded to the input of another and so on until all
stages create a single shift register as shown below.
CD4006b 18-bit serial-in/ serial-out shift register
A CD4031 64-bit serial-in/ serial-out shift register is shown
below. A number of pins are not connected (nc). Both Q and
Q' are available from the 64th stage, actually Qe, and Q'¢,.
There is also a Qe, "delayed" from a half stage which is
delayed by half a clock cycle. A major feature is a data
selector which is at the data input to the shift register.
mode control
Vop ne ne ne ne CLp
Qe, delayed
CD4031 64-bit serial-in/ serial-out shift register
The "mode control" selects between two inputs: data 1 and
data 2. If "mode control" is high, data will be selected from
“data 2" for input to the shift register. In the case of "mode
control" being logic low, the "data 1" is selected. Examples
of this are shown in the two figures below.
mode contro]
= logic high
Cock II IHIINNNNNIININNNNNIININANUITITAONQUNTLTOOUUIVTYNOOOUUNVTTUNOOOUUTEUUAOOOUL TED UAOOO TTT AAE OUTS UALU A L
Qs4
ge 4A Clocks BA Clocks Lg
CD4031 64-bit serial-in/ serial-out shift register recirculating data.
The "data 2" above is wired to the Q¢, output of the shift
register. With "mode control" high, the Qg, output is routed
back to the shifter data input D. Data will recirculate from
output to input. The data will repeat every 64 clock pulses
as shown above. The question that arises is how did this
data pattern get into the shift register in the first place?
Vop
mode control
a logic low
t t&
clock | f | f |
data |
Qe
CD4031 64-bit serial-in/ serial-out shitt register load new data at Data 1.
With "mode control" low, the CD4031 "data 1" is selected for
input to the shifter. The output, Q¢y, is not recirculated
because the lower data selector gate is disabled. By disabled
we mean that the logic low "mode select" inverted twice to a
low at the lower NAND gate prevents it for passing any
signal on the lower pin (data 2) to the gate output. Thus, it
iS disabled.
V,
sweitve, Qi Qun WEn Cly Qem Qus Ds
Qhea Qusa WE, CL Qesa Qs24 Da Vss
CD4517b dual 64-bit serial-in/ serial-out shift register
A CD4517b dual 64-bit shift register is shown above. Note
the taps at the 16th, 32nd, and 48th stages. That means
that shift registers of those lengths can be configured from
one of the 64-bit shifters. Of course, the 64-bit shifters may
be cascaded to yield an 80-bit, 96-bit, 112-bit, or 128-bit
shift register. The clock CL, and CLg need to be paralleled
when cascading the two shifters. WEg and WE, are grounded
for normal shifting operations. The data inputs to the shift
registers A and B are Dy and Dg respectively.
Suppose that we require a 16-bit shift register. Can this be
configured with the CD4517b? How about a 64-shift register
from the same part?
data in
clock Qiea out
——
7 V
Qiea Qisa Tee, CL, le SS
data in . -
clock Q,4, out
CD4517b dual 64-bit serial-in/ serial-out shift register, wired for
16-shift register, 64-bit shift register
Above we show A CD4517b wired as a 16-bit shift register
for section B. The clock for section B is CLg. The data is
clocked in at CLg. And the data delayed by 16-clocks is
picked of off Qigpg. WEg, the write enable, is grounded.
Above we also show the same CD4517b wired as a 64-bit
shift register for the independent section A. The clock for
section A is CLy. The data enters at CLy. The data delayed by
64-clock pulses is picked up from Q¢yy. WEa, the write
enable for section A, is grounded.
Parallel-in, serial-out shift register
Parallel-in/ serial-out shift registers do everything that the
previous serial-in/ serial-out shift registers do plus input data
to all stages simultaneously. The parallel-in/ serial-out shift
register stores data, shifts it on a clock by clock basis, and
delays it by the number of stages times the clock period. In
addition, parallel-in/ serial-out really means that we can load
data in parallel into all stages before any shifting ever
begins. This is a way to convert data from a paralle/ format
to a serial format. By parallel format we mean that the data
bits are present simultaneously on individual wires, one for
each data bit as shown below. By serial format we mean that
the data bits are presented sequentially in time on a single
wire or circuit as in the case of the "data out" on the block
diagram below.
Da Dz De Dp
data in __,| |_, data out
clock __,}
stage A stage B stage C stage D
Parallel-in, serial-out shift register with 4-stages
Below we take a close look at the internal details of a 3-
stage parallel-in/ serial-out shift register. A stage consists of
a type D Flip-Flop for storage, and an AND-OR selector to
determine whether data will load in parallel, or shift stored
data to the right. In general, these elements will be
replicated for the number of stages required. We show three
stages due to space limitations. Four, eight or sixteen bits is
normal for real parts.
[Ps Q Li Q Ls Q
sea +: “oo o 71> Js ee a pot TS D jo SO
St f He" a |) Lea ch | LU " a
7 Tl ; — {be .
CLK
ft [ f
SHIFT/LD =0
Parallel-in/ serial-out shift register showing parallel load path
Above we show the parallel load path when SHIFT/LD' is
logic low. The upper NAND gates serving D, Dp Dc are
enabled, passing data to the D inputs of type D Flip-Flops Qn
Qzp Dc respectively. At the next positive going clock edge,
the data will be clocked from D to Q of the three FFs. Three
bits of data will load into Qa Qp Dc at the same time.
The type of parallel load just described, where the data
loads on a clock pulse is Known as synchronous load
because the loading of data is synchronized to the clock.
This needs to be differentiated from asynchronous load
where loading is controlled by the preset and clear pins of
the Flip-Flops which does not require the clock. Only one of
these load methods is used within an individual device, the
synchronous load being more common in newer devices.
De
a”
\. Dee re
Dz
it oe a 8 oe ip eee so
SL \n “ ¢ : i < C\ 2 aa ;
a _ a — | io
CLK | —
: it f
|
SHIFT/LD = 1
|
lets
|
Parallel-in/ serial-out shift register showing shitt path
The shift path is shown above when SHIFT/LD' is logic high.
The lower AND gates of the pairs feeding the OR gate are
enabled giving us a shift register connection of SI to Da , Qa
to Dg, Qg to Dc, Qc to SO. Clock pulses will cause data to be
right shifted out to SO on successive pulses.
The waveforms below show both parallel loading of three
bits of data and serial shifting of this data. Parallel data at
D, Dg Dc iS converted to serial data at SO.
ae 171 Fi Fini
SHIFT/LD
data in |
Parallel-in/ serial-out shift register load/shift waveforms
What we previously described with words for parallel loading
and shifting is now set down as waveforms above. As an
example we present 101 to the parallel inputs Dan Dgp Dec.
Next, the SHIFT/LD' goes low enabling loading of data as
opposed to shifting of data. It needs to be low a short time
before and after the clock pulse due to setup and hold
requirements. It is considerably wider than it has to be.
Though, with synchronous logic it is convenient to make it
wide. We could have made the active low SHIFT/LD' almost
two clocks wide, low almost a clock before t; and back high
just before t3. The important factor is that it needs to be low
around clock time t, to enable parallel loading of the data
by the clock.
Note that at t; the data 101 at D, Dg Dc is clocked from D to
Q of the Flip-Flops as shown at Qa, Qzp Qc at time t,. This is
the parallel loading of the data synchronous with the clock.
clock
SHIFT/LD
data in
[ \ |
DB It L
Dg
\
a
Parallel-in/ serial-out shift register load/shift waveforms
Now that the data is loaded, we may shift it provided that
SHIFT/LD' is high to enable shifting, which it is prior to t,. At
tz the data O at Q- is shifted out of SO which is the same as
the Qc waveform. It is either shifted into another integrated
circuit, or lost if there is nothing connected to SO. The data
at Qz, a O is shifted to Qc. The 1 at Qy is shifted into Qg. With
"data in" a0, Q, becomes O. After tz, Qn Qg Qc = 010.
After t3, Qa Qg Qc = 001. This 1, which was originally
present at Q, after t,, is now present at SO and Qc. The last
data bit is shifted out to an external integrated circuit if it
exists. After t, all data from the parallel load is gone. At
clock t; we show the shifting in of a data 1 present on the SI,
serial input.
Why provide SI and SO pins on a shift register? These
connections allow us to cascade shift register stages to
provide large shifters than available in a single IC
(Integrated Circuit) package. They also allow serial
connections to and from other ICs like microprocessors.
Parallel-in/serial-out devices
Let's take a closer look at parallel-in/ serial-out shift registers
available as integrated circuits, courtesy of Texas
Instruments. For complete device data sheets follow these
the links.
e SN74ALS166 parallel-in/ serial-out 8-bit shift register,
synchronous load
[=]
SN74ALS165 parallel-in/ serial-out 8-bit shift register,
asynchronous load
bell
e CD4014B parallel-in/ serial-out 8-bit shift register,
synchronous load
lied
SN74LS647 parallel-in/ serial-out 16-bit shift register,
synchronous load
heal
SN74ALS166 Parallel-in/ serial-out 8-bit shift register
The SN7 4ALS166 shown above is the closest match of an
actual part to the previous parallel-in/ serial out shifter
figures. Let us note the minor changes to our figure above.
First of all, there are 8-stages. We only show three. All 8-
stages are shown on the data sheet available at the link
above. The manufacturer labels the data inputs A, B, C, and
so on to H. The SHIFT/LOAD control is called SH/LD'. It is
abbreviated from our previous terminology, but works the
same: parallel load if low, shift if high. The shift input (serial
data in) is SER on the ALS166 instead of SI. The clock CLK is
controlled by an inhibit signal, CLKINH. If CLKINH is high, the
clock is inhibited, or disabled. Otherwise, this "real part" is
the same as what we have looked at in detail.
SN74ALS166 ANSI Symbol
Above is the ANSI (American National Standards Institute)
symbol for the SN74ALS166 as provided on the data sheet.
Once we know how the part operates, it is convenient to
hide the details within a symbol. There are many general
forms of symbols. The advantage of the ANSI symbol is that
the labels provide hints about how the part operates.
The large notched block at the top of the '74ASL166 is the
control section of the ANSI symbol. There is a reset indicted
by R. There are three control signals: M1 (Shift), M2 (Load),
and C3/1 (arrow) (inhibited clock). The clock has two
functions. First, C3 for shifting parallel data wherever a
prefix of 3 appears. Second, whenever M1 is asserted, as
indicated by the 1 of C3/1 (arrow), the data is shifted as
indicated by the right pointing arrow. The slash (/) isa
separator between these two functions. The 8-shift stages,
as indicated by title SRG8, are identified by the external
inputs A, B, C, to H. The internal 2, 3D indicates that data,
D, is controlled by M2 [Load] and C3 clock. In this case, we
can conclude that the parallel data is loaded synchronously
with the clock C3. The upper stage at A is a wider block than
the others to accommodate the input SER. The legend 1,
3D implies that SER is controlled by M1 [Shift] and C3
clock. Thus, we expect to clock in data at SER when shifting
as opposed to parallel loading.
i> > > >
P}
ANSI gate symbols
The ANSI/IEEE basic gate rectangular symbols are provided
above for comparison to the more familiar shape symbols so
that we may decipher the meaning of the symbology
associated with the CLKINH and CLK pins on the previous
ANSI SN74ALS166 symbol. The CLK and CLKINH feed an OR
gate on the SN74ALS166 ANSI symbol. OR is indicated by
=> on the rectangular inset symbol. The long triangle at the
output indicates a clock. If there was a bubble with the
arrow this would have indicated shift on negative clock edge
(high to low). Since there is no bubble with the clock arrow,
the register shifts on the positive (low to high transition)
clock edge. The long arrow, after the legend C3/1 pointing
right indicates shift right, which is down the symbol.
SN74ALS165 Parallel-in/ serial-out 8-bit shift register,
asynchronous load
Part of the internal logic of the SN7 4ALS165 parallel-in/
serial-out, asynchronous load shift register is reproduced
from the data sheet above. See the link at the beginning of
this section the for the full diagram. We have not looked at
asynchronous loading of data up to this point. First of all, the
loading is accomplished by application of appropriate
signals to the Set (preset) and Reset (clear) inputs of the
Flip-Flops. The upper NAND gates feed the Set pins of the
FFs and also cascades into the lower NAND gate feeding the
Reset pins of the FFs. The lower NAND gate inverts the
signal in going from the Set pin to the Reset pin.
First, SH/LD' must be pulled Low to enable the upper and
lower NAND gates. If SH/LD' were at a logic high instead,
the inverter feeding a logic low to all NAND gates would
force a High out, releasing the "active low" Set and Reset
pins of all FFs. There would be no possibility of loading the
FFs.
With SH/LD' held Low, we can feed, for example, a data 1
to parallel input A, which inverts to a zero at the upper
NAND gate output, setting FF Q, toal. The O at the Set
pin is fed to the lower NAND gate where it is inverted toa 1
, releasing the Reset pin of Qa. Thus, a data A=1 sets
Qa=1. Since none of this required the clock, the loading is
asynchronous with respect to the clock. We use an
asynchronous loading shift register if we cannot wait fora
clock to parallel load data, or if it is inconvenient to
generate a single clock pulse.
The only difference in feeding a data O to parallel input A is
that it inverts to a 1 out of the upper gate releasing Set.
This 1 at Set is inverted to a O at the lower gate, pulling
Reset to a Low, which resets Q,=0.
SN74ALS165 ANSI Symbol
The ANSI symbol for the SN74ALS166 above has two
internal controls Cl [LOAD] and C2 clock from the OR
function of (CLKINH, CLK). SRG8 says 8-stage shifter. The
arrow after C2 indicates shifting right or down. SER input is
a function of the clock as indicated by internal label 2D. The
parallel data inputs A, B, C to H are a function of Cl
[LOAD], indicated by internal label 1D. C1 is asserted when
sh/LD' =0 due to the half-arrow inverter at the input.
Compare this to the control of the parallel data inputs by the
clock of the previous synchronous ANSI SN75ALS166. Note
the differences in the ANSI Data labels.
VSH —2eN M1 [Shift]
a M2 [Load]
CD4014B, synchronous load CD4021B, asynchronous load
CMOS Parallel-in/ serial-out shift registers, 8-bit ANSI symbols
On the CD4014B above, M1 is asserted when LD/SH'= 0.
M2 is asserted when LD/SH'= 1. Clock C3/1 is used for
parallel loading data at 2, 3D when M22 is active as
indicated by the 2,3 prefix labels. Pins P3 to P7 are
understood to have the smae internal 2,3 prefix labels as P2
and P8. At SER, the 1,3D prefix implies that M1 and clock
C3 are necessary to input serial data. Right shifting takes
place when M1 active is as indicated by the 1 in C3/1
arrow.
The CD4021B is a similar part except for asynchronous
parallel loading of data as implied by the lack of any 2 prefix
in the data label 1D for pins P1, P2, to P8. Of course, prefix 2
in label 2D at input SER says that data is clocked into this
pin. The OR gate inset shows that the clock is controlled by
LD/SH'.
T4LS674
SN74LS674, parallel-in/serial-out, synchronous load
The above SN74LS67 4 internal label SRG 16 indicates 16-
bit shift register. The MODE input to the control section at
the top of the symbol is labeled 1,2 M3. Internal M3 isa
function of input MODE and G1 and G2 as indicated by the
1,2 preceding M3. The base label G indicates an AND
function of any such G inputs. Input R/W' is internally
labeled G1/2 EN. This is an enable EN (controlled by Gl
AND G2) for tristate devices used elsewhere in the symbol.
We note that CS' on (pin 1) is internal G2. Chip select CS'
also is ANDed with the input CLK to give internal clock C4.
The bubble within the clock arrow indicates that activity is
on the negative (high to low transition) clock edge. The
Slash (/) is a separator implying two functions for the clock.
Before the slash, C4 indicates control of anything with a
prefix of 4. After the slash, the 3’ (arrow) indicates shifting.
The 3' of C4/3' implies shifting when M3 is de-asserted
(MODE= 0). The long arrow indicates shift right (down).
Moving down below the control section to the data section,
we have external inputs PO-P15, pins (7-11, 13-23). The
prefix 3,4 of internal label 3,4D indicates that M3 and the
clock C4 control loading of parallel data. The D stands for
Data. This label is assumed to apply to all the parallel
inputs, though not explicitly written out. Locate the label
3',4D on the right of the PO (pin7) stage. The
complemented-3 indicates that M3= MODE=0 inputs
(shifts) SER/Q 45 (pin5) at clock time, (4 of 3',4D)
corresponding to clock C4. In other words, with MODE=0,
we shift data into Qg from the serial input (pin 6). All other
stages shift right (down) at clock time.
Moving to the bottom of the symbol, the triangle pointing
right indicates a buffer between Q and the output pin. The
Triangle pointing down indicates a tri-state device. We
previously stated that the tristate is controlled by enable
EN, which is actually Gl AND G2 from the control section. If
R/W= 0, the tri-state is disabled, and we can shift data into
Qo via SER (pin 6), a detail we omitted above. We actually
need MODE=0, R/W'=0, CS'=0
The internal logic of the SN74LS674 and a table
summarizing the operation of the control signals is available
in the link in the bullet list, top of section.
If R/W'=1, the tristate is enabled, Qq5 shifts out SER/Q,5
(pin 6) and recirculates to the Qg stage via the right hand
wire to 3',4D. We have assumed that CS' was low giving us
clock C4/3' and G2 to ENable the tri-state.
Practical applications
An application of a parallel-in/ serial-out shift register is to
read data into a microprocessor.
+5V
Serial data
Clock
Gnd
Keypad Alarm
Alarm with remote keypad
The Alarm above is controlled by a remote keypad. The
alarm box supplies +5V and ground to the remote keypad to
power it. The alarm reads the remote keypad every few tens
of milliseconds by sending shift clocks to the keypad which
returns serial data showing the status of the keys viaa
parallel-in/ serial-out shift register. Thus, we read nine key
switches with four wires. How many wires would be required
if we had to run a circuit for each of the nine keys?
microprocessor
Shitt clock
Load/shitt
. 3 p . Serial data in P&8 PSP4 P?
Reading switches into microprocessor
A practical application of a parallel-in/ serial-out shift
register is to read many switch closures into a
microprocessor on just a few pins. Some low end
microprocessors only have 6-l/O (Input/Output) pins
available on an 8-pin package. Or, we may have used most
of the pins on an 84-pin package. We may want to reduce
the number of wires running around a circuit board,
machine, vehicle, or building. This will increase the
reliability of our system. It has been reported that
manufacturers who have reduced the number of wires in an
automobile produce a more reliable product. In any event,
only three microprocessor pins are required to read in 8-bits
of data from the switches in the figure above.
We have chosen an asynchronous loading device, the
CD4021B because it is easier to control the loading of data
without having to generate a single parallel load clock. The
parallel data inputs of the shift register are pulled up to +5V
with a resistor on each input. If all switches are open, all 1s
will be loaded into the shift register when the
microprocessor moves the LD/SH' line from low to high, then
back low in anticipation of shifting. Any switch closures will
apply logic Os to the corresponding parallel inputs. The data
pattern at P1-P7 will be parallel loaded by the LD/SH'=1
generated by the microprocessor software.
The microprocessor generates shift pulses and reads a data
bit for each of the 8-bits. This process may be performed
totally with software, or larger microprocessors may have
one or more serial interfaces to do the task more quickly
with hardware. With LD/SH'=0, the microprocessor
generates a O to 1 transition on the Shift clock line, then
reads a data bit on the Serial data in line. This is repeated
for all 8-bits.
The SER line of the shift register may be driven by another
identical CD4021B circuit if more switch contacts need to be
read. In which case, the microprocessor generates 16-shift
pulses. More likely, it will be driven by something else
compatible with this serial data format, for example, an
analog to digital converter, a temperature sensor, a
keyboard scanner, a serial read-only memory. As for the
switch closures, they may be limit switches on the carriage
of a machine, an over-temperature sensor, a magnetic reed
switch, a door or window switch, an air or water pressure
switch, or a solid state optical interrupter.
Serial-in, parallel-out shift register
A serial-in/parallel-out shift register is similar to the serial-in/
serial-out shift register in that it shifts data into internal
storage elements and shifts data out at the serial-out, data-
out, pin. It is different in that it makes all the internal stages
available as outputs. Therefore, a serial-in/parallel-out shift
register converts data from serial format to parallel format. If
four data bits are shifted in by four clock pulses via a single
wire at data-in, below, the data becomes available
simultaneously on the four Outputs Q, to Qp after the fourth
clock pulse.
data in __,} |_, data out
clock __,}
stage A stage B stage C stage D
Qu Qs Qc Qp
Serial-in, parallel-out shift register with 4-stages
The practical application of the serial-in/parallel-out shift
register is to convert data from serial format on a single wire
to parallel format on multiple wires. Perhaps, we will
illuminate four LEDs (Light Emitting Diodes) with the four
outputs (Q, Qg Qc Qp ).
Qa Qs Qe Qo
Serial-in/ Parallel out shift register details
The above details of the serial-in/parallel-out shift register
are fairly simple. It looks like a serial-in/ serial-out shift
register with taps added to each stage output. Serial data
shifts in at SI (Serial Input). After a number of clocks equal
to the number of stages, the first data bit in appears at SO
(Qp) in the above figure. In general, there is no SO pin. The
last stage (Qp above) serves as SO and is cascaded to the
next package if it exists.
If a serial-in/parallel-out shift register is so similar to a serial-
in/ serial-out shift register, why do manufacturers bother to
offer both types? Why not just offer the serial-in/parallel-out
shift register? They actually only offer the serial-in/parallel-
out shift register, as long as it has no more than 8-bits. Note
that serial-in/ serial-out shift registers come in bigger than 8-
bit lengths of 18 to to 64-bits. It is not practical to offer a 64-
bit serial-in/parallel-out shift register requiring that many
output pins. See waveforms below for above shift register.
Serial-in/ parallel-out shift register waveforms
The shift register has been cleared prior to any data by
CLR’, an active low signal, which clears all type D Flip-Flops
within the shift register. Note the serial data 1011 pattern
presented at the SI input. This data is synchronized with the
clock CLK. This would be the case if it is being shifted in
from something like another shift register, for example, a
parallel-in/ serial-out shift register (not shown here). On the
first clock at t1, the data 1 at SI is shifted from D to Q of the
first shift register stage. After t2 this first data bit is at Qz.
After t3 it is at Qc. After t4 it is at Qp. Four clock pulses
have shifted the first data bit all the way to the last stage
Qp. The second data bit a O is at Q¢- after the 4th clock. The
third data bit a 1 is at Qg. The fourth data bit another 1 is at
Qa. Thus, the serial data input pattern 1011 is contained in
(Qp Qc Qp Qa). It is now available on the four outputs.
It will available on the four outputs from just after clock tg to
just before t,. This parallel data must be used or stored
between these two times, or it will be lost due to shifting out
the Qp stage on following clocks ts to tg as shown above.
Serial-in/ parallel-out devices
Let's take a closer look at Serial-in/ parallel-out shift
registers available as integrated circuits, courtesy of Texas
Instruments. For complete device data sheets follow the
links.
e SN74ALS164A serial-in/ parallel-out 8-bit shift register
[*]
e SN74AHC594 serial-in/ parallel-out 8-bit shift register
with output register
fal
e SN74AHC595 serial-in/ parallel-out 8-bit shift register
with output register
[=]
e CD4094 serial-in/ parallel-out 8-bit shift register with
output register
baal
[=]
Serial-in/ Parallel out shift register details
The 74ALS164A is almost identical to our prior diagram with
the exception of the two serial inputs A and B. The unused
input should be pulled high to enable the other input. We do
not show all the stages above. However, all the outputs are
shown on the ANSI symbol below, along with the pin
numbers.
CLR —2
aK —§
acl
B = Qu
+. Q
_ Q
6 Qn
0 Q
LL 9,
12 Q
13 Qy
SN74ALS164A ANSI Symbol
The CLK input to the control section of the above ANSI
symbol has two internal functions Cl, control of anything
with a prefix of 1. This would be clocking in of data at 1D.
The second function, the arrow after after the slash (/) is
right (down) shifting of data within the shift register. The
eight outputs are available to the right of the eight registers
below the control section. The first stage is wider than the
others to accommodate the A&B input.
Qe Q VW QA Ao
15 l 2 3 4 $3 6 7
74AHC594 Serial-in/ Parallel out 8-bit shift register with output registers
The above internal logic diagram is adapted from the TI
(Texas Instruments) data sheet for the 7 4AHC594. The type
"D" FFs in the top row comprise a Serial-in/ parallel-out shift
register. This section works like the previously described
devices. The outputs (Qq' Q,' to Q,,' ) of the shift register
half of the device feed the type "D" FFs in the lower half in
parallel. Q,,' (pin 9) is shifted out to any optional cascaded
device package.
A single positive clock edge at RCLK will transfer the data
from D to Q of the lower FFs. All 8-bits transfer in parallel to
the output register (a collection of storage elements). The
purpose of the output register is to maintain a constant data
output while new data is being shifted into the upper shift
register section. This is necessary if the outputs drive relays,
valves, motors, solenoids, horns, or buzzers. This feature
may not be necessary when driving LEDs as long as flicker
during shifting is not a problem.
Note that the 74AHC594 has separate clocks for the shift
register (SRCLK) and the output register ( RCLK). Also, the
shifter may be cleared by SRCLR and, the output register by
RCLR. It desirable to put the outputs in a known state at
power-on, in particular, if driving relays, motors, etc. The
waveforms below illustrate shifting and latching of data.
Waveforms for 74AHC594 serial-in/ parallel-out shift registe rwith latch
The above waveforms show shifting of 4-bits of data into the
first four stages of 74AHC594, then the parallel transfer to
the output register. In actual fact, the 74AHC594 is an 8-bit
shift register, and it would take 8-clocks to shift in 8-bits of
data, which would be the normal mode of operation.
However, the 4-bits we show saves space and adequately
illustrates the operation.
We clear the shift register half a clock prior to tg with
SRCLR'=0. SRCLR' must be released back high prior to
shifting. Just prior to tg the output register is cleared by
RCLR'=0. It, too, is released ( RCLR'=11).
Serial data 1011 is presented at the SI pin between clocks
to and ty. It is shifted in by clocks tj tz tz ty appearing at
internal shift stages Qn' Qp Qc' Qp . This data is present at
these stages between ty and ts. After t, the desired data
(1011) will be unavailable on these internal shifter stages.
Between ty and t, we apply a positive going RCLK
transferring data 1011 to register outputs Q, Qg Qc Qp.
This data will be frozen here as more data (Qs) shifts in
during the succeeding SRCLKs (ts to tg). There will not bea
change in data here until another RCLK is applied.
Qe Qn Qe Qe Qa
15 l 2 3 4 5 6 7
74AHC595 Serial-in/ Parallel out 8-bit shift register with output registers
The 74AHC595 is identical to the '594 except that the
RCLR' is replaced by an OE' enabling a tri-state buffer at
the output of each of the eight output register bits. Though
the output register cannot be cleared, the outputs may be
disconnected by OE'=1. This would allow external pull-up or
pull-down resistors to force any relay, solenoid, or valve
drivers to a known state during a system power-up. Once the
system is powered-up and, say, a microprocessor has shifted
and latched data into the '595, the output enable could be
asserted (OE'=0) to drive the relays, solenoids, and valves
with valid data, but, not before that time.
OE
sRcLR —l0__r.
srctk —11
RCLK
SN74AHC594 ANS! Symbol SN74AHC595 ANSI Symbol
Above are the proposed ANSI symbols for these devices. C3
clocks data into the serial input (external SER) as indicate
by the3 prefix of 2,3D. The arrow after C3/ indicates
shifting right (down) of the shift register, the 8-stages to the
left of the '595symbol below the control section. The 2 prefix
of 2,3D and 2D indicates that these stages can be reset by
R2 (external SRCLR’).
The 1 prefix of 1,4D on the '594 indicates that R1 (external
RCLR’') may reset the output register, which is to the right of
the shift register section. The '595, which has an EN at
external OE' cannot reset the output register. But, the EN
enables tristate (inverted triangle) output buffers. The right
pointing triangle of both the '594 and'595 indicates internal
buffering. Both the '594 and'595 output registers are
clocked by C4 as indicated by 4 of 1,4D and 4D
respectively.
CLOCK
STROBE
OUTPUT 15
ENABLE
SERIAL 2
LN
Q
13 Qs
l Q.
LL Qs
9 Qs SERLAL OUT
10 Qs’ SERIAL OUT
CD4094B/ 74HCT4094 ANSI Symbol
The CD4094B is a 3 to 15Vpc- capable latching shift register
alternative to the previous 74AHC594 devices. CLOCK, Cl,
shifts data in at SERIAL IN as implied by the 1 prefix of 1D.
It is also the clock of the right shifting shift register (left half
of the symbol body) as indicated by the /(right-arrow) of
C1/(arrow) at the CLOCK input.
STROBE, C2 is the clock for the 8-bit output register to the
right of the symbol body. The 2 of 2D indicates that C2 is
the clock for the output register. The inverted triangle in the
output latch indicates that the output is tristated, being
enabled by EN3. The 3 preceding the inverted triangle and
the 3 of EN3 are often omitted, as any enable (EN) is
understood to control the tristate outputs.
Qs and Q,' are non-latched outputs of the shift register
stage. Q, could be cascaded to SERIAL IN of a succeeding
device.
Practical applications
A real-world application of the serial-in/ parallel-out shift
register is to output data from a microprocessor to a remote
panel indicator. Or, another remote output device which
accepts serial format data.
+5V
Serial data
Clock
Gnd
Alarm
Remote display
Alarm with remote keypad and display
The figure "Alarm with remote key pad" is repeated here
from the parallel-in/ serial-out section with the addition of
the remote display. Thus, we can display, for example, the
status of the alarm loops connected to the main alarm box. If
the Alarm detects an open window, it can send serial data to
the remote display to let us know. Both the keypad and the
display would likely be contained within the same remote
enclosure, separate from the main alarm box. However, we
will only look at the display panel in this section.
If the display were on the same board as the Alarm, we could
just run eight wires to the eight LEDs along with two wires
for power and ground. These eight wires are much less
desirable on a long run to a remote panel. Using shift
registers, we only need to run five wires- clock, serial data, a
strobe, power, and ground. If the panel were just a few
inches away from the main board, it might still be desirable
to cut down on the number of wires in a connecting cable to
improve reliability. Also, we sometimes use up most of the
available pins on a microprocessor and need to use Serial
techniques to expand the number of outputs. Some
integrated circuit output devices, such as Digital to Analog
converters contain serial-in/ parallel-out shift registers to
receive data from microprocessors. The techniques
illustrated here are applicable to those parts.
4702 x8
Shitt clock
Latch LEDdata
Serial data out
Output to LEDs from microprocessor
We have chosen the 74AHC594 serial-in/ parallel-out shift
register with output register; though, it requires an extra
pin, RCLK, to parallel load the shifted-in data to the output
pins. This extra pin prevents the outputs from changing
while data is shifting in. This is not much of a problem for
LEDs. But, it would be a problem if driving relays, valves,
motors, etc.
Code executed within the microprocessor would start with 8-
bits of data to be output. One bit would be output on the
"Serial data out" pin, driving SER of the remote 7 4AHC594.
Next, the microprocessor generates a low to high transition
on "Shift clock", driving SRCLK of the '595 shift register.
This positive clock shifts the data bit at SER from "D" to "Q"
of the first shift register stage. This has no effect on the Qa,
LED at this time because of the internal 8-bit output register
between the shift register and the output pins (Qa to Qy).
Finally, "Shift clock" is pulled back low by the
microprocessor. This completes the shifting of one bit into
the '595.
The above procedure is repeated seven more times to
complete the shifting of 8-bits of data from the
microprocessor into the 74AHC594 serial-in/ parallel-out
shift register. To transfer the 8-bits of data within the internal
'595 shift register to the output requires that the
microprocessor generate a low to high transition on RCLK,
the output register clock. This applies new data to the LEDs.
The RCLK needs to be pulled back low in anticipation of the
next 8-bit transfer of data.
The data present at the output of the '595 will remain until
the process in the above two paragraphs is repeated for a
new 8-bits of data. In particular, new data can be shifted into
the '595 internal shift register without affecting the LEDs.
The LEDs will only be updated with new data with the
application of the RCLK rising edge.
What if we need to drive more than eight LEDs? Simply
cascade another 74AHC594 SER pin to the Q,,' of the
existing shifter. Parallel the SRCLK and RCLK pins. The
microprocessor would need to transfer 16-bits of data with
16-clocks before generating an RCLK feeding both devices.
The discrete LED indicators, which we show, could be 7-
segment LEDs. Though, there are LSI (Large Scale
Integration) devices capable of driving several 7-segment
digits. This device accepts data from a microprocessor in a
serial format, driving more LED segments than it has pins by
by multiplexing the LEDs. For example, see link below for
MAX6955
fal
Parallel-in, parallel-out, universal
shift register
The purpose of the parallel-in/ parallel-out shift register is to
take in parallel data, shift it, then output it as shown below.
A universal shift register is a do-everything device in
addition to the parallel-in/ parallel-out function.
Da Ds Me Dp
ee ee
data in __,} |, data out
clock _,}
ode _,)
cael stage A stage B stage C stage D
T
! !
Qs Qs Qe Qp
Parallel-in, parallel-out shift register with 4-stages
Above we apply four bit of data to a parallel-in/ parallel-out
shift register at Dx Dg Dc Dp. The mode control, which may
be multiple inputs, controls parallel loading vs shifting. The
mode control may also control the direction of shifting in
some real devices. The data will be shifted one bit position
for each clock pulse. The shifted data is available at the
outputs Qn Qp Qc Qp. The "data in" and "data out" are
provided for cascading of multiple stages. Though, above,
we can only cascade data for right shifting. We could
accommodate cascading of left-shift data by adding a pair of
left pointing signals, "data in" and "data out", above.
The internal details of a right shifting parallel-in/ parallel-out
shift register are shown below. The tri-state buffers are not
strictly necessary to the parallel-in/ parallel-out shift
register, but are part of the real-world device shown below.
D, Dy De Dy
-—s Lr, Lr, ~ Cascade
5S — a ops ee aN
sR | -—~ og / — = a ek =e ae
1 ae | or ne cee | tr]! ina \ Je
+d / <> ea —i> eq —o> q —i>
as! EI |
LE | A Lt ie a |
[up 7] f i! [ ( c
= J > | [ + > J
= be = oe > Le
x oa | al | |
| On ’ O5 | Qe Op
74LS395 parallel-in/ parallel-out shift register with tri-state output
The 74LS395 so closely matches our concept of a
hypothetical right shifting parallel-in/ parallel-out shift
register that we use an overly simplified version of the data
sheet details above. See the link to the full data sheet more
more details, later in this chapter.
LD/SH' controls the AND-OR multiplexer at the data input to
the FF's. If LD/SH'=1, the upper four AND gates are enabled
allowing application of parallel inputs Da Dg Dc Dp to the
four FF data inputs. Note the inverter bubble at the clock
input of the four FFs. This indicates that the 74LS395 clocks
data on the negative going clock, which is the high to low
transition. The four bits of data will be clocked in parallel
from Dag Dg Dc Dp to Qa Op Qc Qp at the next negative
going clock. In this "real part", OC' must be low if the data
needs to be available at the actual output pins as opposed
to only on the internal FFs.
The previously loaded data may be shifted right by one bit
position if LD/SH'=0 for the succeeding negative going
clock edges. Four clocks would shift the data entirely out of
our 4-bit shift register. The data would be lost unless our
device was cascaded from Qp to SER of another device.
D, Dg De Dp D, Ds De Dp
data lL 1 0 61 data Ll Ll o tL
OC, 8 G& ® OQ. OQ Q@ OQ
lad 1 1 O 1 ae Ta ey
shitt 1 1 0 shitt xX L 1 0
shift * x i
_—
Load and shift Load and 2-shitts
Parallel-in/ parallel-out shift register
Above, a data pattern is presented to inputs Da Dg Dc Dp.
The pattern is loaded to Qa Qg Qc Qp. Then it is shifted one
bit to the right. The incoming data is indicated by X,
meaning the we do no know what it is. If the input (SER)
were grounded, for example, we would know what data (0)
was shifted in. Also shown, is right shifting by two positions,
requiring two clocks.
Shift right
The above figure serves as a reference for the hardware
involved in right shifting of data. It is too simple to even
bother with this figure, except for comparison to more
complex figures to follow.
OQ, Os &
load 1 1 0
shift xX 1 1
——e
Load and right shitt
Right shifting of data is provided above for reference to the
previous right shifter.
Shift left
If we need to shift left, the FFs need to be rewired. Compare
to the previous right shifter. Also, SI and SO have been
reversed. SI shifts to Qc. Qc shifts to Qg. Qzg shifts to Qa. Qa
leaves on the SO connection, where it could cascade to
another shifter SI. This left shift sequence is backwards from
the right shift sequence.
QO, O QO
load 1 1 0
shift 1 0 xX
—_——
Load and lett shit
Above we shift the same data pattern left by one bit.
There is one problem with the "shift left" figure above. There
is no market for it. Nobody manufactures a shift-left part. A
"real device" which shifts one direction can be wired
externally to shift the other direction. Or, should we say
there is no left or right in the context of a device which shifts
in only one direction. However, there is a market for a device
which will shift left or right on command by a control line. Of
course, left and right are valid in that context.
Shift left/ right, right action
What we have above is a hypothetical shift register capable
of shifting either direction under the control of L'/R. It is
setup with L'/R=1 to shift the normal direction, right.
L'/R= 1 enables the multiplexer AND gates labeled R. This
allows data to follow the path illustrated by the arrows, when
a clock is applied. The connection path is the same as
the"too simple" "shift right" figure above.
Data shifts in at SR, to Qa, to Qg, to Qc, where it leaves at
SR cascade. This pin could drive SR of another device to
the right.
What if we change L'/R to L'/R=0?
Shift left/ right register, left action
With L'/R=0O, the multiplexer AND gates labeled L are
enabled, yielding a path, shown by the arrows, the same as
the above "shift left" figure. Data shifts in at SL, to Qc, to
Qs, to Qa, where it leaves at SL cascade. This pin could
drive SL of another device to the left.
The prime virtue of the above two figures illustrating the
"shift left/ right register" is simplicity. The operation of the
left right control L'/R=0 is easy to follow. A commercial part
needs the parallel data loading implied by the section title.
This appears in the figure below.
Shift left/ right/ load
Now that we can shift both left and right via L'/R, let us add
SH/LD', shift/ load, and the AND gates labeled "load" to
provide for parallel loading of data from inputs Dag Dg De.
When SH/LD'=0, AND gates R and L are disabled, AND
gates "load" are enabled to pass data D,g Dg Dc to the FF
data inputs. the next clock CLK will clock the data to Qn Qzp
Qc. As long as the same data is present it will be re-loaded
on succeeding clocks. However, data present for only one
clock will be lost from the outputs when it is no longer
present on the data inputs. One solution is to load the data
on one clock, then proceed to shift on the next four clocks.
This problem is remedied in the 74ALS299 by the addition of
another AND gate to the multiplexer.
If SH/LD' is changed to SH/LD'= 1, the AND gates labeled
"load" are disabled, allowing the left/ right control L'/R to set
the direction of shift on the L or R AND gates. Shifting is as
in the previous figures.
The only thing needed to produce a viable integrated device
is to add the fourth AND gate to the multiplexer as alluded
for the 7 4ALS299. This is shown in the next section for that
part.
Parallel-in/ parallel-out and universal devices
Let's take a closer look at Serial-in/ parallel-out shift
registers available as integrated circuits, courtesy of Texas
Instruments. For complete device data sheets, follow the
links.
e SN74LS395A parallel-in/ parallel-out 4-bit shift register
bal
e SN74ALS299 parallel-in/ parallel-out 8-bit universal shift
register
[=]
CLR SRG+
ac
LD/SH —
CLK
M1L(LOAD)
= M2 (SHIFT)
SN74LS3395A ANS! Symbol
We have already looked at the internal details of the
SN74LS395A, see above previous figure, 74LS395 parallel-
in/ parallel-out shift register with tri-state output. Directly
above is the ANSI symbol for the 74LS395.
Why only 4-bits, as indicated by SRG4 above? Having both
parallel inputs, and parallel outputs, in addition to control
and power pins, does not allow for any more I/O
(Input/Output) bits in a 16-pin DIP (Dual Inline Package).
R indicates that the shift register stages are reset by input
CLR' (active low- inverting half arrow at input) of the control
section at the top of the symbol. OC’, when low, (invert
arrow again) will enable (EN4) the four tristate output
buffers (Qn Qp Qc Qp ) in the data section. Load/shift'
(LD/SH') at pin (7) corresponds to internals M1 (load) and
M2 (shift). Look for prefixes of 1 and 2 in the rest of the
symbol to ascertain what is controlled by these.
The negative edge sensitive clock (indicated by the invert
arrow at pin-10) C3/2has two functions. First, the 3 of C3/2
affects any input having a prefix of 3, say 2,3D or 1,3D in
the data section. This would be parallel load at A, B, C, D
attributed to M1 and C3 for 1,3D. Second, 2 of C3/2-right-
arrow indicates data clocking wherever 2 appears in a prefix
(2,3D at pin-2). Thus we have clocking of data at SER into
Qa with mode 2. The right arrow after C3/2 accounts for
shifting at internal shift register stages Qn Qp Qc Qp.
The right pointing triangles indicate buffering; the inverted
triangle indicates tri-state, controlled by the EN4. Note, all
the 4s in the symbol associated with the EN are frequently
omitted. Stages Qp Q¢ are understood to have the same
attributes as Qp. Qp' cascades to the next package's SER to
the right.
Sl SG OE2 oOEF1 tristate
[x 1
x | x [1 x
i 0 i/o
> 0
shift left
shift right
load
The table above, condensed from the data '299 data sheet,
summarizes the operation of the 74ALS299 universal shift/
storage register. Follow the '299 link above for full details.
The Multiplexer gates R, L, load operate as in the previous
"shift left/ right register" figures. The difference is that the
mode inputs S1 and SO select shift left, shift right, and load
with mode set to $1 SO = to O1, 10, and 1llrespectively as
shown in the table, enabling multiplexer gates L, R, and
load respectively. See table. A minor difference is the
parallel load path from the tri-state outputs. Actually the tri-
state buffers are (must be) disabled by $1 SO = 11 to float
the I/O bus for use as inputs. A bus is a collection of similar
signals. The inputs are applied to A, B through H (same pins
as Qa, Qz, through Q,,) and routed to the load gate in the
multiplexers, and on the the D inputs of the FFs. Data Is
parallel load on a clock pulse.
The one new multiplexer gate is the AND gate labeled hold,
enabled by S1 SO = OO. The hold gate enables a path from
the Q output of the FF back to the hold gate, to the D input
of the same FF. The result is that with mode S1 SO = OO, the
output is continuously re-loaded with each new clock pulse.
Thus, data is held. This is summarized in the table.
To read data from outputs Qa, Qg, through Qy, the tri-state
buffers must be enabled by OE2', OE1l' =00 and mode =S1
SO = 00, O1, or 10. That is, mode is anything except load.
See second table.
bos
=
ee}
a
Goce
>
z in
i
——PCk
R
| r
6-stages H/Q,
‘2 omitted
74ALS299 universal shift/ storage register with tri-state outputs
Right shift data from a package to the left, shifts in on the
SR input. Any data shifted out to the right from stage Q,,
cascades to the right via Q,,'. This output is unaffected by
the tri-state buffers. The shift right sequence for S1 SO = 10
S:
SR > Qa > Qp > Qc > Qn & Qe > OF > QG & Quy (Qu)
Left shift data from a package to the right shifts in on the SL
input. Any data shifted out to the left from stage Qa,
cascades to the left via Q,', also unaffected by the tri-state
buffers. The shift left sequence for S1 SO = OL is:
(Qy') Qa < Qp < Qc < Qp < Qe < OF < QG < Quy (Qs,')
Shifting may take place with the tri-state buffers disabled by
one of OE2' or OEl' = 1. Though, the register contents
outputs will not be accessible. See table.
SN74ALS299 ANSI Symbol
The "clean" ANSI symbol for the SN7 4ALS299 parallel-in/
parallel-out 8-bit universal shift register with tri-state output
is shown for reference above.
SRG8 uec ~ mode? function SO St M
ENL3=mode3 & OBL & OE2 [hold
enable tn-state buffers
CLR
OEL
0H = , 2 shift right
shift lett
SO
SL
CLK prefix 3,4D implies mode-3 parallel
bad by C4
SR
4as a prefix (4D) implies clocking of
data by C4, as opposed to shifting
Z5, Z6 to Z12 are tri-state outputs of the-
shift register stages associated with the
'© pins A’/Qa, B/Qp, to Q'Qy as implied
by prefixes 5,13; 6,13; to 12,13 respectively.
data, equivalent to the input (no arrow)
and output (single arrow).
[> is buffer
Vv is tr-state
SN74ALS299 ANSI Symbol, annotated
The annotated version of the ANSI symbol is shown to clarify
the terminology contained therein. Note that the ANSI mode
(SO S1) is reversed from the order (S1 SO) used in the
previous table. That reverses the decimal mode numbers (1
& 2). In any event, we are in complete agreement with the
official data sheet, copying this inconsistency.
Practical applications
The Alarm with remote keypad block diagram is repeated
below. Previously, we built the keypad reader and the
remote display as separate units. Now we will combine both
the keypad and display into a single unit using a universal
shift register. Though separate in the diagram, the Keypad
and Display are both contained within the same remote
enclosure.
+5V
Serial data
Clock
Gnd
Alarm
Remote display
Alarm with remote keypad and display
We will parallel load the keyboard data into the shift register
on a single clock pulse, then shift it out to the main alarm
box. At the same time , we will shift LED data from the main
alarm to the remote shift register to illuminate the LEDs. We
will be simultaneously shifting keyboard data out and LED
data into the shift register.
fe
PA
5
tig
o
aol D>
z6
|_|
= Pal
aah |
2a== =
T-—)— Ra
O © O © SL 18 17 he
- _
6 ¢ ¢ L_™= 750 hoae
18 1 16 15 l4 GB 2 fT Oo O {hold
et a Oo 1 dL
1 oO IR
LN ALZNALNALN a Bo
/\ /\; AS 1 \ T4ALSS41
3s ss
74ALS8299 universal shift register reads switches, drives LEDs
Eight LEDs and current limiting resistors are connected to
the eight I/O pins of the 74ALS299 universal shift register.
The LEDS can only be driven during Mode 3 with S1=0
SO= 0. The OE1' and OE2' tristate enables are grounded to
permenantly enable the tristate outputs during modes O, 1,
2. That will cause the LEDS to light (flicker) during shifting.
If this were a problem the EN1' and EN2' could be
ungrounded and paralleled with S1 and SO respectively to
only enable the tristate buffers and light the LEDS during
hold, mode 3. Let's keep it simple for this example.
During parallel loading, SO=1 inverted to a O, enables the
octal tristate buffers to ground the switch wipers. The upper,
open, switch contacts are pulled up to logic high by the
resister-LED combination at the eight inputs. Any switch
closure will short the input low. We parallel load the switch
data into the '299 at clock tO when both SO and S1 are
high. See waveforms below.
tO t1 tf t t4 th t6 t7 t8 t9 t10 t11
S150 pode] §
z
1a) 1 IL S
1 O {R
ae El m7 si shift right ————————-+— hold —
load
Load (tO) & shift (t1-t8) switches out of Q,’, shift LED data into SR
Once SO goes low, eight clocks (tO tot8) shift switch closure
data out of the '299 via the Q,. pin. At the same time, new
LED data is shifted in at SR of the 299 by the same eight
clocks. The LED data replaces the switch closure data as
shifting proceeds.
After the 8th shift clock, t8, S1 goes low to yield hold mode
(S1 SO = OO). The data in the shift register remains the
same even if there are more clocks, for example, T9, t10,
etc. Where do the waveforms come from? They could be
generated by a microprocessor if the clock rate were not
over 100 kHz, in which case, it would be inconvenient to
generate any clocks after t8. If the clock was in the
megahertz range, the clock would run continuously. The
clock, $1 and SO would be generated by digital logic, not
shown here.
Ring counters
If the output of a shift register is fed back to the input. a ring
counter results. The data pattern contained within the shift
register will recirculate as long as clock pulses are applied.
For example, the data pattern will repeat every four clock
pulses in the figure below. However, we must load a data
pattern. All O's or all 1's doesn't count. Is a continuous logic
level from such a condition useful?
data out
data in
clock __,} ~
Qo
stage A stage B stage C stage D
Ring Counter, shift register output fed back to input
We make provisions for loading data into the parallel-in/
serial-out shift register configured as a ring counter below.
Any random pattern may be loaded. The most generally
useful pattern is a single 1.
data in data out
clock
stage A stage B stage C stage D
Parallel-in, serial-out shift register configured as
a ring counter
Loading binary 1000 into the ring counter, above, prior to
shifting yields a viewable pattern. The data pattern for a
single stage repeats every four clock pulses in our 4-stage
example. The waveforms for all four stages look the same,
except for the one clock time delay from one stage to the
next. See figure below.
t ob tt t ts ty
cock Lf LF LFLILILIU UU UU UU
sunt |__|
Q, ji | ; | ; |
ed oe i a re en ||
Q 0 ay es es ; [|
QO, 0 | | | |
Load 1000 into 4-stage ring counter and shift
The circuit above is a divide by 4 counter. Comparing the
clock input to any one of the outputs, shows a frequency
ratio of 4:1. How may stages would we need for a divide by
10 ring counter? Ten stages would recirculate the 1 every 10
clock pulses.
SET
O|
CLOCK
Set one stage. clear three stages
An alternate method of initializing the ring counter to 1000
is shown above. The shift waveforms are identical to those
above, repeating every fourth clock pulse. The requirement
for initialization is a disadvantage of the ring counter over a
conventional counter. At a minimum, it must be initialized at
power-up since there is no way to predict what state flip-
flops will power up in. In theory, initialization should never
be required again. In actual practice, the flip-flops could
eventually be corrupted by noise, destroying the data
pattern. A "self correcting" counter, like a conventional
synchronous binary counter would be more reliable.
The above binary synchronous counter needs only two
stages, but requires decoder gates. The ring counter had
more stages, but was self decoding, saving the decode gates
above. Another disadvantage of the ring counter is that it is
not "self starting". If we need the decoded outputs, the ring
counter looks attractive, in particular, if most of the logic is
in a single shift register package. If not, the conventional
binary counter is less complex without the decoder.
Compare to binary synchronous counter with decode. waveforms
The waveforms decoded from the synchronous binary
counter are identical to the previous ring counter
waveforms. The counter sequence is (Qa Qg) = (00 01 10
11).
Johnson counters
The switch-tail ring counter, also know as the Johnson
counter, overcomes some of the limitations of the ring
counter. Like a ring counter a Johnson counter is a shift
register fed back on its' self. It requires half the stages of a
comparable ring counter for a given division ratio. If the
complement output of a ring counter is fed back to the input
instead of the true output, a Johnson counter results. The
difference between a ring counter and a Johnson counter is
which output of the last stage is fed back (Q or Q'). Carefully
compare the feedback connection below to the previous ring
counter.
~— . D Q D ; |
oa0d0 | at |
noe Cc Cc Cl. ch.
110aa — — > —>
5 a Soe Fe
4 4.4 2 a | fe} q
go1ii1 |
aoo11)| RESET I I |
qaqa. Y ; 4 i}
= CLOCK
Johnson counter (note the Qp to D, feedback connection)
This "reversed" feedback connection has a profound effect
upon the behavior of the otherwise similar circuits.
Recirculating a single 1 around a ring counter divides the
input clock by a factor equal to the number of stages.
Whereas, a Johnson counter divides by a factor equal to
twice the number of stages. For example, a 4-stage ring
counter divides by 4. A 4-stage Johnson counter divides by
8.
Start a Johnson counter by clearing all stages to Os before
the first clock. This is often done at power-up time. Referring
to the figure below, the first clock shifts three Os from ( Qa
Qs Q,) to the right into (Qg Qc Qp). The 1 at Qp' (the
complement of Q) is shifted back into Q,. Thus, we start
shifting 1s to the right, replacing the Os. Where a ring
counter recirculated a single 1, the 4-stage Johnson counter
recirculates four Os then four Ls for an 8-bit pattern, then
repeats.
clock
RESET | |
a, | | |
Qn
Qc i J
‘ | r |
Four stage Johnson counter waveforms
The above waveforms illustrates that multi-phase square
waves are generated by a Johnson counter. The 4-stage unit
above generates four overlapping phases of 50% duty cycle.
How many stages would be required to generate a set of
three phase waveforms? For example, a three stage Johnson
counter, driven by a 360 Hertz clock would generate three
120° phased square waves at 60 Hertz.
The outputs of the flop-flops in a Johnson counter are easy to
decode to a single state. Below for example, the eight states
of a 4-stage Johnson counter are decoded by no more than a
two input gate for each of the states. In our example, eight
of the two input gates decode the states for our example
Johnson counter.
OoOoOrRrFFrFFO
Johnson counter with decoder (CD4022B)
No matter how long the Johnson counter, only 2-input
decoder gates are needed. Note, we could have used
uninverted inputs to the AND gates by changing the gate
inputs from true to inverted at the FFs, Q to Q’, (and vice
versa). However, we are trying to make the diagram above
match the data sheet for the CD4022B, as closely as
practical.
Go=Q4 Qn
G,=Q,.25 |
Gr=QgQ¢
Gy=QQp | |
G,=Q, Qn
Gs=Q, Qs | |
G.=Q,2-
G7=Q-Qp
Four stage (8-state) Johnson counter decoder waveforms
Above, our four phased square waves Qa to Qp are decoded
to eight signals (Gg to Gz) active during one clock period out
of a complete 8-clock cycle. For example, Gg is active high
when both Qa and Qp are low. Thus, pairs of the various
register outputs define each of the eight states of our
Johnson counter example.
3 10
2 4
AS pe ‘ )
CLOCK : C4 . .
a “oF?
13
CLOCK TY YT YT ane
ENABLE
4G
Qa Qp
D Q D Q D Q D Q
; . > ' —> 5 a. a) > 4
Cc C C Cc
b> |p
(0) (0) [e] Q
be
RESET
“~Y ~\
Uy
o
»
fe)
w
fe)
ny
&P
D
aAnAnnaAnH
wee
woe ww
1 & G3
5 7 12
°
°
-
FRE
ap
NOR gate unused state detector: Q, Q, Q. = 010 forces the 1 to a@
CD4022B modulo-8 Johnson counter with unused state detector
Above is the more complete internal diagram of the
CD4022B Johnson counter. See the manufacturers’ data
sheet for minor details omitted. The major new addition to
the diagram as compared to previous figures is the
disallowed state detector composed of the two NOR gates.
Take a look at the inset state table. There are 8-permissible
states as listed in the table. Since our shifter has four flip-
flops, there are a total of 16-states, of which there are 8-
disallowed states. That would be the ones not listed in the
table.
In theory, we will not get into any of the disallowed states as
long as the shift register is RESET before first use. However,
in the "real world" after many days of continuous operation
due to unforeseen noise, power line disturbances, near
lightning strikes, etc, the Johnson counter could get into one
of the disallowed states. For high reliability applications, we
need to plan for this slim possibility. More serious is the case
where the circuit is not cleared at power-up. In this case
there is no way to know which of the 16-states the circuit
will power up in. Once in a disallowed state, the Johnson
counter will not return to any of the permissible states
without intervention. That is the purpose of the NOR gates.
Examine the table for the sequence (Q,q Qg Q-) = (010).
Nowhere does this sequence appear in the table of allowed
states. Therefore (O10) is disallowed. It should never occur.
If it does, the Johnson counter is in a disallowed state, which
it needs to exit to any allowed state. Suppose that (Q, Qp
Q-) = (010). The second NOR gate will replace Qg = 1 with
a 0 at the D input to FF Q¢. In other words, the offending
010 is replaced by 000. And 000, which does appear in the
table, will be shifted right. There are may triple-O sequences
in the table. This is how the NOR gates get the Johnson
counter out of a disallowed state to an allowed state.
Not all disallowed states contain a 010 sequence. However,
after a few clocks, this sequence will appear so that any
disallowed states will eventually be escaped. If the circuit is
powered-up without a RESET, the outputs will be
unpredictable for a few clocks until an allowed state is
reached. If this is a problem for a particular application, be
sure to RESET on power-up.
Johnson counter devices
A pair of integrated circuit Johnson counter devices with the
output states decoded is available. We have already looked
at the CD4017 internal logic in the discussion of Johnson
counters. The 4000 series devices can operate from 3V to
15V power supplies. The the 74HC' part, designed for a TTL
compatiblity, can operate from a 2V to 6V supply, count
faster, and has greater output drive capability. For complete
device data sheets, follow the links.
e CD4017 Johnson counter with 10 decoded outputs
CD4022 Johnson counter with 8 decoded outputs
baal
e 74HC4017 Johnson counter, 10 decoded outputs
fad
CTR DLV Lay :
DEC 0 : Q
L ; Qa CTR DIV 8
= a OCT: .
CEN 13 nt - Qa OCTAL i Qo
: Y& 7
14 : LO = 3 bas
cK —— JI + Qa 13 2 Q
Ll = CKEN ——«& 7 ‘
LS 5 Q; P 3 Q
CLR ———) CT=0 5 a 14 ? LL 7
6 Qs CK I 4 : Qa
= 6 ~ ~
Q: 15 5 Qs
9 - CLR CT=0 5 7
8 Qs 6 Qe
LL i a 10
9 Qo 7 Q,;
CDe«s 1? co cD<4 = co
CD4017B, 74HC4017 CD4022B
The ANSI symbols for the modulo-10 (divide by 10) and
modulo-8 Johnson counters are shown above. The symbol
takes on the characteristics of a counter rather than a shift
register derivative, which it is. Waveforms for the CD4022
modulo-8 and operation were shown previously. The
CD4017B/ 74HC4017 decade counter is a 5-stage Johnson
counter with ten decoded outputs. The operation and
waveforms are similar to the CD4017. In fact, the CD4017
and CD4022 are both detailed on the same data sheet. See
above links. The 74HC4017 is a more modern version of the
decade counter.
These devices are used where decoded outputs are needed
instead of the binary or BCD (Binary Coded Decimal) outputs
found on normal counters. By decoded, we mean one line
out of the ten lines is active at a time for the '4017 in place
of the four bit BCD code out of conventional counters. See
previous waveforms for 1-of-8 decoding for the '4022 Octal
Johnson counter.
Practical applications
sv
ul
16) = i
aver
crow | smltl]/ [yy S "
= Ras20Nx10 yy
—— “- |e
Dy
hap
1.46
(R, +2R,)C
T4HC+4017 at
Decoded ring counter drives walking LED
The above Johnson counter shifts a lighted LED each fifth of
a second around the ring of ten. Note that the 74HC4017 is
used instead of the '40017 because the former part has
more current drive capability. From the data sheet, (at the
link above) operating at V¢c= 5V, the Voy= 4.6V at 4ma. In
other words, the outputs can supply 4 ma at 4.6 V to drive
the LEDs. Keep in mind that LEDs are normally driven with
10 to 20 ma of current. Though, they are visible down to 1
ma. This simple circuit illustrates an application of the
'HC4017. Need a bright display for an exhibit? Then, use
inverting buffers to drive the cathodes of the LEDs pulled up
to the power supply by lower value anode resistors.
The 555 timer, serving as an astable multivibrator,
generates a clock frequency determined by R, R> Cy. This
drives the 74HC4017 a step per clock as indicated by a
single LED illuminated on the ring. Note, if the 555 does not
reliably drive the clock pin of the '4015, run it through a
single buffer stage between the 555 and the '4017.A
variable Ry could change the step rate. The value of
decoupling capacitor C, is not critical. A similar capacitor
should be applied across the power and ground pins of the
‘4017.
CLOCK
Disallowed state
CLOCK
HTT
Three phase square/ sine wave generator.
The Johnson counter above generates 3-phase square
waves, phased 60° apart with respect to (Qa Qgp Qc).
However, we need 120° phased waveforms of power
applications (see Volume II, AC). Choosing Py=Qan P2=Qc
P3=Q,' yields the 120° phasing desired. See figure below. If
these (P 1 P> P3) are low-pass filtered to sine waves and
amplified, this could be the beginnings of a 3-phase power
supply. For example, do you need to drive a small 3-phase
400 Hz aircraft motor? Then, feed 6x 400Hz to the above
circuit CLOCK. Note that all these waveforms are 50% duty
cycle.
clock |
Qa __| EE
Qp —__i_] po =
Qe ae ee
P.=Q, ——A Loi ————}
P=Qc |______ SS
P.=Qp’
3-stage Johnson counter generates 3-0 waveform.
The circuit below produces 3-phase nonoverlapping, less
than 50% duty cycle, waveforms for driving 3-phase stepper
motors.
CLOCK
Vuotor
Disallowed state
P >=Q:Qp
P 1=QnQc
Po= QQ
u3Cc LN2003 =
3-stage (6-state) Johnson counter decoded for 3-@ stepper
motor.
Above we decode the overlapping outputs Qa Qp Q¢ to non-
overlapping outputs Pg P, Pz as shown below. These
waveforms drive a 3-phase stepper motor after suitable
amplification from the milliamp level to the fractional amp
level using the ULN2003 drivers shown above, or the
discrete component Darlington pair driver shown in the
circuit which follow. Not counting the motor driver, this
circuit requires three IC (Integrated Circuit) packages: two
dual type "D" FF packages and a quad NAND gate.
to ty ts t, 5 ts t ts
4 ty t
dock FLP LP LE LE LELILILILU Lu
PU@ Lo LCS
Pe@Qe PL CCL
Pewee [| Loi Jf 2. Li _
3-stage Johnson counter generates 3-0 stepper
waveform.
0
L
2
3
4
5
6
7
8
9
CD4017B, 74HC4017
Johnson sequence terminated early by reset at Qs, which is high.
for nano seconds
A single CD4017, above, generates the required 3-phase
stepper waveforms in the circuit above by clearing the
Johnson counter at count 3. Count 3 persists for less than a
microsecond before it clears its' self. The other counts
(Qo= Go Qy= Gy Qa= G2) remain for a full clock period each.
The Darlington bipolar transistor drivers shown above are a
substitute for the internal circuitry of the ULN2003. The
design of drivers is beyond the scope of this digital
electronics chapter. Either driver may be used with either
waveform generator circuit.
clock
Qo=Gp=Q,. Qn r 7 ay +L : Td
QaG-ae —! Leff Le f Le J Le.
QiHG=QpQc Po | z= i
G.=Q-Qp
CD4017B 5-stage (10-state) Johnson counter resetting
at Q, Qp Q,=100 generates 3- stepper waveform.
The above waceforms make the most sense in the context of
the internal logic of the CD4017 shown earlier in this
section. Though, the AND gating equations for the internal
decoder are shown. The signals Qqg Qg Q¢ are Johnson
counter direct shift register outputs not available on pin-
outs. The Qp waveform shows resetting of the '4017 every
three clocks. Qg Qq Qo, etc. are decoded outputs which
actually are available at output pins.
0
l
3
4
5
6
7
8
9
CD4017B, 74HC4017
Johnson counter drives unipolar stepper motor.
Above we generate waveforms for driving a unipolar stepper
motor, which only requires one polarity of driving signal.
That is, we do not have to reverse the polarity of the drive to
the windings. This simplifies the power driver between the
‘4017 and the motor. Darlington pairs from a prior diagram
may be substituted for the ULN3003.
clock |
Qo | | [ | | |
Q [| _ J [|
Q je “i . | =
Q | | |
Johnson counter unipolar stepper motor waveforms.
Once again, the CD4017B generates the required waveforms
with a reset after the teminal count. The decoded outputs Qy
Q, Q> Q3 sucessively drive the stepper motor windings, with
Q, reseting the counter at the end of each group of four
pulses.
references
DataSheetCatalog.com http://www.datasheetcatalog.com/
http://www.st.com/stonline/psearch/index.htm select
standard logics
http://www.st.com/stonline/books/pdf/docs/2069.pdf
http://www.ti.com/ (Products, Logic, Product Tree)
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
Lessons In Electric Circuits
-- Volume IV
Chapter 13
DIGITAL-ANALOG
CONVERSION
e Introduction
e The R/2"R DAC
The R/2R DAC
Flash ADC
Digital ramp ADC
Successive approximation ADC
Tracking ADC
Slope (integrating) ADC
Delta-Sigma (AZ) ADC
Practical considerations of ADC circuits
Introduction
Connecting digital circuitry to sensor devices is simple if the
sensor devices are inherently digital themselves. Switches,
relays, and encoders are easily interfaced with gate circuits
due to the on/off nature of their signals. However, when
analog devices are involved, interfacing becomes much
more complex. What is needed is a way to electronically
translate analog signals into digital (binary) quantities, and
vice versa. An analog-to-digital converter, or ADC, performs
the former task while a digital-to-analog converter, or DAC,
performs the latter.
An ADC inputs an analog electrical signal such as voltage or
current and outputs a binary number. In block diagram form,
it can be represented as such:
Vdd
Analog ADC Binary
signal output
input -_
A DAC, on the other hand, inputs a binary number and
outputs an analog voltage or current signal. In block
diagram form, it looks like this:
Vdd
Binary An ao
inpui DAC signa
_ {output
Together, they are often used in digital systems to provide
complete interface with analog sensors and output devices
for control systems such as those used in automotive engine
controls:
Digital control system with
analog I/O
anal Control
eins computer
signal
input 7
It is much easier to convert a digital signal into an analog
signal than it is to do the reverse. Therefore, we will begin
with DAC circuitry and then move to ADC circuitry.
The R/2"R DAC
This DAC circuit, otherwise known as the binary-weighted-
input DAC, is a variation on the inverting summer op-amp
circuit. If you recall, the classic inverting summer circuit is
an operational amplifier using negative feedback for
controlled gain, with several voltage inputs and one voltage
output. The output voltage is the inverted (opposite
polarity) sum of all input voltages:
Inverting summer circuit
—+— 1,+L+1
out
Vout =-(V, + V>+ V5)
out — ~
For a simple inverting summer circuit, all resistors must be
of equal value. If any of the input resistors were different,
the input voltages would have different degrees of effect on
the output, and the output voltage would not be a true sum.
Let's consider, however, intentionally setting the input
resistors at different values. Suppose we were to set the
input resistor values at multiple powers of two: R, 2R, and
4R, instead of all the same value R:
R ~«— ],
Vv,
Starting from V, and going through V3, this would give each
input voltage exactly half the effect on the output as the
voltage before it. In other words, input voltage V; has a1:1
effect on the output voltage (gain of 1), while input voltage
V> has half that much effect on the output (a gain of 1/2),
and V3 half of that (a gain of 1/4). These ratios are were not
arbitrarily chosen: they are the same ratios corresponding to
place weights in the binary numeration system. If we drive
the inputs of this circuit with digital gates so that each input
is either O volts or full supply voltage, the output voltage
will be an analog representation of the binary value of these
three bits.
MSB
Bina
inpu
LSB
If we chart the output voltages for all eight combinations of
binary bits (000 through 111) input to this circuit, we will
get the following progression of voltages:
| O11 | -3.75 V |
| 10 | 5.00V |
[ a0. |) ~@25v |
( Vie yo 7.50V |
i ae? a, 8.75V |
Note that with each step in the binary count sequence, there
results a 1.25 volt change in the output. This circuit is very
easy to simulate using SPICE. In the following simulation, |
set up the DAC circuit with a binary input of 110 (note the
first node numbers for resistors Rj, Ro, and R3: a node
number of "1" connects it to the positive side of a 5 volt
battery, and a node number of "0" connects it to ground).
The output voltage appears on node 6 in the simulation:
Rteedbk 6
binary-weighted dac
vl 10dc5
rbogus 1 0 99k
rl 15 1k
r2 15 2k
r3 05 4k
rfeedbk 5 6 1k
el 6 0 5 0 999k
.end
node voltage node voltage node voltage
(1) 5.0000 (5) 0.0000 (6) -7.5000
We can adjust resistors values in this circuit to obtain output
voltages directly corresponding to the binary input. For
example, by making the feedback resistor 800 Q instead of 1
kQ, the DAC will output -1 volt for the binary input 001, -4
volts for the binary input 100, -7 volts for the binary input
111, and so on.
(with feedback resistor set at 800 ohms)
| Binary | Output voltage |
| 00 | | 0.00V |
| oo. | | -1.00V |
| 0 | 2.00V |
| ou | 3.00V |
| 100 | | -4.00V |
| 1. | 5.00V |
If we wish to expand the resolution of this DAC (add more
bits to the input), all we need to do is add more input
resistors, holding to the same power-of-two sequence of
values:
6-bit binary-weighted DAC
R
MSB
Rpedback
Bina
inpu
LSB
It should be noted that all logic gates must output exactly
the same voltages when in the "high" state. If one gate is
outputting +5.02 volts for a "high" while another is
outputting only +4.86 volts, the analog output of the DAC
will be adversely affected. Likewise, all "low" voltage levels
should be identical between gates, ideally 0.00 volts exactly.
It is recommended that CMOS output gates are used, and
that input/feedback resistor values are chosen so as to
minimize the amount of current each gate has to source or
sink.
The R/2R DAC
An alternative to the binary-weighted-input DAC is the so-
called R/2R DAC, which uses fewer unique resistor values. A
disadvantage of the former DAC design was its requirement
of several different precise input resistor values: one unique
value per binary input bit. Manufacture may be simplified if
there are fewer different resistor values to purchase, stock,
and sort prior to assembly.
Of course, we could take our last DAC circuit and modify it to
use a Single input resistance value, by connecting multiple
resistors together in series:
MSB
Bina
iINpu
LSB
Unfortunately, this approach merely substitutes one type of
complexity for another: volume of components over
diversity of component values. There is, however, a more
efficient design methodology.
By constructing a different kind of resistor network on the
input of our summing circuit, we can achieve the same kind
of binary weighting with only two kinds of resistor values,
and with only a modest increase in resistor count. This
"ladder" network looks like this:
R/2R "ladder" DAC
MSB
Bina
iINpu
LSB
Mathematically analyzing this ladder network is a bit more
complex than for the previous circuit, where each input
resistor provided an easily-calculated gain for that bit. For
those who are interested in pursuing the intricacies of this
circuit further, you may opt to use Thevenin's theorem for
each binary input (remember to consider the effects of the
virtual ground), and/or use a simulation program like SPICE
to determine circuit response. Either way, you should obtain
the following table of figures:
| Binary | Output voltage |
| 006 | 0.00V |
| ool | -1.25V |
| oo | 2.50V |
| 101 | -6.25 V |
| 110 | -7.50 V |
| 111 | -8.75 V |
As was the case with the binary-weighted DAC design, we
can modify the value of the feedback resistor to obtain any
"span" desired. For example, if we're using +5 volts for a
"high" voltage level and 0 volts for a "low" voltage level, we
can obtain an analog output directly corresponding to the
binary input (O11 = -3 volts, 101 = -5 volts, 111 = -7 volts,
etc.) by using a feedback resistance with a value of 1.6R
instead of 2R.
Flash ADC
Also called the para/le/ A/D converter, this circuit is the
simplest to understand. It is formed of a series of
comparators, each one comparing the input signal toa
unique reference voltage. The comparator outputs connect
to the inputs of a priority encoder circuit, which then
produces a binary output. The following illustration shows a
3-bit flash ADC circuit:
8-line to
3-line
priority
encoder
Binary output
Vref iS a Stable reference voltage provided by a precision
voltage regulator as part of the converter circuit, not shown
in the schematic. As the analog input voltage exceeds the
reference voltage at each comparator, the comparator
outputs will sequentially saturate to a high state. The
priority encoder generates a binary number based on the
highest-order active input, ignoring all other active inputs.
When operated, the flash ADC produces an output that looks
something like this:
Analog
input
ia
Time —~
Digital
output
Time —>
For this particular application, a regular priority encoder
with all its inherent complexity isn't necessary. Due to the
nature of the sequential comparator output states (each
comparator saturating "high" in sequence from lowest to
highest), the same "highest-order-input selection" effect
may be realized through a set of Exclusive-OR gates,
allowing the use of a simpler, non-priority encoder:
8-line to
3-line
encoder
Binary output
And, of course, the encoder circuit itself can be made from a
matrix of diodes, demonstrating just how simply this
converter design may be constructed:
Binary output
Pulldown
resistors
Not only is the flash converter the simplest in terms of
operational theory, but it is the most efficient of the ADC
technologies in terms of speed, being limited only in
comparator and gate propagation delays. Unfortunately, it is
the most component-intensive for any given number of
output bits. This three-bit flash ADC requires seven
comparators. A four-bit version would require 15
comparators. With each additional output bit, the number of
required comparators doubles. Considering that eight bits is
generally considered the minimum necessary for any
practical ADC (255 comparators needed!), the flash
methodology quickly shows its weakness.
An additional advantage of the flash converter, often
overlooked, is the ability for it to produce a non-linear
output. With equal-value resistors in the reference voltage
divider network, each successive binary count represents
the same amount of analog signal increase, providing a
proportional response. For special applications, however, the
resistor values in the divider network may be made non-
equal. This gives the ADC a custom, nonlinear response to
the analog input signal. No other ADC design is able to grant
this signal-conditioning behavior with just a few component
value changes.
Digital ramp ADC
Also known as the stairstep-ramp, or simply counter A/D
converter, this is also fairly easy to understand but
unfortunately suffers from several limitations.
The basic idea is to connect the output of a free-running
binary counter to the input of a DAC, then compare the
analog output of the DAC with the analog input signal to be
digitized and use the comparator's output to tell the counter
when to stop counting and reset. The following schematic
shows the basic idea:
aa
ees of ot
CTR [softer
[tte
Bina
output
As the counter counts up with each clock pulse, the DAC
outputs a slightly higher (more positive) voltage. This
voltage is compared against the input voltage by the
comparator. If the input voltage is greater than the DAC
output, the comparator's output will be high and the counter
will continue counting normally. Eventually, though, the DAC
output will exceed the input voltage, causing the
comparator's output to go low. This will cause two things to
happen: first, the high-to-low transition of the comparator's
output will cause the shift register to "load" whatever binary
count is being output by the counter, thus updating the ADC
circuit's output; secondly, the counter will receive a low
signal on the active-low LOAD input, causing it to reset to
00000000 on the next clock pulse.
The effect of this circuit is to produce a DAC output that
ramps up to whatever level the analog input signal is at,
output the binary number corresponding to that level, and
start over again. Plotted over time, it looks like this:
Analog
input
/ View
Time —~
Digital
Time —
Note how the time between updates (new digital output
values) changes depending on how high the input voltage
is. For low signal levels, the updates are rather close-spaced.
For higher signal levels, they are spaced further apart in
time:
Digital
i fata eee —
For many ADC applications, this variation in update
frequency (Sample time) would not be acceptable. This, and
the fact that the circuit's need to count all the way from 0 at
the beginning of each count cycle makes for relatively slow
sampling of the analog signal, places the digital-ramp ADC
at a disadvantage to other counter strategies.
Successive approximation ADC
One method of addressing the digital ramp ADC's
shortcomings is the so-called successive-approximation
ADC. The only change in this design is a very special
counter circuit Known as a successive-approximation
register. Instead of counting up in binary sequence, this
register counts by trying all values of bits starting with the
most-significant bit and finishing at the least-significant bit.
Throughout the count process, the register monitors the
comparator's output to see if the binary count is less than or
greater than the analog signal input, adjusting the bit
values accordingly. The way the register counts is identical
to the "trial-and-fit" method of decimal-to-binary conversion,
whereby different values of bits are tried from MSB to LSB to
get a binary number that equals the original decimal
number. The advantage to this counting strategy is much
faster results: the DAC output converges on the analog
signal input in much larger steps than with the O-to-full
count sequence of a regular counter.
Without showing the inner workings of the successive-
approximation register (SAR), the circuit looks like this:
&
———_——__
SAR —t
Eee
>/<
tee
ny a A a a aT 4
mn aaa
Bina
output
It should be noted that the SAR is generally capable of
outputting the binary number in seria/ (one bit at a time)
format, thus eliminating the need for a shift register. Plotted
over time, the operation of a successive-approximation ADC
looks like this:
Analog
input
il
Time —>
Digital
— eS ee
Time —>
Note how the updates for this ADC occur at regular intervals,
unlike the digital ramp ADC circuit.
Tracking ADC
A third variation on the counter-DAC-based converter theme
is, in my estimation, the most elegant. Instead of a regular
"up" counter driving the DAC, this circuit uses an up/down
counter. The counter is continuously clocked, and the
up/down control line is driven by the output of the
comparator. So, when the analog input signal exceeds the
DAC output, the counter goes into the "count up" mode.
When the DAC output exceeds the analog input, the counter
switches into the "count down" mode. Either way, the DAC
output always counts in the proper direction to track the
input signal.
4
tec]
Cent _- -'ttec
nn 62744
ip fe
hee e nee
Vin >
Bina
output
Notice how no shift register is needed to buffer the binary
count at the end of a cycle. Since the counter's output
continuously tracks the input (rather than counting to meet
the input and then resetting back to zero), the binary output
is legitimately updated with every clock pulse.
An advantage of this converter circuit is soeed, since the
counter never has to reset. Note the behavior of this circuit:
Analog
input
Time —~
Digital
output ar ee
Time —~
Note the much faster update time than any of the other
"counting" ADC circuits. Also note how at the very beginning
of the plot where the counter had to "catch up" with the
analog signal, the rate of change for the output was
identical to that of the first counting ADC. Also, with no shift
register in this circuit, the binary output would actually
ramp up rather than jump from zero to an accurate count as
it did with the counter and successive approximation ADC
circuits.
Perhaps the greatest drawback to this ADC design is the fact
that the binary output is never stable: it always switches
between counts with every clock pulse, even with a
perfectly stable analog input signal. This phenomenon is
informally known as bit bobble, and it can be problematic in
some digital systems.
This tendency can be overcome, though, through the
creative use of a shift register. For example, the counter's
output may be latched through a parallel-in/parallel-out shift
register only when the output changes by two or more steps.
Building a circuit to detect two or more successive counts in
the same direction takes a little ingenuity, but is worth the
effort.
Slope (integrating) ADC
So far, we've only been able to escape the sheer volume of
components in the flash converter by using a DAC as part of
our ADC circuitry. However, this is not our only option. It is
possible to avoid using a DAC if we substitute an analog
ramping circuit and a digital counter with precise timing.
The is the basic idea behind the so-called single-s/ope, or
integrating ADC. Instead of using a DAC with a ramped
output, we use an op-amp circuit called an integrator to
generate a sawtooth waveform which is then compared
against the analog input by a comparator. The time it takes
for the sawtooth waveform to exceed the input signal
voltage level is measured by means of a digital counter
clocked with a precise-frequency square wave (usually from
a crystal oscillator). The basic schematic diagram is shown
here:
Binar
outpu
The IGFET capacitor-discharging transistor scheme shown
here is a bit oversimplified. In reality, a latching circuit timed
with the clock signal would most likely have to be connected
to the IGFET gate to ensure full discharge of the capacitor
when the comparator's output goes high. The basic idea,
however, is evident in this diagram. When the comparator
output is low (input voltage greater than integrator output),
the integrator is allowed to charge the capacitor in a linear
fashion. Meanwhile, the counter is counting up at a rate
fixed by the precision clock frequency. The time it takes for
the capacitor to charge up to the same voltage level as the
input depends on the input signal level and the combination
Of -V af, R, and C. When the capacitor reaches that voltage
level, the comparator output goes high, loading the
counter's output into the shift register for a final output. The
IGFET is triggered "on" by the comparator's high output,
discharging the capacitor back to zero volts. When the
integrator output voltage falls to zero, the comparator
output switches back to a low state, clearing the counter
and enabling the integrator to ramp up voltage again.
This ADC circuit behaves very much like the digital ramp
ADC, except that the comparator reference voltage is a
smooth sawtooth waveform rather than a "stairstep:"
Analog
input
Time —>
Digital
Time —~
The single-slope ADC suffers all the disadvantages of the
digital ramp ADC, with the added drawback of ca/ibration
drift. The accurate correspondence of this ADC's output with
its input is dependent on the voltage slope of the integrator
being matched to the counting rate of the counter (the clock
frequency). With the digital ramp ADC, the clock frequency
had no effect on conversion accuracy, only on update time.
In this circuit, since the rate of integration and the rate of
count are independent of each other, variation between the
two is inevitable as it ages, and will result in a loss of
accuracy. The only good thing to say about this circuit is that
it avoids the use of a DAC, which reduces circuit complexity.
An answer to this calibration drift dilemma is found ina
design variation called the dua/-s/ope converter. In the dual-
slope converter, an integrator circuit is driven positive and
negative in alternating cycles to ramp down and then up,
rather than being reset to O volts at the end of every cycle.
In one direction of ramping, the integrator is driven by the
positive analog input signal (producing a negative, variable
rate of output voltage change, or output s/ope) for a fixed
amount of time, as measured by a counter with a precision
frequency clock. Then, in the other direction, with a fixed
reference voltage (producing a fixed rate of output voltage
change) with time measured by the same counter. The
counter stops counting when the integrator's output reaches
the same voltage as it was when it started the fixed-time
portion of the cycle. The amount of time it takes for the
integrator's capacitor to discharge back to its original output
voltage, as measured by the magnitude accrued by the
counter, becomes the digital output of the ADC circuit.
The dual-slope method can be thought of analogously in
terms of a rotary spring such as that used in a mechanical
clock mechanism. Imagine we were building a mechanism to
measure the rotary speed of a shaft. Thus, shaft speed is our
"input signal" to be measured by this device. The
measurement cycle begins with the spring in a relaxed state.
The spring is then turned, or "wound up," by the rotating
Shaft (input signal) for a fixed amount of time. This places
the spring in a certain amount of tension proportional to the
shaft speed: a greater shaft soeed corresponds to a faster
rate of winding. and a greater amount of spring tension
accumulated over that period of time. After that, the spring
is uncoupled from the shaft and allowed to unwind at a fixed
rate, the time for it to unwind back to a relaxed state
measured by a timer device. The amount of time it takes for
the spring to unwind at that fixed rate will be directly
proportional to the speed at which it was wound (input
signal magnitude) during the fixed-time portion of the cycle.
This technique of analog-to-digital conversion escapes the
calibration drift problem of the single-slope ADC because
both the integrator's integration coefficient (or "gain") and
the counter's rate of speed are in effect during the entire
"winding" and "unwinding" cycle portions. If the counter's
clock speed were to suddenly increase, this would shorten
the fixed time period where the integrator "winds up"
(resulting in a lesser voltage accumulated by the integrator),
but it would also mean that it would count faster during the
period of time when the integrator was allowed to "unwind"
at a fixed rate. The proportion that the counter is counting
faster will be the same proportion as the integrator's
accumulated voltage is diminished from before the clock
speed change. Thus, the clock speed error would cancel
itself out and the digital output would be exactly what it
should be.
Another important advantage of this method is that the
input signal becomes averaged as it drives the integrator
during the fixed-time portion of the cycle. Any changes in
the analog signal during that period of time have a
cumulative effect on the digital output at the end of that
cycle. Other ADC strategies merely "capture" the analog
signal level at a single point in time every cycle. If the
analog signal is "noisy" (contains significant levels of
Spurious voltage spikes/dips), one of the other ADC
converter technologies may occasionally convert a spike or
dip because it captures the signal repeatedly at a single
point in time. A dual-slope ADC, on the other hand, averages
together all the spikes and dips within the integration
period, thus providing an output with greater noise
immunity. Dual-slope ADCs are used in applications
demanding high accuracy.
Delta-Sigma (Az) ADC
One of the more advanced ADC technologies is the so-called
delta-sigma, or AZ (using the proper Greek letter notation).
In mathematics and physics, the capital Greek letter delta
(A) represents difference or change, while the capital letter
sigma (2) represents summation: the adding of multiple
terms together. Sometimes this converter is referred to by
the same Greek letters in reverse order: sigma-delta, or 2A.
In a AX converter, the analog input voltage signal is
connected to the input of an integrator, producing a voltage
rate-of-change, or slope, at the output corresponding to
input magnitude. This ramping voltage is then compared
against ground potential (0 volts) by a comparator. The
comparator acts as a sort of 1-bit ADC, producing 1 bit of
output ("high" or "low") depending on whether the
integrator output is positive or negative. The comparator's
output is then latched through a D-type flip-flop clocked ata
high frequency, and fed back to another input channel on
the integrator, to drive the integrator in the direction of a O
volt output. The basic circuit looks like this:
The leftmost op-amp is the (Summing) integrator. The next
op-amp the integrator feeds into is the comparator, or 1-bit
ADC. Next comes the D-type flip-flop, which latches the
comparator's output at every clock pulse, sending either a
"high" or "low" signal to the next comparator at the top of
the circuit. This final comparator is necessary to convert the
single-polarity OV / 5V logic level output voltage of the flip-
flop into a +V / -V voltage signal to be fed back to the
integrator.
If the integrator output is positive, the first comparator will
output a "high" signal to the D input of the flip-flop. At the
next clock pulse, this "high" signal will be output from the Q
line into the noninverting input of the last comparator. This
last comparator, seeing an input voltage greater than the
threshold voltage of 1/2 +V, saturates in a positive direction,
sending a full +V signal to the other input of the integrator.
This +V feedback signal tends to drive the integrator output
in a negative direction. If that output voltage ever becomes
negative, the feedback loop will send a corrective signal (-V)
back around to the top input of the integrator to drive it ina
positive direction. This is the delta-sigma concept in action:
the first comparator senses a difference (A) between the
integrator output and zero volts. The integrator sums (z) the
comparator's output with the analog input signal.
Functionally, this results in a serial stream of bits output by
the flip-flop. If the analog input is zero volts, the integrator
will have no tendency to ramp either positive or negative,
except in response to the feedback voltage. In this scenario,
the flip-flop output will continually oscillate between "high"
and "low," as the feedback system "hunts" back and forth,
trying to maintain the integrator output at zero volts:
AS converter operation with
0 volt analog input
Flip-flop output
Oe EO De BOF 2 Pe ie tbe ft
a a ee ie Me a a
Integrator output
If, however, we apply a negative analog input voltage, the
integrator will have a tendency to ramp its output ina
positive direction. Feedback can only add to the integrator's
ramping by a fixed voltage over a fixed time, and so the bit
stream output by the flip-flop will not be quite the same:
AS converter operation with
small negative analog input
Flip-flop output
0o;1;0;1/0;1 1/0 0; 1/0; 1/0) 1
Integrator output
By applying a larger (negative) analog input signal to the
integrator, we force its output to ramp more steeply in the
positive direction. Thus, the feedback system has to output
more 1's than before to bring the integrator output back to
zero volts:
A converter operation with
medium negative analog input
Flip-flop output
O;1;O;1 1/0; 1;/0}]1 1/0]; 1 1/0
PL LANL
Integrator output
As the analog input signal increases in magnitude, so does
the occurrence of 1's in the digital output of the flip-flop:
A converter operation with
large negative analog input
Flip-flop output
Integrator output
A parallel binary number output is obtained from this circuit
by averaging the serial stream of bits together. For example,
a counter circuit could be designed to collect the total
number of 1's output by the flip-flop in a given number of
clock pulses. This count would then be indicative of the
analog input voltage.
Variations on this theme exist, employing multiple integrator
stages and/or comparator circuits outputting more than 1
bit, but one concept common to all AX converters is that of
oversampling. Oversampling is when multiple samples of an
analog signal are taken by an ADC (in this case, a 1-bit
ADC), and those digitized samples are averaged. The end
result is an effective increase in the number of bits resolved
from the signal. In other words, an oversampled 1-bit ADC
can do the same job as an 8-bit ADC with one-time
sampling, albeit at a slower rate.
Practical considerations of ADC
circuits
Perhaps the most important consideration of an ADC is its
resolution. Resolution is the number of binary bits output by
the converter. Because ADC circuits take in an analog signal,
which is continuously variable, and resolve it into one of
many discrete steps, it is important to know how many of
these steps there are in total.
For example, an ADC with a 10-bit output can represent up
to 1024 (21°) unique conditions of signal measurement.
Over the range of measurement from 0% to 100%, there will
be exactly 1024 unique binary numbers output by the
converter (from 0000000000 to 1111111111, inclusive). An
11-bit ADC will have twice as many states to its output
(2048, or 2!1), representing twice as many unique
conditions of signal measurement between 0% and 100%.
Resolution is very important in data acquisition systems
(circuits designed to interpret and record physical
measurements in electronic form). Suppose we were
measuring the height of water in a 40-foot tall storage tank
using an instrument with a 10-bit ADC. 0 feet of water in the
tank corresponds to 0% of measurement, while 40 feet of
water in the tank corresponds to 100% of measurement.
Because the ADC is fixed at 10 bits of binary data output, it
will interpret any given tank level as one out of 1024
possible states. To determine how much physical water level
will be represented in each step of the ADC, we need to
divide the 40 feet of measurement span by the number of
steps in the 0-to-1024 range of possibilities, which is 1023
(one less than 1024). Doing this, we obtain a figure of
0.039101 feet per step. This equates to 0.46921 inches per
step, a little less than half an inch of water level represented
for every binary count of the ADC.
Water
40 ft tank
30 ft
20 ft Level "transmitter"
10 ft Pe. A-to-D converter
O ft 10-bit
output
Binary output: Equivalent measurement:
1111111111, = 40 feet of water level
1024 states 0000000010, = 0.07820 feet of water level
1 step 0000000001, = 0.039101 feet of water level
0000000000, =O feet of water level
This step value of 0.039101 feet (0.46921 inches)
represents the smallest amount of tank level change
detectable by the instrument. Admittedly, this is a small
amount, less than 0.1% of the overall measurement span of
40 feet. However, for some applications it may not be fine
enough. Suppose we needed this instrument to be able to
indicate tank level changes down to one-tenth of an inch. In
order to achieve this degree of resolution and still maintain
a measurement span of 40 feet, we would need an
instrument with more than ten ADC bits.
To determine how many ADC bits are necessary, we need to
first determine how many 1/10 inch steps there are in 40
feet. The answer to this is 40/(0.1/12), or 4800 1/10 inch
steps in 40 feet. Thus, we need enough bits to provide at
least 4800 discrete steps in a binary counting sequence. 10
bits gave us 1023 steps, and we knew this by calculating 2
to the power of 10 (21° = 1024) and then subtracting one.
Following the same mathematical procedure, 211-1 = 2047,
212-1 = 4095, and 213-1 = 8191. 12 bits falls shy of the
amount needed for 4800 steps, while 13 bits is more than
enough. Therefore, we need an instrument with at least 13
bits of resolution.
Another important consideration of ADC circuitry is its
sample frequency, or conversion rate. This is simply the
speed at which the converter outputs a new binary number.
Like resolution, this consideration is linked to the specific
application of the ADC. If the converter is being used to
measure slow-changing signals such as level in a water
storage tank, it could probably have a very slow sample
frequency and still perform adequately. Conversely, if it is
being used to digitize an audio frequency signal cycling at
several thousand times per second, the converter needs to
be considerably faster.
Consider the following illustration of ADC conversion rate
versus signal type, typical of a successive-approximation
ADC with regular sample intervals:
Analog
input
Time —~
Digital
output Pr tf++e+ 1 | pl
Time —~
Here, for this slow-changing signal, the sample rate is more
than adequate to capture its general trend. But consider this
example with the same sample time:
Analog
input
HANES
Time —~
Digital
output tt] | ir Hy
Time —~
When the sample period is too long (too slow), substantial
details of the analog signal will be missed. Notice how,
especially in the latter portions of the analog signal, the
digital output utterly fails to reproduce the true shape. Even
in the first section of the analog waveform, the digital
reproduction deviates substantially from the true shape of
the wave.
It is imperative that an ADC's sample time is fast enough to
capture essential changes in the analog waveform. In data
acquisition terminology, the highest-frequency waveform
that an ADC can theoretically capture is the so-called
Nyquist frequency, equal to one-half of the ADC's sample
frequency. Therefore, if an ADC circuit has a sample
frequency of 5000 Hz, the highest-frequency waveform it
can successfully resolve will be the Nyquist frequency of
2500 Hz.
If an ADC is subjected to an analog input signal whose
frequency exceeds the Nyquist frequency for that ADC, the
converter will output a digitized signal of falsely low
frequency. This phenomenon is known as aliasing. Observe
the following illustration to see how aliasing occurs:
Aliasing
Analog
input
Time —>
Digital
output +tPFtiyReyt
Time —~
Note how the period of the output waveform is much longer
(slower) than that of the input waveform, and how the two
waveform shapes aren't even similar:
Analog
input
AVA \ISN\IN\I NI NT
Digital
It should be understood that the Nyquist frequency is an
absolute maximum frequency limit for an ADC, and does not
represent the highest practical frequency measurable. To be
safe, one shouldn't expect an ADC to successfully resolve
any frequency greater than one-fifth to one-tenth of its
Sample frequency.
A practical means of preventing aliasing is to place a low-
pass filter before the input of the ADC, to block any signal
frequencies greater than the practical limit. This way, the
ADC circuitry will be prevented from seeing any excessive
frequencies and thus will not try to digitize them. It is
generally considered better that such frequencies go
unconverted than to have them be "aliased" and appear in
the output as false signals.
Yet another measure of ADC performance is something
called step recovery. This is a measure of how quickly an
ADC changes its output to match a large, sudden change in
the analog input. In some converter technologies especially,
step recovery is a serious limitation. One example is the
tracking converter, which has a typically fast update period
but a disproportionately slow step recovery.
An ideal ADC has a great many bits for very fine resolution,
samples at lightning-fast speeds, and recovers from steps
instantly. It also, unfortunately, doesn't exist in the real
world. Of course, any of these traits may be improved
through additional circuit complexity, either in terms of
increased component count and/or special circuit designs
made to run at higher clock speeds. Different ADC
technologies, though, have different strengths. Here is a
summary of them ranked from best to worst:
Resolution/complexity ratio:
Single-slope integrating, dual-slope integrating, counter,
tracking, Successive approximation, flash.
Speed:
Flash, tracking, successive approximation, single-slope
integrating & counter, dual-slope integrating.
Step recovery:
Flash, successive-approximation, single-slope integrating &
counter, dual-slope integrating, tracking.
Please bear in mind that the rankings of these different ADC
technologies depend on other factors. For instance, how an
ADC rates on step recovery depends on the nature of the
step change. A tracking ADC is equally slow to respond to all
step changes, whereas a single-slope or counter ADC will
register a high-to-low step change quicker than a low-to-
high step change. Successive-approximation ADCs are
almost equally fast at resolving any analog signal, buta
tracking ADC will consistently beat a successive-
approximation ADC if the signal is changing slower than one
resolution step per clock pulse. | ranked integrating
converters as having a greater resolution/complexity ratio
than counter converters, but this assumes that precision
analog integrator circuits are less complex to design and
manufacture than precision DACs required within counter-
based converters. Others may not agree with this
assumption.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4]
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 14
DIGITAL COMMUNICATION
e Introduction
e Networks and busses
o Short-distance busses
o Extended-distance networks
Data flow
Electrical signal types
Optical data communication
Network topology
o Point-to-point
o Bus
o Star
o Ring
e Network protocols
e Practical considerations
Introduction
In the design of large and complex digital systems, it is often
necessary to have one device communicate digital
information to and from other devices. One advantage of
digital information is that it tends to be far more resistant to
transmitted and interpreted errors than information
symbolized in an analog medium. This accounts for the
Clarity of digitally-encoded telephone connections, compact
audio disks, and for much of the enthusiasm in the
engineering community for digital communications
technology. However, digital communication has its own
unique pitfalls, and there are multitudes of different and
incompatible ways in which it can be sent. Hopefully, this
chapter will enlighten you as to the basics of digital
communication, its advantages, disadvantages, and
practical considerations.
Suppose we are given the task of remotely monitoring the
level of a water storage tank. Our job is to design a system
to measure the level of water in the tank and send this
information to a distant location so that other people may
monitor it. Measuring the tank's level is quite easy, and can
be accomplished with a number of different types of
instruments, such as float switches, pressure transmitters,
ultrasonic level detectors, capacitance probes, strain
gauges, or radar level detectors.
For the sake of this illustration, we will use an analog level-
measuring device with an output signal of 4-20 mA. 4 mA
represents a tank level of 0%, 20 mA represents a tank level
of 100%, and anything in between 4 and 20 mA represents a
tank level proportionately between 0% and 100%. If we
wanted to, we could simply send this 4-20 milliamp analog
current signal to the remote monitoring location by means of
a pair of copper wires, where it would drive a panel meter of
some sort, the scale of which was calibrated to reflect the
depth of water in the tank, in whatever units of
measurement preferred.
Analog tank-level measurement "loop"
Water
tank
Panel
meter
"Transmitter"
24 VDC
This analog communication system would be simple and
robust. For many applications, it would suffice for our needs
perfectly. But, it is not the on/y way to get the job done. For
the purposes of exploring digital techniques, we'll explore
other methods of monitoring this hypothetical tank, even
though the analog method just described might be the most
practical.
The analog system, as simple as it may be, does have its
limitations. One of them is the problem of analog signal
interference. Since the tank's water level is symbolized by
the magnitude of DC current in the circuit, any "noise" in
this signal will be interpreted as a change in the water level.
With no noise, a plot of the current signal over time for a
steady tank level of 50% would look like this:
Plot of signal at 50% tank level
12 mA
O mA
Time —>
If the wires of this circuit are arranged too close to wires
carrying 60 Hz AC power, for example, inductive and
Capacitive coupling may create a false "noise" signal to be
introduced into this otherwise DC circuit. Although the low
impedance of a 4-20 mA loop (250 Q, typically) means that
small noise voltages are significantly loaded (and thereby
attenuated by the inefficiency of the capacitive/inductive
coupling formed by the power wires), such noise can be
significant enough to cause measurement problems:
Plot of signal at 50% tank level
(with 60 Hz interference)
12MA NW INN IDO
O mA
Time —>
The above example is a bit exaggerated, but the concept
should be clear: any electrical noise introduced into an
analog measurement system will be interpreted as changes
in the measured quantity. One way to combat this problem is
to symbolize the tank's water level by means of a digital
signal instead of an analog signal. We can do this really
crudely by replacing the analog transmitter device with a
set of water level switches mounted at different heights on
the tank:
Tank level measurement with switches
L l L ri |
Water mA
tank
Each of these switches is wired to close a circuit, sending
Current to individual lamps mounted on a panel at the
monitoring location. As each switch closed, its respective
lamp would light, and whoever looked at the panel would
see a 5-lamp representation of the tank's level.
Being that each lamp circuit is digital in nature -- either
100% on or 100% off -- electrical interference from other
wires along the run have much less effect on the accuracy of
measurement at the monitoring end than in the case of the
analog signal. A huge amount of interference would be
required to cause an "off" signal to be interpreted as an "on"
signal, or vice versa. Relative resistance to electrical
interference is an advantage enjoyed by all forms of digital
communication over analog.
Now that we know digital signals are far more resistant to
error induced by "noise," let's improve on this tank level
measurement system. For instance, we could increase the
resolution of this tank gauging system by adding more
switches, for more precise determination of water level.
Suppose we install 16 switches along the tank's height
instead of five. This would significantly improve our
measurement resolution, but at the expense of greatly
increasing the quantity of wires needing to be strung
between the tank and the monitoring location. One way to
reduce this wiring expense would be to use a priority
encoder to take the 16 switches and generate a binary
number which represented the same information:
Q;Q,Q; Q,
16-line 0000
to
4-line
priority
encoder
Switch 0 eS
Switch 15 =
Now, only 4 wires (plus any ground and power wires
necessary) are needed to communicate the information, as
opposed to 16 wires (plus any ground and power wires). At
the monitoring location, we would need some kind of display
device that could accept the 4-bit binary data and generate
an easy-to-read display for a person to view. A decoder,
wired to accept the 4-bit data as its input and light 1-of-16
output lamps, could be used for this task, or we could use a
4-bit decoder/driver circuit to drive some kind of numerical
digit display.
et et st ts tt tL OOOOOOC
— St St HOO OS SS St I OOO
323 00=]=00==00==00
—- 0-0 $+ Oo $+ OO = 0 S| CO = OC
Still, a resolution of 1/16 tank height may not be good
enough for our application. To better resolve the water level,
we need more bits in our binary output. We could add still
more switches, but this gets impractical rather quickly. A
better option would be to re-attach our original analog
transmitter to the tank and electronically convert its 4-20
milliamp analog output into a binary number with far more
bits than would be practical using a set of discrete level
switches. Since the electrical noise we're trying to avoid is
encountered along the long run of wire from the tank to the
monitoring location, this A/D conversion can take place at
the tank (where we have a "clean" 4-20 mA signal). There
are a variety of methods to convert an analog signal to
digital, but we'll skip an in-depth discussion of those
techniques and concentrate on the digital signal
communication itself.
The type of digital information being sent from our tank
instrumentation to the monitoring instrumentation is
referred to as paralle/ digital data. That is, each binary bit is
being sent along its own dedicated wire, so that all bits
arrive at their destination simultaneously. This obviously
necessitates the use of at least one wire per bit to
communicate with the monitoring location. We could further
reduce our wiring needs by sending the binary data along a
single channel (one wire + ground), so that each bit is
communicated one at a time. This type of information is
referred to as seria/ digital data.
We could use a multiplexer or a shift register to take the
parallel data from the A/D converter (at the tank
transmitter), and convert it to serial data. At the receiving
end (the monitoring location) we could use a demultiplexer
or another shift register to convert the serial data to parallel
again for use in the display circuitry. The exact details of
how the mux/demux or shift register pairs are maintained in
synchronization is, like A/D conversion, a topic for another
lesson. Fortunately, there are digital IC chips called UARTs
(Universal Asynchronous Receiver-Transmitters) that handle
all these details on their own and make the designer's life
much simpler. For now, we must continue to focus our
attention on the matter at hand: how to communicate the
digital information from the tank to the monitoring location.
Networks and busses
This collection of wires that | keep referring to between the
tank and the monitoring location can be called a busora
network. The distinction between these two terms is more
semantic than technical, and the two may be used
interchangeably for all practical purposes. In my experience,
the term "bus" is usually used in reference to a set of wires
connecting digital components within the enclosure of a
computer device, and "network" is for something that is
physically more widespread. In recent years, however, the
word "bus" has gained popularity in describing networks
that specialize in interconnecting discrete instrumentation
sensors over long distances ("Fieldbus" and "Profibus" are
two examples). In either case, we are making reference to
the means by which two or more digital devices are
connected together so that data can be communicated
between them.
Names like "Fieldbus" or "Profibus" encompass not only the
physical wiring of the bus or network, but also the specified
voltage levels for communication, their timing sequences
(especially for serial data transmission), connector pinout
specifications, and all other distinguishing technical features
of the network. In other words, when we speak of a certain
type of bus or network by name, we're actually speaking of a
communications standard, roughly analogous to the rules
and vocabulary of a written language. For example, before
two or more people can become pen-pals, they must be able
to write to one another in a common format. To merely have
a mail system that is able to deliver their letters to each
other is not enough. If they agree to write to each other in
French, they agree to hold to the conventions of character
set, vocabulary, spelling, and grammar that is specified by
the standard of the French language. Likewise, if we connect
two Profibus devices together, they will be able to
communicate with each other only because the Profibus
standard has specified such important details as voltage
levels, timing sequences, etc. Simply having a set of wires
strung between multiple devices is not enough to construct
a working system (especially if the devices were built by
different manufacturers!).
To illustrate in detail, let's design our own bus standard.
Taking the crude water tank measurement system with five
switches to detect varying levels of water, and using (at
least) five wires to conduct the signals to their destination,
we can lay the foundation for the mighty BogusBus:
BogusBus™
LS5 5 Lamp
<= 5
Lamp
4
Lamp
2
Lamp
2
Connector
Lamp
1
The physical wiring for the BogusBus consists of seven wires
between the transmitter device (switches) and the receiver
device (lamps). The transmitter consists of all components
and wiring connections to the left of the leftmost connectors
(the "-->>--" symbols). Each connector symbol represents a
complementary male and female element. The bus wiring
consists of the seven wires between the connector pairs.
Finally, the receiver and all of its constituent wiring lies to
the right of the rightmost connectors. Five of the network
wires (labeled 1 through 5) carry the data while two of those
wires (labeled +V and -V) provide connections for DC power
supplies. There is a standard for the 7-pin connector plugs,
as well. The pin layout is asymmetrical to prevent
"backward" connection.
In order for manufacturers to receive the awe-inspiring
“BogusBus-compliant" certification on their products, they
would have to comply with the specifications set by the
designers of BogusBus (most likely another company, which
designed the bus for a specific task and ended up marketing
it for a wide variety of purposes). For instance, all devices
must be able to use the 24 Volt DC supply power of
BogusBus: the switch contacts in the transmitter must be
rated for switching that DC voltage, the lamps must
definitely be rated for being powered by that voltage, and
the connectors must be able to handle it all. Wiring, of
course, must be in compliance with that same standard:
lamps 1 through 5, for example, must be wired to the
appropriate pins so that when LS4 of Manufacturer XYZ's
transmitter closes, lamp 4 of Manufacturer ABC's receiver
lights up, and so on. Since both transmitter and receiver
contain DC power supplies rated at an output of 24 Volts, all
transmitter/receiver combinations (from all certified
manufacturers) must have power supplies that can be safely
wired in parallel. Consider what could happen if
Manufacturer XYZ made a transmitter with the negative (-)
side of their 24VDC power supply attached to earth ground
and Manufacturer ABC made a receiver with the positive (+)
side of their 24VDC power supply attached to earth ground.
If both earth grounds are relatively "solid" (that is, a low
resistance between them, such as might be the case if the
two grounds were made on the metal structure of an
industrial building), the two power supplies would short-
circuit each other!
BogusBus, of course, is a completely hypothetical and very
impractical example of a digital network. It has incredibly
poor data resolution, requires substantial wiring to connect
devices, and communicates in only a single direction (from
transmitter to receiver). It does, however, suffice as a tutorial
example of what a network is and some of the
considerations associated with network selection and
operation.
There are many types of buses and networks that you might
come across in your profession. Each one has its own
applications, advantages, and disadvantages. It is
worthwhile to associate yourself with some of the "alphabet
soup" that is used to label the various designs:
Short-distance busses
PC/AT Bus used in early IBM-compatible computers to
connect peripheral devices such as disk drive and sound
cards to the motherboard of the computer.
PCI Another bus used in personal computers, but not limited
to IBM-compatibles. Much faster than PC/AT. Typical data
transfer rate of 100 Mbytes/second (32 bit) and 200
Mbytes/second (64 bit).
PCMCIA A bus designed to connect peripherals to laptop
and notebook sized personal computers. Has a very small
physical "footprint," but is considerably slower than other
popular PC buses.
VME A high-performance bus (co-designed by Motorola, and
based on Motorola's earlier Versa-Bus standard) for
constructing versatile industrial and military computers,
where multiple memory, peripheral, and even
microprocessor cards could be plugged in to a passive "rack"
or "card cage" to facilitate custom system designs. Typical
data transfer rate of 50 Mbytes/second (64 bits wide).
VXI Actually an expansion of the VME bus, VXI (VME
eXtension for Instrumentation) includes the standard VME
bus along with connectors for analog signals between cards
in the rack.
S-100 Sometimes called the Altair bus, this bus standard
was the product of a conference in 1976, intended to serve
as an interface to the Intel 8080 microprocessor chip. Similar
in philosophy to the VME, where multiple function cards
could be plugged in to a passive "rack," facilitating the
construction of custom systems.
MC6800 The Motorola equivalent of the Intel-centric S-100
bus, designed to interface peripheral devices to the popular
Motorola 6800 microprocessor chip.
STD Stands for Simple-To-Design, and is yet another passive
"rack" similar to the PC/AT bus, and lends itself well toward
designs based on IBM-compatible hardware. Designed by
Pro-Log, it is 8 bits wide (parallel), accommodating relatively
small (4.5 inch by 6.5 inch) circuit cards.
Multibus | and Il Another bus intended for the flexible
design of custom computer systems, designed by Intel. 16
bits wide (parallel).
CompactPCl An industrial adaptation of the personal
computer PCI standard, designed as a higher-performance
alternative to the older VME bus. At a bus clock speed of 66
MHz, data transfer rates are 200 Mbytes/ second (32 bit) or
400 Mbytes/sec (64 bit).
Microchannel Yet another bus, this one designed by IBM for
their ill-fated PS/2 series of computers, intended for the
interfacing of PC motherboards to peripheral devices.
IDE A bus used primarily for connecting personal computer
hard disk drives with the appropriate peripheral cards.
Widely used in today's personal computers for hard drive
and CD-ROM drive interfacing.
SCSI An alternative (technically superior to IDE) bus used
for personal computer disk drives. SCSI stands for Smal//
Computer System Interface. Used in some IBM-compatible
PC's, as well as Macintosh (Apple), and many mini and
mainframe business computers. Used to interface hard
drives, CD-ROM drives, floppy disk drives, printers, scanners,
modems, and a host of other peripheral devices. Speeds up
to 1.5 Mbytes per second for the original standard.
GPIB (IEEE 488) General Purpose Interface Bus, also known
as HPIB or IEEE 488, which was intended for the interfacing
of electronic test equipment such as oscilloscopes and
multimeters to personal computers. 8 bit wide address/data
"path" with 8 additional lines for communications control.
Centronics parallel Widely used on personal computers
for interfacing printer and plotter devices. Sometimes used
to interface with other peripheral devices, such as external
ZIP (100 Mbyte floppy) disk drives and tape drives.
USB Universal Serial Bus, which is intended to interconnect
many external peripheral devices (such as keyboards,
modems, mice, etc.) to personal computers. Long used on
Macintosh PC's, it is now being installed as new equipment
on IBM-compatible machines.
FireWire (IEEE 1394) A high-speed serial network capable
of operating at 100, 200, or 400 Mbps with versatile features
such as "hot swapping" (adding or removing devices with
the power on) and flexible topology. Designed for high-
performance personal computer interfacing.
Bluetooth A radio-based communications network
designed for office linking of computer devices. Provisions
for data security designed into this network standard.
Extended-distance networks
20 mA current loop Not to be confused with the common
instrumentation 4-20 mA analog standard, this is a digital
communications network based on interrupting a 20 mA (or
sometimes 60 mA) current loop to represent binary data.
Although the low impedance gives good noise immunity, it
is susceptible to wiring faults (Such as breaks) which would
fail the entire network.
RS-232C The most common serial network used in
computer systems, often used to link peripheral devices
such as printers and mice to a personal computer. Limited in
speed and distance (typically 45 feet and 20 kbps, although
higher speeds can be run with shorter distances). I've been
able to run RS-232 reliably at soeeds in excess of 100 kbps,
but this was using a cable only 6 feet long! RS-232C is often
referred to simply as RS-232 (no "C").
RS-422A/RS-485 Two serial networks designed to overcome
some of the distance and versatility limitations of RS-232C.
Used widely in industry to link serial devices together in
electrically "noisy" plant environments. Much greater
distance and speed limitations than RS-232C, typically over
half a mile and at speeds approaching 10 Mbps.
Ethernet (IEEE 802.3) A high-speed network which links
computers and some types of peripheral devices together.
"Normal" Ethernet runs at a speed of 10 million bits/second,
and "Fast" Ethernet runs at 100 million bits/second. The
slower (10 Mbps) Ethernet has been implemented in a
variety of means on copper wire (thick coax = "LOBASE5",
thin coax = "LOBASE2", twisted-pair = "1OBASE-T"), radio,
and on optical fiber ("lOBASE-F"). The Fast Ethernet has also
been implemented on a few different means (twisted-pair, 2
pair = 1OOBASE-TX; twisted-pair, 4 pair = LOOBASE-T4;
optical fiber = LOOBASE-FX).
Token ring Another high-speed network linking computer
devices together, using a philosophy of communication that
is much different from Ethernet, allowing for more precise
response times from individual network devices, and greater
immunity to network wiring damage.
FDDI A very high-speed network exclusively implemented
on fiber-optic cabling.
Modbus/Modbus Plus Originally implemented by the
Modicon corporation, a large maker of Programmable Logic
Controllers (PLCs) for linking remote I/O (Input/Output) racks
with a PLC processor. Still quite popular.
Profibus Originally implemented by the Siemens
corporation, another large maker of PLC equipment.
Foundation Fieldbus A high-performance bus expressly
designed to allow multiple process instruments
(transmitters, controllers, valve positioners) to communicate
with host computers and with each other. May ultimately
displace the 4-20 mA analog signal as the standard means
of interconnecting process control instrumentation in the
future.
Data flow
Buses and networks are designed to allow communication to
occur between individual devices that are interconnected.
The flow of information, or data, between nodes can take a
variety of forms:
Simplex communication
Transmitter | ———\—\——~ | Receiver
With simplex communication, all data flow is unidirectional:
from the designated transmitter to the designated receiver.
BogusBus is an example of simplex communication, where
the transmitter sent information to the remote monitoring
location, but no information is ever sent back to the water
tank. If all we want to do is send information one-way, then
simplex is just fine. Most applications, however, demand
more:
Duplex communication
Receiver / | ——————— | Receiver /
Transmitter |~—————— | Transmitter
With duplex communication, the flow of information is
bidirectional for each device. Duplex can be further divided
into two sub-categories:
Half-duplex
Receiver / Receiver /
Transmitter Transmitter
take turns)
Full-duplex
Receiver | ~————————_ | Transmitter
Transmitter | —————————> Receiver
(simultaneous)
Half-duplex communication may be likened to two tin cans
on the ends of a single taut string: Either can may be used
to transmit or receive, but not at the same time. Full-duplex
communication is more like a true telephone, where two
people can talk at the same time and hear one another
simultaneously, the mouthpiece of one phone transmitting
the the earpiece of the other, and vice versa. Full-duplex is
often facilitated through the use of two separate channels or
networks, with an individual set of wires for each direction of
communication. It is sometimes accomplished by means of
multiple-frequency carrier waves, especially in radio links,
where one frequency is reserved for each direction of
communication.
Electrical signal types
With BogusBus, our signals were very simple and
straightforward: each signal wire (1 through 5) carried a
single bit of digital data, 0 Volts representing "off" and 24
Volts DC representing "on." Because all the bits arrived at
their destination simultaneously, we would call BogusBus a
paralle/ network technology. If we were to improve the
performance of BogusBus by adding binary encoding (to the
transmitter end) and decoding (to the receiver end), so that
more steps of resolution were available with fewer wires, it
would still be a parallel network. If, however, we were to add
a parallel-to-serial converter at the transmitter end and a
serial-to-parallel converter at the receiver end, we would
have something quite different.
It is primarily with the use of serial technology that we are
forced to invent clever ways to transmit data bits. Because
serial data requires us to send all data bits through the same
wiring channel from transmitter to receiver, it necessitates a
potentially high frequency signal on the network wiring.
Consider the following illustration: a modified BogusBus
system is communicating digital data in parallel, binary-
encoded form. Instead of 5 discrete bits like the original
BogusBus, we're sending 8 bits from transmitter to receiver.
The A/D converter on the transmitter side generates a new
output every second. That makes for 8 bits per second of
data being sent to the receiver. For the sake of illustration,
let's say that the transmitter is bouncing between an output
of 10101010 and 10101011 every update (once per
second):
1 second
Sa —=
PPO th. = = i, — 1
Bitte
Bit 2
Bit 3
Bit 4
Bit 5
Bit 6
Bit 7
Since only the least significant bit (Bit 1) is changing, the
frequency on that wire (to ground) is only 1/2 Hertz. In fact,
no matter what numbers are being generated by the A/D
converter between updates, the frequency on any wire in
this modified BogusBus network cannot exceed 1/2 Hertz,
because that's how fast the A/D updates its digital output.
1/2 Hertz is pretty slow, and should present no problems for
our network wiring.
On the other hand, if we used an 8-bit serial network, all
data bits must appear on the single channel in sequence.
And these bits must be output by the transmitter within the
1-second window of time between A/D converter updates.
Therefore, the alternating digital output of 10101010 and
10101011 (once per second) would look something like this:
1 second
> —~——_
10101010 10101010
Serial data JUUULJUUU LIUUULJUUU L
10101011 10101011
The frequency of our BogusBus signal is now approximately
4 Hertz instead of 1/2 Hertz, an eightfold increase! While 4
Hertz is still fairly slow, and does not constitute an
engineering problem, you should be able to appreciate what
might happen if we were transmitting 32 or 64 bits of data
per update, along with the other bits necessary for parity
checking and signal synchronization, at an update rate of
thousands of times per second! Serial data network
frequencies start to enter the radio range, and simple wires
begin to act as antennas, pairs of wires as transmission lines,
with all their associated quirks due to inductive and
Capacitive reactances.
What is worse, the signals that we're trying to communicate
along a serial network are of a square-wave shape, being
binary bits of information. Square waves are peculiar things,
being mathematically equivalent to an infinite series of sine
waves of diminishing amplitude and increasing frequency. A
simple square wave at 10 kHz is actually "seen" by the
Capacitance and inductance of the network as a series of
multiple sine-wave frequencies which extend into the
hundreds of kHz at significant amplitudes. What we receive
at the other end of a long 2-conductor network won't look
like a clean Square wave anymore, even under the best of
conditions!
When engineers speak of network bandwidth, they're
referring to the practical frequency limit of a network
medium. In serial communication, bandwidth is a product of
data volume (binary bits per transmitted "word") and data
speed ("words" per second). The standard measure of
network bandwidth is bits per second, or bps. An obsolete
unit of bandwidth known as the baud is sometimes falsely
equated with bits per second, but is actually the measure of
signal level changes per second. Many serial network
standards use multiple voltage or current level changes to
represent a single bit, and so for these applications bps and
baud are not equivalent.
The general BogusBus design, where all bits are voltages
referenced to a common "ground" connection, is the worst-
case situation for high-frequency square wave data
communication. Everything will work well for short
distances, where inductive and capacitive effects can be
held to a minimum, but for long distances this method will
surely be problematic:
Ground-referenced voltage signal
Transmitter Receiver
B signal wire be
Input Output
Signal [ Signal
| ground wire [
Stray capacitance
an eee
A robust alternative to the common ground signal method is
the differential voltage method, where each bit is
represented by the difference of voltage between a ground-
isolated pair of wires, instead of a voltage between one wire
and a common ground. This tends to limit the capacitive and
inductive effects imposed upon each signal and the
tendency for the signals to be corrupted due to outside
electrical interference, thereby significantly improving the
practical distance of a serial network:
Differential voltage signal
Transmitter Receiver
signal wire
Input
Output
Signal
! Signal
signal wire | [
= Both sone! salbch isolated
rom ground! ’
g Capacitance through ground
minimized due to series-
diminishing effect.
The triangular amplifier symbols represent differential
amplifiers, which output a voltage signal between two wires,
neither one electrically common with ground. Having
eliminated any relation between the voltage signal and
ground, the only significant capacitance imposed on the
signal voltage is that existing between the two signal wires.
Capacitance between a signal wire and a grounded
conductor is of much less effect, because the capacitive
path between the two signal wires via a ground connection
is two capacitances in series (from signal wire #1 to ground,
then from ground to signal wire #2), and series capacitance
values are always less than any of the individual
Capacitances. Furthermore, any "noise" voltage induced
between the signal wires and earth ground by an external
source will be ignored, because that noise voltage will likely
be induced on both signal wires in equal measure, and the
receiving amplifier only responds to the differential voltage
between the two signal wires, rather than the voltage
between any one of them and earth ground.
RS-232C is a prime example of a ground-referenced serial
network, while RS-422A is a prime example of a differential
voltage serial network. RS-232C finds popular application in
office environments where there is little electrical
interference and wiring distances are short. RS-422A is more
widely used in industrial applications where longer wiring
distances and greater potential for electrical interference
from AC power wiring exists.
However, a large part of the problem with digital network
signals is the square-wave nature of such voltages, as was
previously mentioned. If only we could avoid square waves
all together, we could avoid many of their inherent
difficulties in long, high-frequency networks. One way of
doing this is to modulate a sine wave voltage signal with our
digital data. "Modulation" means that magnitude of one
signal has control over some aspect of another signal. Radio
technology has incorporated modulation for decades now, in
allowing an audio-frequency voltage signal to control either
the amplitude (AM) or frequency (FM) of a much higher
frequency "carrier" voltage, which is then send to the
antenna for transmission. The frequency-modulation (FM)
technique has found more use in digital networks than
amplitude-modulation (AM), except that its referred to as
Frequency Shift Keying (FSK). With simple FSK, sine waves of
two distinct frequencies are used to represent the two binary
states, 1 and 0:
(high)
1
<— 0 (low) ——>=— * > —+~+— 0 (low) ——
I
Due to the practical problems of getting the low/high
frequency sine waves to begin and end at the zero crossover
points for any given combination of 0's and 1's, a variation
of FSK called phase-continuous FSK is sometimes used,
where the consecutive combination of a low/high frequency
represents one binary state and the combination of a
high/low frequency represents the other. This also makes for
a situation where each bit, whether it be 0 or 1, takes
exactly the same amount of time to transmit along the
network:
$$ — 0 (low) ————> + 11 (high) >
a
With sine wave signal voltages, many of the problems
encountered with square wave digital signals are minimized,
although the circuitry required to modulate (and
demodulate) the network signals is more complex and
expensive.
Optical data communication
A modern alternative to sending (binary) digital information
via electric voltage signals is to use optical (light) signals.
Electrical signals from digital circuits (high/low voltages)
may be converted into discrete optical signals (light or no
light) with LEDs or solid-state lasers. Likewise, light signals
can be translated back into electrical form through the use
of photodiodes or phototransistors for introduction into the
inputs of gate circuits.
Transmitter Receiver
[UL UL
I N —_—_—_ —_ 7
os Light pulses _L =
Transmitting digital information in optical form may be done
in open air, simply by aiming a laser at a photodetector at a
remote distance, but interference with the beam in the form
of temperature inversion layers, dust, rain, fog, and other
obstructions can present significant engineering problems:
Transmitter Receiver
TUL rv
1° &@\\\. 8 4
One way to avoid the problems of open-air optical data
transmission is to send the light pulses down an ultra-pure
glass fiber. Glass fibers will "conduct" a beam of light much
as a copper wire will conduct electrons, with the advantage
of completely avoiding all the associated problems of
inductance, capacitance, and external interference plaguing
electrical signals. Optical fibers keep the light beam
contained within the fiber core by a phenomenon known as
total internal reflectance.
An optical fiber is composed of two layers of ultra-pure glass,
each layer made of glass with a slightly different refractive
index, or capacity to "bend" light. With one type of glass
concentrically layered around a central glass core, light
introduced into the central core cannot escape outside the
fiber, but is confined to travel within the core:
Cladding
Core !
Q
\
Cladding
These layers of glass are very thin, the outer "cladding"
typically 125 microns (1 micron = 1 millionth of a meter, or
10° meter) in diameter. This thinness gives the fiber
considerable flexibility. To protect the fiber from physical
damage, it is usually given a thin plastic coating, placed
inside of a plastic tube, wrapped with kevlar fibers for tensile
strength, and given an outer sheath of plastic similar to
electrical wire insulation. Like electrical wires, optical fibers
are often bundled together within the same sheath to form a
single cable.
Optical fibers exceed the data-handling performance of
copper wire in almost every regard. They are totally immune
to electromagnetic interference and have very high
bandwidths. However, they are not without certain
weaknesses.
One weakness of optical fiber is a phenomenon Known as
microbending. This is where the fiber is bend around too
small of a radius, causing light to escape the inner core,
through the cladding:
Microbending
Escaping
Sharp light
bend
Reflected
light
Not only does microbending lead to diminished signal
strength due to the lost light, but it also constitutes a
security weakness in that a light sensor intentionally placed
on the outside of a sharp bend could intercept digital data
transmitted over the fiber.
Another problem unique to optical fiber is signal distortion
due to multiple light paths, or modes, having different
distances over the length of the fiber. When light is emitted
by a source, the photons (light particles) do not all travel the
exact same path. This fact is patently obvious in any source
of light not conforming to a straight beam, but is true even
in devices such as lasers. If the optical fiber core is large
enough in diameter, it will support multiple pathways for
photons to travel, each of these pathways having a slightly
different length from one end of the fiber to the other. This
type of optical fiber is called mu/timode fiber:
"Modes" of light traveling in a fiber
A light pulse emitted by the LED taking a shorter path
through the fiber will arrive at the detector sooner than light
pulses taking longer paths. The result is distortion of the
square-wave's rising and falling edges, called pulse
stretching. This problem becomes worse as the overall fiber
length is increased:
"Pulse-stretching" in optical fiber
Transmitted Received
pulse
However, if the fiber core is made small enough (around 5
microns in diameter), light modes are restricted to a single
pathway with one length. Fiber so designed to permit only a
single mode of light is known as single-mode fiber. Because
single-mode fiber escapes the problem of pulse stretching
experienced in long cables, it is the fiber of choice for long-
distance (several miles or more) networks. The drawback, of
course, is that with only one mode of light, single-mode
fibers do not conduct as as much light as multimode fibers.
Over long distances, this exacerbates the need for
"repeater" units to boost light power.
Network topology
If we want to connect two digital devices with a network, we
would have a kind of network known as "point-to-point:"
Point-to-Point topology
For the sake of simplicity, the network wiring is symbolized
as a Single line between the two devices. In actuality, it may
be a twisted pair of wires, a coaxial cable, an optical fiber, or
even a seven-conductor BogusBus. Right now, we're merely
focusing on the "shape" of the network, technically known
as its topology.
If we want to include more devices (sometimes called nodes)
on this network, we have several options of network
configuration to choose from:
Bus topology
|
Device ue fetal Pare
1
Star topology Hub
Device Device Device Device
1 2 3 4
Ring topology
|_| Device |____ | Device |____ | Device |____ | Device L_!
1 2 3 4
Many network standards dictate the type of topology which
is used, while others are more versatile. Ethernet, for
example, is commonly implemented in a "bus" topology but
can also be implemented in a "star" or "ring" topology with
the appropriate interconnecting equipment. Other networks,
such as RS-232C, are almost exclusively point-to-point; and
token ring (as you might have guessed) is implemented
solely in a ring topology.
Different topologies have different pros and cons associated
with them:
Point-to-point
Quite obviously the only choice for two nodes.
Bus
Very simple to install and maintain. Nodes can be easily
added or removed with minimal wiring changes. On the
other hand, the one bus network must handle a//
communication signals from a// nodes. This is Known as
broadcast networking, and is analogous to a group of people
talking to each other over a single telephone connection,
where only one person can talk at a time (limiting data
exchange rates), and everyone can hear everyone else when
they talk (which can be a data security issue). Also, a break
in the bus wiring can lead to nodes being isolated in groups.
Star
With devices known as "gateways" at branching points in
the network, data flow can be restricted between nodes,
allowing for private communication between specific groups
of nodes. This addresses some of the speed and security
issues of the simple bus topology. However, those branches
could easily be cut off from the rest of the "star" network if
one of the gateways were to fail. Can also be implemented
with "switches" to connect individual nodes to a larger
network on demand. Such a switched network is similar to
the standard telephone system.
Ring
This topology provides the best reliability with the least
amount of wiring. Since each node has two connection
points to the ring, a single break in any part of the ring
doesn't affect the integrity of the network. The devices,
however, must be designed with this topology in mind. Also,
the network must be interrupted to install or remove nodes.
As with bus topology, ring networks are broadcast by nature.
As you might suspect, two or more ring topologies may be
combined to give the "best of both worlds" in a particular
application. Quite often, industrial networks end up in this
fashion over time, simply from engineers and technicians
joining multiple networks together for the benefit of plant-
wide information access.
Network protocols
Aside from the issues of the physical network (signal types
and voltage levels, connector pinouts, cabling, topology,
etc.), there needs to be a standardized way in which
communication is arbitrated between multiple nodes ina
network, even if its as simple as a two-node, point-to-point
system. When a node "talks" on the network, it is generating
a signal on the network wiring, be it high and low DC
voltage levels, some kind of modulated AC carrier wave
signal, or even pulses of light in a fiber. Nodes that "listen"
are simply measuring that applied signal on the network
(from the transmitting node) and passively monitoring it. If
two or more nodes "talk" at the same time, however, their
output signals may clash (imagine two logic gates trying to
apply opposite signal voltages to a single line on a bus!),
corrupting the transmitted data.
The standardized method by which nodes are allowed to
transmit to the bus or network wiring is called a protocol.
There are many different protocols for arbitrating the use of
a common network between multiple nodes, and I'll cover
just a few here. However, its good to be aware of these few,
and to understand why some work better for some purposes
than others. Usually, a specific protocol is associated with a
standardized type of network. This is merely another "layer"
to the set of standards which are specified under the titles of
various networks.
The International Standards Organization (ISO) has specified
a general architecture of network specifications in their
DIS7498 model (applicable to most any digital network).
Consisting of seven "layers," this outline attempts to
categorize all levels of abstraction necessary to
communicate digital data.
Level 1: Physical Specifies electrical and mechanical
details of communication: wire type, connector design,
Signal types and levels.
Level 2: Data link Defines formats of messages, how
data is to be addressed, and error detection/correction
techniques.
Level 3: Network Establishes procedures for
encapsulation of data into "packets" for transmission
and reception.
Level 4: Transport Among other things, the transport
layer defines how complete data files are to be handled
over a network.
Level 5: Session Organizes data transfer in terms of
beginning and end of a specific transmission. Analogous
to job contro/ on a multitasking computer operating
system.
Level 6: Presentation Includes definitions for
character sets, terminal control, and graphics commands
so that abstract data can be readily encoded and
decoded between communicating devices.
Level 7: Application The end-user standards for
generating and/or interpreting communicated data in its
final form. In other words, the actual computer programs
using the communicated data.
Some established network protocols only cover one or a few
of the DIS7 498 levels. For example, the widely used RS-
232C serial communications protocol really only addresses
the first ("physical") layer of this seven-layer model. Other
protocols, such as the X-windows graphical client/server
system developed at MIT for distributed graphic-user-
interface computer systems, cover all seven layers.
Different protocols may use the same physical layer
standard. An example of this is the RS-422A and RS-485
protocols, both of which use the same differential-voltage
transmitter and receiver circuitry, using the same voltage
levels to denote binary 1's and 0's. On a physical level,
these two communication protocols are identical. However,
on a more abstract level the protocols are different: RS-422A
iS point-to-point only, while RS-485 supports a bus topology
“multidrop" with up to 32 addressable nodes.
Perhaps the simplest type of protocol is the one where there
is Only one transmitter, and all the other nodes are merely
receivers. Such is the case for BogusBus, where a single
transmitter generates the voltage signals impressed on the
network wiring, and one or more receiver units (with 5 lamps
each) light up in accord with the transmitter's output. This is
always the case with a simplex network: there's only one
talker, and everyone else listens!
When we have multiple transmitting nodes, we must
orchestrate their transmissions in such a way that they don't
conflict with one another. Nodes shouldn't be allowed to talk
when another node is talking, so we give each node the
ability to "listen" and to refrain from talking until the
network is silent. This basic approach is called Carrier Sense
Multiple Access (CSMA), and there exists a few variations on
this theme. Please note that CSMA is not a standardized
protocol in itself, but rather a methodology that certain
protocols follow.
One variation is to simply let any node begin to talk as soon
as the network is silent. This is analogous to a group of
people meeting at a round table: anyone has the ability to
start talking, so long as they don't interrupt anyone else. As
soon as the last person stops talking, the next person
waiting to talk will begin. So, what happens when two or
more people start talking at once? In a network, the
simultaneous transmission of two or more nodes is called a
collision. With CSMA/CD (CSMA/Collision Detection), the
nodes that collide simply reset themselves with a random
delay timer circuit, and the first one to finish its time delay
tries to talk again. This is the basic protocol for the popular
Ethernet network.
Another variation of CSMA is CSMA/BA (CSMA/Bitwise
Arbitration), where colliding nodes refer to pre-set priority
numbers which dictate which one has permission to speak
first. In other words, each node has a "rank" which settles
any dispute over who gets to start talking first after a
collision occurs, much like a group of people where
dignitaries and common citizens are mixed. If a collision
occurs, the dignitary is generally allowed to speak first and
the common person waits afterward.
In either of the two examples above (CSMA/CD and
CSMA/BA), we assumed that any node could initiate a
conversation so long as the network was silent. This is
referred to as the "unsolicited" mode of communication.
There is a variation called "solicited" mode for either
CSMA/CD or CSMA/BA where the initial transmission is only
allowed to occur when a designated master node requests
(solicits) a reply. Collision detection (CD) or bitwise
arbitration (BA) applies only to post-collision arbitration as
multiple nodes respond to the master device's request.
An entirely different strategy for node communication is the
Master/Sl/ave protocol, where a single master device allots
time slots for all the other nodes on the network to transmit,
and schedules these time slots so that multiple nodes
cannot collide. The master device addresses each node by
name, one at a time, letting that node talk for a certain
amount of time. When it is finished, the master addresses
the next node, and so on, and so on.
Yet another strategy is the Token-Passing protocol, where
each node gets a turn to talk (one at a time), and then
grants permission for the next node to talk when its done.
Permission to talk is passed around from node to node as
each one hands off the "token" to the next in sequential
order. The token itself is not a physical thing: it is a series of
binary 1's and 0's broadcast on the network, carrying a
specific address of the next node permitted to talk. Although
token-passing protocol is often associated with ring-topology
networks, it is not restricted to any topology in particular.
And when this protocol is implemented in a ring network,
the sequence of token passing does not have to follow the
physical connection sequence of the ring.
Just as with topologies, multiple protocols may be joined
together over different segments of a heterogeneous
network, for maximum benefit. For instance, a dedicated
Master/Slave network connecting instruments together on
the manufacturing plant floor may be linked through a
gateway device to an Ethernet network which links multiple
desktop computer workstations together, one of those
computer workstations acting as a gateway to link the data
to an FDDI fiber network back to the plant's mainframe
computer. Each network type, topology, and protocol serves
different needs and applications best, but through gateway
devices, they can all share the same data.
It is also possible to blend multiple protocol strategies into a
new hybrid within a single network type. Such is the case for
Foundation Fieldbus, which combines Master/Slave with a
form of token-passing. A Link Active Scheduler (LAS) device
sends scheduled "Compel Data" (CD) commands to query
Slave devices on the Fieldbus for time-critical information. In
this regard, Fieldbus is a Master/Slave protocol. However,
when there's time between CD queries, the LAS sends out
"tokens" to each of the other devices on the Fieldbus, one at
a time, giving them opportunity to transmit any
unscheduled data. When those devices are done
transmitting their information, they return the token back to
the LAS. The LAS also probes for new devices on the
Fieldbus with a "Probe Node" (PN) message, which is
expected to produce a "Probe Response" (PR) back to the
LAS. The responses of devices back to the LAS, whether by
PR message or returned token, dictate their standing ona
"Live List" database which the LAS maintains. Proper
operation of the LAS device is absolutely critical to the
functioning of the Fieldbus, so there are provisions for
redundant LAS operation by assigning "Link Master" status
to some of the nodes, empowering them to become alternate
Link Active Schedulers if the operating LAS fails.
Other data communications protocols exist, but these are
the most popular. | had the opportunity to work on an old
(circa 1975) industrial control system made by Honeywell
where a master device called the Highway Traffic Director, or
HTD, arbitrated all network communications. What made
this network interesting is that the signal sent from the HTD
to all slave devices for permitting transmission was not
communicated on the network wiring itself, but rather on
sets of individual twisted-pair cables connecting the HTD
with each slave device. Devices on the network were then
divided into two categories: those nodes connected to the
HTD which were allowed to initiate transmission, and those
nodes not connected to the HTD which could only transmit
in response to a query sent by one of the former nodes.
Primitive and slow are the only fitting adjectives for this
communication network scheme, but it functioned
adequately for its time.
Practical considerations
A principal consideration for industrial control networks,
where the monitoring and control of real-life processes must
often occur quickly and at set times, is the guaranteed
maximum communication time from one node to another. If
you're controlling the position of a nuclear reactor coolant
valve with a digital network, you need to be able to
guarantee that the valve's network node will receive the
proper positioning signals from the control computer at the
right times. If not, very bad things could happen!
The ability for a network to guarantee data "throughput" is
called determinism. A deterministic network has a
guaranteed maximum time delay for data transfer from node
to node, whereas a non-deterministic network does not. The
preeminent example of a non-deterministic network is
Ethernet, where the nodes rely on random time-delay
circuits to reset and re-attempt transmission after a collision.
Being that a node's transmission of data could be delayed
indefinitely from a long series of re-sets and re-tries after
repeated collisions, there is no guarantee that its data will
ever get sent out to the network. Realistically though, the
odds are so astronomically great that such a thing would
happen that it is of little practical concern in a lightly-loaded
network.
Another important consideration, especially for industrial
control networks, is network fault tolerance: that is, how
susceptible is a particular network's signaling, topology,
and/or protocol to failures? We've already briefly discussed
some of the issues surrounding topology, but protocol
impacts reliability just as much. For example, a Master/Slave
network, while being extremely deterministic (a good thing
for critical controls), is entirely dependent upon the master
node to keep everything going (generally a bad thing for
critical controls). If the master node fails for any reason,
none of the other nodes will be able to transmit any data at
all, because they'll never receive their alloted time slot
permissions to do so, and the whole system will fail.
A similar issue surrounds token-passing systems: what
happens if the node holding the token were to fail before
passing the token on to the next node? Some token-passing
systems address this possibility by having a few designated
nodes generate a new token if the network is silent for too
long. This works fine if a node holding the token dies, but it
causes problems if part of a network falls silent because a
cable connection comes undone: the portion of the network
that falls silent generates its own token after awhile, and
you essentially are left with two smaller networks with one
token that's getting passed around each of them to sustain
communication. Trouble occurs, however, if that cable
connection gets plugged back in: those two segmented
networks are joined in to one again, and now there's two
tokens being passed around one network, resulting in nodes'
transmissions colliding!
There is no "perfect network" for all applications. The task of
the engineer and technician is to know the application and
know the operations of the network(s) available. Only then
can efficient system design and maintenance become a
reality.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—|/|+4]l\—
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 15
DIGITAL STORAGE
(MEMORY)
Why_digital?
Digital memory terms and concepts
Modern nonmechanical memory
Historical, nonmechanical memory technologies
Read-only memory
Memory with moving_parts: "Drives"
Why digital?
Although many textbooks provide good introductions to
digital memory technology, | intend to make this chapter
unique in presenting both past and present technologies to
some degree of detail. While many of these memory designs
are obsolete, their foundational principles are still quite
interesting and educational, and may even find re-
application in the memory technologies of the future.
The basic goal of digital memory is to provide a means to
store and access binary data: sequences of 1's and O's. The
digital storage of information holds advantages over analog
techniques much the same as digital communication of
information holds advantages over analog communication.
This is not to say that digital data storage is unequivocally
superior to analog, but it does address some of the more
common problems associated with analog techniques and
thus finds immense popularity in both consumer and
industrial applications. Digital data storage also
complements digital computation technology well, and thus
finds natural application in the world of computers.
The most evident advantage of digital data storage is the
resistance to corruption. Suppose that we were going to
store a piece of data regarding the magnitude of a voltage
signal by means of magnetizing a small chunk of magnetic
material. Since many magnetic materials retain their
strength of magnetization very well over time, this would be
a logical media candidate for long-term storage of this
particular data (in fact, this is precisely how audio and video
tape technology works: thin plastic tape is impregnated with
particles of iron-oxide material, which can be magnetized or
demagnetized via the application of a magnetic field from
an electromagnet coil. The data is then retrieved from the
tape by moving the magnetized tape past another coil of
wire, the magnetized spots on the tape inducing voltage in
that coil, reproducing the voltage waveform initially used to
magnetize the tape).
If we represent an analog signal by the strength of
magnetization on spots of the tape, the storage of data on
the tape will be susceptible to the smallest degree of
degradation of that magnetization. As the tape ages and the
magnetization fades, the analog signal magnitude
represented on the tape will appear to be less than what it
was when we first recorded the data. Also, if any spurious
magnetic fields happen to alter the magnetization on the
tape, even if its only by a small amount, that altering of field
strength will be interpreted upon re-play as an altering (or
corruption) of the signal that was recorded. Since analog
signals have infinite resolution, the smallest degree of
change will have an impact on the integrity of the data
storage.
If we were to use that same tape and store the data in binary
digital form, however, the strength of magnetization on the
tape would fall into two discrete levels: "high" and "low,"
with no valid in-between states. As the tape aged or was
exposed to spurious magnetic fields, those same locations
on the tape would experience slight alteration of magnetic
field strength, but unless the alterations were extreme, no
data corruption would occur upon re-play of the tape. By
reducing the resolution of the signal impressed upon the
magnetic tape, we've gained significant immunity to the
kind of degradation and "noise" typically plaguing stored
analog data. On the other hand, our data resolution would
be limited to the scanning rate and the number of bits
output by the A/D converter which interpreted the original
analog signal, so the reproduction wouldn't necessarily be
"better" than with analog, merely more rugged. With the
advanced technology of modern A/D's, though, the tradeoff
is acceptable for most applications.
Also, by encoding different types of data into specific binary
number schemes, digital storage allows us to archive a wide
variety of information that is often difficult to encode in
analog form. Text, for example, is represented quite easily
with the binary ASCII code, seven bits for each character,
including punctuation marks, spaces, and carriage returns. A
wider range of text is encoded using the Unicode standard,
in like manner. Any kind of numerical data can be
represented using binary notation on digital media, and any
kind of information that can be encoded in numerical form
(which almost any kind can!) is storable, too. Techniques
such as parity and checksum error detection can be
employed to further guard against data corruption, in ways
that analog does not lend itself to.
Digital memory terms and concepts
When we store information in some kind of circuit or device,
we not only need some way to store and retrieve it, but also
to locate precisely where in the device that it is. Most, if not
all, memory devices can be thought of as a series of mail
boxes, folders in a file cabinet, or some other metaphor
where information can be located in a variety of places.
When we refer to the actual information being stored in the
memory device, we usually refer to it as the data. The
location of this data within the storage device is typically
called the address, in a manner reminiscent of the postal
service.
With some types of memory devices, the address in which
certain data is stored can be called up by means of parallel
data lines in a digital circuit (we'll discuss this in more detail
later in this lesson). With other types of devices, data is
addressed in terms of an actual physical location on the
surface of some type of media (the tracks and sectors of
circular computer disks, for instance). However, some
memory devices such as magnetic tapes have a one-
dimensional type of data addressing: if you want to play
your favorite song in the middle of a cassette tape album,
you have to fast-forward to that spot in the tape, arriving at
the proper spot by means of trial-and-error, judging the
approximate area by means of a counter that keeps track of
tape position, and/or by the amount of time it takes to get
there from the beginning of the tape. The access of data
from a storage device falls roughly into two categories:
random access and sequential access. Random access
means that you can quickly and precisely address a specific
data location within the device, and non-random simply
means that you cannot. A vinyl record platter is an example
of a random-access device: to skip to any song, you just
position the stylus arm at whatever location on the record
that you want (compact audio disks so the same thing, only
they do it automatically for you). Cassette tape, on the other
hand, is sequential. You have to wait to go past the other
songs in sequence before you can access or address the
song that you want to skip to.
The process of storing a piece of data to a memory device is
called writing, and the process of retrieving data is called
reading. Memory devices allowing both reading and writing
are equipped with a way to distinguish between the two
tasks, so that no mistake is made by the user (writing new
information to a device when all you wanted to do is see
what was stored there). Some devices do not allow for the
writing of new data, and are purchased "pre-written" from
the manufacturer. Such is the case for vinyl records and
compact audio disks, and this is typically referred to in the
digital world as read-only memory, or ROM. Cassette audio
and video tape, on the other hand, can be re-recorded (re-
written) or purchased blank and recorded fresh by the user.
This is often called read-write memory.
Another distinction to be made for any particular memory
technology is its volatility, or data storage permanence
without power. Many electronic memory devices store binary
data by means of circuits that are either latched in a "high"
or "low" state, and this latching effect holds only as long as
electric power is maintained to those circuits. Such memory
would be properly referred to as volatile. Storage media such
as magnetized disk or tape is nonvolatile, because no source
of power is needed to maintain data storage. This is often
confusing for new students of computer technology,
because the volatile electronic memory typically used for
the construction of computer devices is commonly and
distinctly referred to as RAM (Random Access Memory).
While RAM memory is typically randomly-accessed, so is
virtually every other kind of memory device in the
computer! What "RAM" really refers to is the vo/atility of the
memory, and not its mode of access. Nonvolatile memory
integrated circuits in personal computers are commonly
(and properly) referred to as ROM (Read-Only Memory), but
their data contents are accessed randomly, just like the
volatile memory circuits!
Finally, there needs to be a way to denote how much data
can be stored by any particular memory device. This,
fortunately for us, is very simple and straightforward: just
count up the number of bits (or bytes, 1 byte = 8 bits) of
total data storage space. Due to the high capacity of modern
data storage devices, metric prefixes are generally affixed to
the unit of bytes in order to represent storage space: 1.6
Gigabytes is equal to 1.6 billion bytes, or 12.8 billion bits, of
data storage capacity. The only caveat here is to be aware of
rounded numbers. Because the storage mechanisms of
many random-access memory devices are typically arranged
so that the number of "cells" in which bits of data can be
stored appears in binary progression (powers of 2), a "one
kilobyte" memory device most likely contains 1024 (2 to the
power of 10) locations for data bytes rather than exactly
1000. A "64 kbyte" memory device actually holds 65,536
bytes of data (2 to the 16th power), and should probably be
called a "66 Kbyte" device to be more precise. When we
round numbers in our base-10 system, we fall out of step
with the round equivalents in the base-2 system.
Modern nonmechanical memory
Now we can proceed to studying specific types of digital
storage devices. To start, | want to explore some of the
technologies which do not require any moving parts. These
are not necessarily the newest technologies, as one might
suspect, although they will most likely replace moving-part
technologies in the future.
A very simple type of electronic memory is the bistable
multivibrator. Capable of storing a single bit of data, it is
volatile (requiring power to maintain its memory) and very
fast. The D-latch is probably the simplest implementation of
a bistable multivibrator for memory usage, the D input
serving as the data "write" input, the Q output serving as
the "read" output, and the enable input serving as the
read/write control line:
Data write _D Q_Dataread
. E
Write/Read
Q
If we desire more than one bit's worth of storage (and we
probably do), we'll have to have many latches arranged in
some kind of an array where we can selectively address
which one (or which set) we're reading from or writing to.
Using a pair of tristate buffers, we can connect both the data
write input and the data read output to a common data bus
line, and enable those buffers to either connect the Q output
to the data line (READ), connect the D input to the data line
(WRITE), or keep both buffers in the High-Z state to
disconnect D and Q from the data line (unaddressed mode).
One memory "cell" would look like this, internally:
Memory cell circuit
Data
in/out
Write/Read
Address
Enable
When the address enable input is 0, both tristate buffers will
be placed in high-Z mode, and the latch will be
disconnected from the data input/output (bus) line. Only
when the address enable input is active (1) will the latch be
connected to the data bus. Every latch circuit, of course, will
be enabled with a different "address enable" (AE) input line,
which will come from a 1-of-n output decoder:
16 x 1 bit memory
Memory cell 15 1-bit
data
16-line bus
decoder
P>Pr>
ono
Write/Read
In the above circuit, 16 memory cells are individually
addressed with a 4-bit binary code input into the decoder. If
a cell is not addressed, it will be disconnected from the 1-bit
data bus by its internal tristate buffers: consequently, data
cannot be either written or read through the bus to or from
that cell. Only the cell circuit that is addressed by the 4-bit
decoder input will be accessible through the data bus.
This simple memory circuit is random-access and volatile.
Technically, it is known as a Static RAM. Its total memory
capacity is 16 bits. Since it contains 16 addresses and has a
data bus that is 1 bit wide, it would be designated as a 16 x
1 bit static RAM circuit. As you can see, it takes an incredible
number of gates (and multiple transistors per gate!) to
construct a practical static RAM circuit. This makes the static
RAM a relatively low-density device, with less capacity than
most other types of RAM technology per unit IC chip space.
Because each cell circuit consumes a certain amount of
power, the overall power consumption for a large array of
cells can be quite high. Early static RAM banks in personal
computers consumed a fair amount of power and generated
a lot of heat, too. CMOS IC technology has made it possible
to lower the specific power consumption of static RAM
circuits, but low storage density is still an issue.
To address this, engineers turned to the capacitor instead of
the bistable multivibrator as a means of storing binary data.
A tiny capacitor could serve as a memory cell, complete with
a single MOSFET transistor for connecting it to the data bus
for charging (writing a 1), discharging (writing a 0), or
reading. Unfortunately, such tiny capacitors have very small
Capacitances, and their charge tends to "leak" away through
any circuit impedances quite rapidly. To combat this
tendency, engineers designed circuits internal to the RAM
memory chip which would periodically read all cells and
recharge (or "refresh") the capacitors as needed. Although
this added to the complexity of the circuit, it still required
far less componentry than a RAM built of multivibrators.
They called this type of memory circuit a dynamic RAM,
because of its need of periodic refreshing.
Recent advances in IC chip manufacturing has led to the
introduction of flash memory, which works on a capacitive
storage principle like the dynamic RAM, but uses the
insulated gate of a MOSFET as the capacitor itself.
Before the advent of transistors (especially the MOSFET),
engineers had to implement digital circuitry with gates
constructed from vacuum tubes. As you can imagine, the
enormous comparative size and power consumption of a
vacuum tube as compared to a transistor made memory
circuits like static and dynamic RAM a practical impossibility.
Other, rather ingenious, techniques to store digital data
without the use of moving parts were developed.
Historical, nonmechanical memory
technologies
Perhaps the most ingenious technique was that of the de/ay
line. A delay line is any kind of device which delays the
propagation of a pulse or wave signal. If you've ever heard a
sound echo back and forth through a canyon or cave, you've
experienced an audio delay line: the noise wave travels at
the speed of sound, bouncing off of walls and reversing
direction of travel. The delay line "stores" data on a very
temporary basis if the signal is not strengthened
periodically, but the very fact that it stores data at allisa
phenomenon exploitable for memory technology.
Early computer delay lines used long tubes filled with liquid
mercury, which was used as the physical medium through
which sound waves traveled along the length of the tube. An
electrical/sound transducer was mounted at each end, one
to create sound waves from electrical impulses, and the
other to generate electrical impulses from sound waves. A
stream of serial binary data was sent to the transmitting
transducer as a voltage signal. The sequence of sound
waves would travel from left to right through the mercury in
the tube and be received by the transducer at the other end.
The receiving transducer would receive the pulses in the
Same order as they were transmitted:
Mercury tube delay-line memory
Amplifier Data oo ineneag = gine of sping Amplifier
ae 1) 8) RARE) BB IC
~ ~« ~ ~ ~~
Data pulses moving at speed of light
A feedback circuit connected to the receiving transducer
would drive the transmitting transducer again, sending the
Same sequence of pulses through the tube as sound waves,
storing the data as long as the feedback circuit continued to
function. The delay line functioned like a first-in-first-out
(FIFO) shift register, and external feedback turned that shift
register behavior into a ring counter, cycling the bits around
indefinitely.
The delay line concept suffered numerous limitations from
the materials and technology that were then available. The
EDVAC computer of the early 1950's used 128 mercury-filled
tubes, each one about 5 feet long and storing a maximum of
384 bits. Temperature changes would affect the speed of
sound in the mercury, thus skewing the time delay in each
tube and causing timing problems. Later designs replaced
the liquid mercury medium with solid rods of glass, quartz,
or special metal that delayed torsional (twisting) waves
rather than longitudinal (lengthwise) waves, and operated
at much higher frequencies.
One such delay line used a special nickel-iron-titanium wire
(chosen for its good temperature stability) about 95 feet in
length, coiled to reduce the overall package size. The total
delay time from one end of the wire to the other was about
9.8 milliseconds, and the highest practical clock frequency
was 1 MHz. This meant that approximately 9800 bits of data
could be stored in the delay line wire at any given time.
Given different means of delaying signals which wouldn't be
so susceptible to environmental variables (Such as serial
pulses of light within a long optical fiber), this approach
might someday find re-application.
Another approach experimented with by early computer
engineers was the use of a cathode ray tube (CRT), the type
commonly used for oscilloscope, radar, and television
viewscreens, to store binary data. Normally, the focused and
directed electron beam in a CRT would be used to make bits
of phosphor chemical on the inside of the tube glow, thus
producing a viewable image on the screen. In this
application, however, the desired result was the creation of
an electric charge on the glass of the screen by the impact
of the electron beam, which would then be detected by a
metal grid placed directly in front of the CRT. Like the delay
line, the so-called Williams Tube memory needed to be
periodically refreshed with external circuitry to retain its
data. Unlike the delay line mechanisms, it was virtually
immune to the environmental factors of temperature and
vibration. The IBM model 701 computer sported a Williams
Tube memory with 4 Kilobyte capacity and a bad habit of
"overcharging" bits on the tube screen with successive re-
writes so that false "1" states might overflow to adjacent
spots on the screen.
The next major advance in computer memory came when
engineers turned to magnetic materials as a means of
storing binary data. It was discovered that certain
compounds of iron, namely "ferrite," possessed hysteresis
curves that were almost square:
Hysteresis curve for ferrite
Flux density
(B)
Field intensity (H)
Shown on a graph with the strength of the applied magnetic
field on the horizontal axis (fie/d intensity), and the actual
magnetization (orientation of electron spins in the ferrite
material) on the vertical axis (flux density), ferrite won't
become magnetized one direction until the applied field
exceeds a critical threshold value. Once that critical value is
exceeded, the electrons in the ferrite "snap" into magnetic
alignment and the ferrite becomes magnetized. If the
applied field is then turned off, the ferrite maintains full
magnetism. To magnetize the ferrite in the other direction
(polarity), the applied magnetic field must exceed the
critical value in the opposite direction. Once that critical
value is exceeded, the electrons in the ferrite "snap" into
magnetic alignment in the opposite direction. Once again, if
the applied field is then turned off, the ferrite maintains full
magnetism. To put it simply, the magnetization of a piece of
ferrite is "bistable."
Exploiting this strange property of ferrite, we can use this
natural magnetic "latch" to store a binary bit of data. To set
or reset this "latch," we can use electric current through a
wire or coil to generate the necessary magnetic field, which
will then be applied to the ferrite. Jay Forrester of MIT
applied this principle in inventing the magnetic "core"
memory, which became the dominant computer memory
technology during the 1970's.
Column wire drivers
8x8
magnetic
corememory \/ ... ..
array
Row
wire
drivers
WWE WW WANZINE:
REAR RA
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A grid of wires, electrically insulated from one another,
crossed through the center of many ferrite rings, each of
which being called a "core." As DC current moved through
any wire from the power supply to ground, a circular
magnetic field was generated around that energized wire.
The resistor values were set so that the amount of current at
the regulated power supply voltage would produce slightly
more than 1/2 the critical magnetic field strength needed to
magnetize any one of the ferrite rings. Therefore, if column
#4 wire was energized, all the cores on that column would
be subjected to the magnetic field from that one wire, but it
would not be strong enough to change the magnetization of
any of those cores. However, if column #4 wire and row #5
wire were both energized, the core at that intersection of
column #4 and row #5 would be subjected to a sum of those
two magnetic fields: a magnitude strong enough to "set" or
"reset" the magnetization of that core. In other words, each
core was addressed by the intersection of row and column.
The distinction between "set" and "reset" was the direction
of the core's magnetic polarity, and that bit value of data
would be determined by the polarity of the voltages (with
respect to ground) that the row and column wires would be
energized with.
The following photograph shows a core memory board from
a Data General brand, "Nova" model computer, circa late
1960's or early 1970's. It had a total storage capacity of 4
kbytes (that's k//jobytes, not megabytes!). A ball-point pen is
shown for size comparison:
ats
i
i
i
5
5
a
— pperens Sane! SF
aveear
24
oon omen
pes ogee
ebrnboes 7
erpes eye eet
_” apeqeneppenal
The electronic components seen around the periphery of this
board are used for "driving" the column and row wires with
current, and also to read the status of a core. A close-up
photograph reveals the ring-shaped cores, through which
the matrix wires thread. Again, a ball-point pen is shown for
size Comparison:
fr SOY 0 Ki)
A core memory board of later design (circa 1971) is shown in
the next photograph. Its cores are much smaller and more
densely packed, giving more memory storage capacity than
the former board (8 kbytes instead of 4 kbytes):
And, another close-up of the cores:
UU LLL LLL
Writing data to core memory was easy enough, but reading
that data was a bit of a trick. To facilitate this essential
function, a "read" wire was threaded through a// the cores in
a memory matrix, one end of it being grounded and the
other end connected to an amplifier circuit. A pulse of
voltage would be generated on this "read" wire if the
addressed core changed states (from 0 to 1, or 1 to O). In
other words, to read a core's value, you had to write either a
1 or a O to that core and monitor the voltage induced on the
read wire to see if the core changed. Obviously, if the core's
state was changed, you would have to re-set it back to its
original state, or else the data would have been lost. This
process is known as a destructive read, because data may
be changed (destroyed) as it is read. Thus, refreshing is
necessary with core memory, although not in every case
(that is, in the case of the core's state not changing when
either a 1 or a O was written to it).
One major advantage of core memory over delay lines and
Williams Tubes was nonvolatility. The ferrite cores
maintained their magnetization indefinitely, with no power
or refreshing required. It was also relatively easy to build,
denser, and physically more rugged than any of its
predecessors. Core memory was used from the 1960's until
the late 1970's in many computer systems, including the
computers used for the Apollo space program, CNC machine
tool control computers, business ("mainframe") computers,
and industrial control systems. Despite the fact that core
memory is long obsolete, the term "core" is still used
sometimes with reference to a computer's RAM memory.
All the while that delay lines, Williams Tube, and core
memory technologies were being invented, the simple static
RAM was being improved with smaller active component
(vacuum tube or transistor) technology. Static RAM was
never totally eclipsed by its competitors: even the old ENIAC
computer of the 1950's used vacuum tube ring-counter
circuitry for data registers and computation. Eventually
though, smaller and smaller scale IC chip manufacturing
technology gave transistors the practical edge over other
technologies, and core memory became a museum piece in
the 1980's.
One last attempt at a magnetic memory better than core
was the bubble memory. Bubble memory took advantage of
a peculiar phenomenon in a mineral called garnet, which,
when arranged in a thin film and exposed to a constant
magnetic field perpendicular to the film, supported tiny
regions of oppositely-magnetized "bubbles" that could be
nudged along the film by prodding with other external
magnetic fields. "Tracks" could be laid on the garnet to focus
the movement of the bubbles by depositing magnetic
material on the surface of the film. A continuous track was
formed on the garnet which gave the bubbles a long loop in
which to travel, and motive force was applied to the bubbles
with a pair of wire coils wrapped around the garnet and
energized with a 2-phase voltage. Bubbles could be created
or destroyed with a tiny coil of wire strategically placed in
the bubbles’ path.
The presence of a bubble represented a binary "1" and the
absence of a bubble represented a binary "0." Data could be
read and written in this chain of moving magnetic bubbles
as they passed by the tiny coil of wire, much the same as the
read/write "head" in a cassette tape player, reading the
magnetization of the tape as it moves. Like core memory,
bubble memory was nonvolatile: a permanent magnet
supplied the necessary background field needed to support
the bubbles when the power was turned off. Unlike core
memory, however, bubble memory had phenomenal storage
density: millions of bits could be stored on a chip of garnet
only a couple of square inches in size. What killed bubble
memory as a viable alternative to static and dynamic RAM
was its slow, sequential data access. Being nothing more
than an incredibly long serial shift register (ring counter),
access to any particular portion of data in the serial string
could be quite slow compared to other memory
technologies.
An electrostatic equivalent of the bubble memory is the
Charge-Coupled Device (CCD) memory, an adaptation of the
CCD devices used in digital photography. Like bubble
memory, the bits are serially shifted along channels on the
substrate material by clock pulses. Unlike bubble memory,
the electrostatic charges decay and must be refreshed. CCD
memory is therefore volatile, with high storage density and
sequential access. Interesting, isn't it? The old Williams Tube
memory was adapted from CRT viewing technology, and
CCD memory from video recording technology.
Read-only memory
Read-only memory (ROM) is similar in design to static or
dynamic RAM circuits, except that the "latching" mechanism
is made for one-time (or limited) operation. The simplest
type of ROM is that which uses tiny "fuses" which can be
selectively blown or left alone to represent the two binary
states. Obviously, once one of the little fuses is blown, it
cannot be made whole again, so the writing of such ROM
circuits is one-time only. Because it can be written
(programmed) once, these circuits are sometimes referred to
as PROMs (Programmable Read-Only Memory).
However, not all writing methods are as permanent as blown
fuses. If a transistor latch can be made which is resettable
only with significant effort, a memory device that's
something of a cross between a RAM and a ROM can be
built. Such a device is given a rather oxymoronic name: the
EPROM (Erasable Programmable Read-Only Memory).
EPROMs come in two basic varieties: Electrically-erasable
(EEPROM) and Ultraviolet-erasable (UV/EPROM). Both types
of EPROMs use capacitive charge MOSFET devices to latch
on or off. UV/EPROMs are "cleared" by long-term exposure to
ultraviolet light. They are easy to identify: they havea
transparent glass window which exposes the silicon chip
material to light. Once programmed, you must cover that
glass window with tape to prevent ambient light from
degrading the data over time. EPROMs are often
programmed using higher signal voltages than what is used
during "read-only" mode.
Memory with moving parts: "Drives"
The earliest forms of digital data storage involving moving
parts was that of the punched paper card. Joseph Marie
Jacquard invented a weaving loom in 1780 which
automatically followed weaving instructions set by carefully
placed holes in paper cards. This same technology was
adapted to electronic computers in the 1950's, with the
cards being read mechanically (metal-to-metal contact
through the holes), pneumatically (air blown through the
holes, the presence of a hole sensed by air nozzle
backpressure), or optically (light shining through the holes).
An improvement over paper cards is the paper tape, still
used in some industrial environments (notably the CNC
machine tool industry), where data storage and speed
demands are low and ruggedness is highly valued. Instead
of wood-fiber paper, mylar material is often used, with
optical reading of the tape being the most popular method.
Magnetic tape (very similar to audio or video cassette tape)
was the next logical improvement in storage media. It is still
widely used today, as a means to store "backup" data for
archiving and emergency restoration for other, faster
methods of data storage. Like paper tape, magnetic tape is
sequential access, rather than random access. In early home
computer systems, regular audio cassette tape was used to
store data in modulated form, the binary 1's and O's
represented by different frequencies (similar to FSK data
communication). Access speed was terribly slow (if you were
reading ASCII text from the tape, you could almost keep up
with the pace of the letters appearing on the computer's
screen!), but it was cheap and fairly reliable.
Tape suffered the disadvantage of being sequential access.
To address this weak point, magnetic storage "drives" with
disk- or drum-shaped media were built. An electric motor
provided constant-speed motion. A movable read/write coil
(also Known as a "head") was provided which could be
positioned via servo-motors to various locations on the
height of the drum or the radius of the disk, giving access
that is almost random (you might still have to wait for the
drum or disk to rotate to the proper position once the
read/write coil has reached the right location).
The disk shape lent itself best to portable media, and thus
the floppy disk was born. Floppy disks (so-called because
the magnetic media is thin and flexible) were originally
made in 8-inch diameter formats. Later, the 5-1/4 inch
variety was introduced, which was made practical by
advances in media particle density. All things being equal, a
larger disk has more space upon which to write data.
However, storage density can be improved by making the
little grains of iron-oxide material on the disk substrate
smaller. Today, the 3-1/2 inch floppy disk is the preeminent
format, with a capacity of 1.44 Mbytes (2.88 Mbytes on SCSI
drives). Other portable drive formats are becoming popular,
with loMega's 100 Mbyte "ZIP" and 1 Gbyte "JAZ" disks
appearing as original equipment on some personal
computers.
Still, floppy drives have the disadvantage of being exposed
to harsh environments, being constantly removed from the
drive mechanism which reads, writes, and spins the media.
The first disks were enclosed units, sealed from all dust and
other particulate matter, and were definitely not portable.
Keeping the media in an enclosed environment allowed
engineers to avoid dust altogether, as well as spurious
magnetic fields. This, in turn, allowed for much closer
Spacing between the head and the magnetic material,
resulting in a much tighter-focused magnetic field to write
data to the magnetic material.
The following photograph shows a hard disk drive "platter"
of approximately 30 Mbytes storage capacity. A ball-point
pen has been set near the bottom of the platter for size
reference:
Modern disk drives use multiple platters made of hard
material (hence the name, "hard drive") with multiple
read/write heads for every platter. The gap between head
and platter is much smaller than the diameter of a human
hair. If the hermetically-sealed environment inside a hard
disk drive is contaminated with outside air, the hard drive
will be rendered useless. Dust will lodge between the heads
and the platters, causing damage to the surface of the
media.
Here is a hard drive with four platters, although the angle of
the shot only allows viewing of the top platter. This unit is
complete with drive motor, read/write heads, and associated
electronics. It has a storage capacity of 340 Mbytes, and is
about the same length as the ball-point pen shown in the
previous photograph:
While it is inevitable that non-moving-part technology will
replace mechanical drives in the future, current state-of-the-
art electromechanical drives continue to rival "solid-state"
nonvolatile memory devices in storage density, and ata
lower cost. In 1998, a 250 Mbyte hard drive was announced
that was approximately the size of a quarter (smaller than
the metal platter hub in the center of the last hard disk
photograph)! In any case, storage density and reliability will
undoubtedly continue to improve.
An incentive for digital data storage technology
advancement was the advent of digitally encoded music. A
joint venture between Sony and Phillips resulted in the
release of the "compact audio disc" (CD) to the public in the
late 1980's. This technology is a read-only type, the media
being a transparent plastic disc backed by a thin film of
aluminum. Binary bits are encoded as pits in the plastic
which vary the path length of a low-power laser beam. Data
is read by the low-power laser (the beam of which can be
focused more precisely than normal light) reflecting off the
aluminum to a photocell receiver.
The advantages of CDs over magnetic tape are legion. Being
digital, the information is highly resistant to corruption.
Being non-contact in operation, there is no wear incurred
through playing. Being optical, they are immune to
magnetic fields (which can easily corrupt data on magnetic
tape or disks). It is possible to purchase CD "burner" drives
which contain the high-power laser necessary to write to a
blank disc.
Following on the heels of the music industry, the video
entertainment industry has leveraged the technology of
optical storage with the introduction of the Digital Video
Disc, or DVD. Using a similar-sized plastic disc as the music
CD, a DVD employs closer spacing of pits to achieve much
greater storage density. This increased density allows
feature-length movies to be encoded on DVD media,
complete with trivia information about the movie, director's
notes, and so on.
Much effort is being directed toward the development of a
practical read/write optical disc (CD-W). Success has been
found in using chemical substances whose color may be
changed through exposure to bright laser light, then "read"
by lower-intensity light. These optical discs are immediately
identified by their characteristically colored surfaces, as
opposed to the silver-colored underside of a standard CD.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | +4]
—/ | 4]
Lessons In Electric Circuits
-- Volume IV
Chapter 16
PRINCIPLES OF DIGITAL
COMPUTING
e A binary adder
e Look-up tables
e Finite-state machines
Microprocessors
Microprocessor programming
A binary adder
Suppose we wanted to build a device that could add two
binary bits together. Such a device is known as a half-adder,
and its gate circuit looks like this:
stb
x
B
C
out
The 2 symbol represents the "sum" output of the half-adder,
the sum's least significant bit (LSB). C,,; represents the
"carry" output of the half-adder, the sum's most significant
bit (MSB).
If we were to implement this same function in ladder (relay)
logic, it would look like this:
Either circuit is capable of adding two binary digits together.
The mathematical "rules" of how to add bits together are
intrinsic to the hard-wired logic of the circuits. If we wanted
to perform a different arithmetic operation with binary bits,
such as multiplication, we would have to construct another
circuit. The above circuit designs will only perform one
function: add two binary bits together. To make them do
something else would take re-wiring, and perhaps different
componentry.
In this sense, digital arithmetic circuits aren't much different
from analog arithmetic (operational amplifier) circuits: they
do exactly what they're wired to do, no more and no less. We
are not, however, restricted to designing digital computer
circuits in this manner. It is possible to embed the
mathematical "rules" for any arithmetic operation in the
form of digital data rather than in hard-wired connections
between gates. The result is unparalleled flexibility in
operation, giving rise to a whole new kind of digital device:
the programmable computer.
While this chapter is by no means exhaustive, it provides
what | believe is a unique and interesting look at the nature
of programmable computer devices, starting with two
devices often overlooked in introductory textbooks: /ook-up
table memories and finite-state machines.
Look-up tables
Having learned about digital memory devices in the last
chapter, we know that it is possible to store binary data
within solid-state devices. Those storage "cells" within solid-
state memory devices are easily addressed by driving the
"address" lines of the device with the proper binary value(s).
Suppose we had a ROM memory circuit written, or
programmed, with certain data, such that the address lines
of the ROM served as inputs and the data lines of the ROM
served as outputs, generating the characteristic response of
a particular logic function. Theoretically, we could program
this ROM chip to emulate whatever logic function we wanted
without having to alter any wire connections or gates.
Consider the following example of a 4 x 2 bit ROM memory
(a very small memory!) programmed with the functionality
of a half adder:
Address | Data
4x 2 ROM
| on 3
out
If this ROM has been written with the above data
(representing a half-adder's truth table), driving the A and B
address inputs will cause the respective memory cells in the
ROM chip to be enabled, thus outputting the corresponding
data as the Z (Sum) and C,,; bits. Unlike the half-adder
circuit built of gates or relays, this device can be set up to
perform any logic function at all with two inputs and two
outputs, not just the half-adder function. To change the logic
function, all we would need to do is write a different table of
data to another ROM chip. We could even use an EPROM
chip which could be re-written at will, giving the ultimate
flexibility in function.
It is vitally important to recognize the significance of this
principle as applied to digital circuitry. Whereas the half-
adder built from gates or relays processes the input bits to
arrive at a specific output, the ROM simply remembers what
the outputs should be for any given combination of inputs.
This is not much different from the "times tables" memorized
in grade school: rather than having to calculate the product
of 5 times 6(5+5+5+5+4+5+5 = 30), school-children
are taught to remember that 5 x 6 = 30, and then expected
to recall this product from memory as needed. Likewise,
rather than the logic function depending on the functional
arrangement of hard-wired gates or relays (hardware), it
depends solely on the data written into the memory
(software).
Such a simple application, with definite outputs for every
input, is called a /ook-up table, because the memory device
simply "looks up" what the output(s) should to be for any
given combination of inputs states.
This application of a memory device to perform logical
functions is significant for several reasons:
e Software is much easier to change than hardware.
e Software can be archived on various kinds of memory
media (disk, tape), thus providing an easy way to
document and manipulate the function in a "virtual"
form; hardware can only be "archived" abstractly in the
form of some kind of graphical drawing.
e Software can be copied from one memory device (such
as the EPROM chip) to another, allowing the ability for
one device to "learn" its function from another device.
e Software such as the logic function example can be
designed to perform functions that would be extremely
difficult to emulate with discrete logic gates (or relays!).
The usefulness of a look-up table becomes more and more
evident with increasing complexity of function. Suppose we
wanted to build a 4-bit adder circuit using a ROM. We'd
require a ROM with 8 address lines (two 4-bit numbers to be
added together), plus 4 data lines (for the signed output):
Ay
First A,
4-bit
number A;
A 4-bit
3 result
A,
Second As
4-bit
number Ag
>
“4
With 256 addressable memory locations in this ROM chip,
we would have a fair amount of programming to do, telling it
what binary output to generate for each and every
combination of binary inputs. We would also run the risk of
making a mistake in our programming and have it output an
incorrect sum, if we weren't careful. However, the flexibility
of being able to configure this function (or any function)
through software alone generally outweighs that costs.
Consider some of the advanced functions we could
implement with the above "adder." We know that when we
add two sets of numbers in 2's complement signed notation,
we risk having the answer overflow. For instance, if we try to
add 0111 (decimal 7) to 0110 (decimal 6) with only a 4-bit
number field, the answer we'll get is 1001 (decimal -7 )
instead of the correct value, 13 (7 + 6), which cannot be
expressed using 4 signed bits. If we wanted to, we could
avoid the strange answers given in overflow conditions by
programming this look-up table circuit to output something
else in conditions where we know overflow will occur (that is,
in any case where the real sum would exceed +7 or -8). One
alternative might be to program the ROM to output the
quantity 0111 (the maximum positive value that can be
represented with 4 signed bits), or any other value that we
determined to be more appropriate for the application than
the typical overflowed "error" value that a regular adder
circuit would output. It's all up to the programmer to decide
what he or she wants this circuit to do, because we are no
longer limited by the constraints of logic gate functions.
The possibilities don't stop at customized logic functions,
either. By adding more address lines to the 256 x 4 ROM
chip, we can expand the look-up table to include multiple
functions:
o
_
number
ho
4-bit
result
wo
Second
ol
-bi
number
fos)
~
co
Function
control
A
A
A
A
A,
A
A
A
A
A
wo
With two more address lines, the ROM chip will have 4 times
as many addresses as before (1024 instead of 256). This
ROM could be programmed so that when A8 and AY were
both low, the output data represented the sum of the two 4-
bit binary numbers input on address lines AO through A7,
just as we had with the previous 256 x 4 ROM circuit. For the
addresses A8=1 and A9=0, it could be programmed to
output the difference (subtraction) between the first 4-bit
binary number (AO through A3) and the second binary
number (A4 through A7). For the addresses A8=0 and A9=1,
we could program the ROM to output the difference
(subtraction) of the two numbers in reverse order (Second -
first rather than first - second), and finally, for the addresses
A8=1 and A9=1, the ROM could be programmed to compare
the two inputs and output an indication of equality or
inequality. What we will have then is a device that can
perform four different arithmetical operations on 4-bit binary
numbers, all by "looking up" the answers programmed into
it.
If we had used a ROM chip with more than two additional
address lines, we could program it with a wider variety of
functions to perform on the two 4-bit inputs. There area
number of operations peculiar to binary data (such as parity
check or Exclusive-ORing of bits) that we might find useful
to have programmed in such a look-up table.
Devices such as this, which can perform a variety of
arithmetical tasks as dictated by a binary input code, are
known as Arithmetic Logic Units (ALUs), and they comprise
one of the essential components of computer technology.
Although modern ALUs are more often constructed from very
complex combinational logic (gate) circuits for reasons of
speed, it should be comforting to know that the exact same
functionality may be duplicated with a "dumb" ROM chip
programmed with the appropriate look-up table(s). In fact,
this exact approach was used by IBM engineers in 1959 with
the development of the IBM 1401 and 1620 computers,
which used look-up tables to perform addition, rather than
binary adder circuitry. The machine was fondly known as the
"CADET," which stood for "Can't Add, Doesn't Even Try."
A very common application for look-up table ROMs is in
control systems where a custom mathematical function
needs to be represented. Such an application is found in
computer-controlled fuel injection systems for automobile
engines, where the proper air/fuel mixture ratio for efficient
and clean operation changes with several environmental
and operational variables. Tests performed on engines in
research laboratories determine what these ideal ratios are
for varying conditions of engine load, ambient air
temperature, and barometric air pressure. The variables are
measured with sensor transducers, their analog outputs
converted to digital signals with A/D circuitry, and those
parallel digital signals used as address inputs to a high-
Capacity ROM chip programmed to output the optimum
digital value for air/fuel ratio for any of these given
conditions.
Sometimes, ROMs are used to provide one-dimensional look-
up table functions, for "correcting" digitized signal values so
that they more accurately represent their real-world
significance. An example of such a device is a thermocouple
transmitter, which measures the millivoltage signal
generated by a junction of dissimilar metals and outputs a
signal which is supposed to direct/y correspond to that
junction temperature. Unfortunately, thermocouple
junctions do not have perfectly linear temperature/voltage
responses, and so the raw voltage signal is not perfectly
proportional to temperature. By digitizing the voltage signal
(A/D conversion) and sending that digital value to the
address of a ROM programmed with the necessary correction
values, the ROM's programming could eliminate some of the
nonlinearity of the thermocouple's temperature-to-
millivoltage relationship, so that the final output of the
device would be more accurate. The popular
instrumentation term for such a look-up table is a digital
cCharacterizer.
ND FF SJ DIA 4-20 mA
converter [ 4 converter analo
— = signa
Another application for look-up tables is in special code
translation. A 128 x 8 ROM, for instance, could be used to
translate 7 -bit ASCII code to 8-bit EBCDIC code:
o
—
ASCII
in
w
D
D
D,
D
D
EBCDIC
out
Again, all that is required is for the ROM chip to be properly
programmed with the necessary data so that each valid
ASCII input will produce a corresponding EBCDIC output
code.
Finite-state machines
Feedback is a fascinating engineering principle. It can turn a
rather simple device or process into something substantially
more complex. We've seen the effects of feedback
intentionally integrated into circuit designs with some rather
astounding effects:
e Comparator + negative feedback ----------- > controllable-
gain amplifier
e Comparator + positive feedback ----------- > comparator
with hysteresis
e Combinational logic + positive feedback -->
multivibrator
In the field of process instrumentation, feedback is used to
transform a simple measurement system into something
capable of control:
e Measurement system + negative feedback ---> closed-
loop control system
Feedback, both positive and negative, has the tendency to
add whole new dynamics to the operation of a device or
system. Sometimes, these new dynamics find useful
application, while other times they are merely interesting.
With look-up tables programmed into memory devices,
feedback from the data outputs back to the address inputs
creates a whole new type of device: the Finite State
Machine, or FSM:
A crude Finite State Machine
~— Feedback
oO
OO OO DO
po —
w
The above circuit illustrates the basic idea: the data stored
at each address becomes the next storage location that the
ROM gets addressed to. The result is a specific sequence of
binary numbers (following the sequence programmed into
the ROM) at the output, over time. To avoid signal timing
problems, though, we need to connect the data outputs
back to the address inputs through a 4-bit D-type flip-flop,
so that the sequence takes place step by step to the beat of
a controlled clock pulse:
An improved Finite State Machine
~— Feedback
An analogy for the workings of such a device might be an
array of post-office boxes, each one with an identifying
number on the door (the address), and each one containing
a piece of paper with the address of another P.O. box written
on it (the data). A person, opening the first P.O. box, would
find in it the address of the next P.O. box to open. By storing
a particular pattern of addresses in the P.O. boxes, we can
dictate the sequence in which each box gets opened, and
therefore the sequence of which paper gets read.
Having 16 addressable memory locations in the ROM, this
Finite State Machine would have 16 different stable "states"
in which it could latch. In each of those states, the identity
of the next state would be programmed in to the ROM,
awaiting the signal of the next clock pulse to be fed back to
the ROM as an address. One useful application of such an
FSM would be to generate an arbitrary count sequence, such
as Gray Code:
Address” ----- > Data Gray Code count sequence:
0000 ------- > 0001 0 0000
0001 = ------- > 0011 1 0001
0010 ------- > 0110 2 0011
0011 ------- > 0010 3 0010
0100 ------- > 1100 4 0110
0101 ------- > 0100 5 6111
0110 ------- > 0111 6 0101
O111 ------- > 0101 7 ~~ =0100
1000 —------- > 0000 8 1100
1001 ------- > 1000 9 1101
1010 —- ------ > 1011 10 1111
1011 ------- > 1001 11 1110
1100 =------- > 1101 12 1010
1101 ------- > 1111 13 1011
1110 —------- > 1010 14 =1001
1111 ------- > 1110 15 1000
Try to follow the Gray Code count sequence as the FSM
would do it: starting at 0000, follow the data stored at that
address (0001) to the next address, and so on (0011), and
so on (0010), and so on (0110), etc. The result, for the
program table shown, is that the sequence of addressing
jumps around from address to address in what looks like a
haphazard fashion, but when you check each address that is
accessed, you will find that it follows the correct order for 4-
bit Gray code. When the FSM arrives at its last programmed
state (address 1000), the data stored there is 0000, which
starts the whole sequence over again at address 0000 in
step with the next clock pulse.
We could expand on the capabilities of the above circuit by
using a ROM with more address lines, and adding more
programming data:
~«— Feedback
"function control” Clock
Now, just like the look-up table adder circuit that we turned
into an Arithmetic Logic Unit (+4, -, x, / functions) by utilizing
more address lines as "function control" inputs, this FSM
counter can be used to generate more than one count
sequence, a different sequence programmed for the four
feedback bits (AO through A3) for each of the two function
control line input combinations (A4 = 0 or 1).
Address” ----- > Data Address’ ----- > Data
00000 ------- > 0001 10000 ------- > 0001
00001 ------- > 0010 10001 ------- > 0011
00010 ------- > 0011 10010 ------- > 0110
00011 ------- > 0100 10011 ------- > 0010
00100 ------- > 0101 10100 ------- > 1100
00101 ------- > 0110 10101 ------- > 0100
00110 ------- > 0111 10110 ------- > 0111
00111 ------- > 1000 10111 ------- > 0101
01000 ------- > 1001 11000 ------- > 0000
01001 ------- > 1010 11001 ------- > 1000
01010 ------- > 1011 11010 ------- > 1011
01011 ------- > 1100 11011 ------- > 1001
01100 ------- > 1101 11100 ------- > 1101
01101 ------- > 1110 11101 ------- > 1111
01110 ------- > 1111 11110 ------- > 1010
Q1111 ------- > 0000 11111 ------- > 1110
If A4 is O, the FSM counts in binary; if A4 is 1, the FSM
counts in Gray Code. In either case, the counting sequence
is arbitrary: determined by the whim of the programmer. For
that matter, the counting sequence doesn't even have to
have 16 steps, as the programmer may decide to have the
sequence recycle to 0000 at any one of the steps at all. It is
a completely flexible counting device, the behavior strictly
determined by the software (programming) in the ROM.
We can expand on the capabilities of the FSM even more by
utilizing a ROM chip with additional address input and data
output lines. Take the following circuit, for example:
~«— Feedback
Inputs Outputs
Clock
Here, the DO through D3 data outputs are used exclusively
for feedback to the AO through A3 address lines. Date output
lines D4 through D7 can be programmed to output
something other than the FSM's "state" value. Being that
four data output bits are being fed back to four address bits,
this is still a 16-state device. However, having the output
data come from other data output lines gives the
programmer more freedom to configure functions than
before. In other words, this device can do far more than just
count! The programmed output of this FSM is dependent not
only upon the state of the feedback address lines (AO
through A3), but also the states of the input lines (A4
through A7). The D-type flip/flop's clock signal input does
not have to come from a pulse generator, either. To make
things more interesting, the flip/flop could be wired up to
clock on some external event, so that the FSM goes to the
next state only when an input signal tells it to.
Now we have a device that better fulfills the meaning of the
word "programmable." The data written to the ROM isa
program in the truest sense: the outputs follow a pre-
established order based on the inputs to the device and
which "step" the device is on in its sequence. This is very
close to the operating design of the Turing Machine, a
theoretical computing device invented by Alan Turing,
mathematically proven to be able to solve any known
arithmetic problem, given enough memory capacity.
Microprocessors
Early computer science pioneers such as Alan Turing and
John Von Neumann postulated that for a computing device
to be really useful, it not only had to be able to generate
specific outputs as dictated by programmed instructions,
but it also had to be able to write data to memory, and be
able to act on that data later. Both the program steps and
the processed data were to reside in a common memory
"pool," thus giving way to the label of the stored-program
computer. Turing's theoretical machine utilized a sequential-
access tape, which would store data for a control circuit to
read, the control circuit re-writing data to the tape and/or
moving the tape to a new position to read more data.
Modern computers use random-access memory devices
instead of sequential-access tapes to accomplish essentially
the same thing, except with greater capability.
A helpful illustration is that of early automatic machine tool
control technology. Called open-loop, or sometimes just NC
(numerical control), these control systems would direct the
motion of a machine tool such as a lathe or a mill by
following instructions programmed as holes in paper tape.
The tape would be run one direction through a "read"
mechanism, and the machine would blindly follow the
instructions on the tape without regard to any other
conditions. While these devices eliminated the burden of
having to have a human machinist direct every motion of
the machine tool, it was limited in usefulness. Because the
machine was blind to the real world, only following the
instructions written on the tape, it could not compensate for
changing conditions such as expansion of the metal or wear
of the mechanisms. Also, the tape programmer had to be
acutely aware of the sequence of previous instructions in the
machine's program to avoid troublesome circumstances
(such as telling the machine tool to move the drill bit
laterally while it is still inserted into a hole in the work),
since the device had no memory other than the tape itself,
which was read-only. Upgrading from a simple tape reader to
a Finite State control design gave the device a sort of
memory that could be used to keep track of what it had
already done (through feedback of some of the data bits to
the address bits), so at least the programmer could decide to
have the circuit remember "states" that the machine tool
could be in (such as "coolant on," or tool position). However,
there was still room for improvement.
The ultimate approach is to have the program give
instructions which would include the writing of new data to
a read/write (RAM) memory, which the program could easily
recall and process. This way, the control system could record
what it had done, and any sensor-detectable process
changes, much in the same way that a human machinist
might jot down notes or measurements on a scratch-pad for
future reference in his or her work. This is what is referred to
as CNC, or Closed-loop Numerical Control.
Engineers and computer scientists looked forward to the
possibility of building digital devices that could modify their
own programming, much the same as the human brain
adapts the strength of inter-neural connections depending
on environmental experiences (that is why memory
retention improves with repeated study, and behavior is
modified through consequential feedback). Only if the
computer's program were stored in the same writable
memory "pool" as the data would this be practical. It is
interesting to note that the notion of a self-modifying
program is still considered to be on the cutting edge of
computer science. Most computer programming relies on
rather fixed sequences of instructions, with a separate field
of data being the only information that gets altered.
To facilitate the stored-program approach, we require a
device that is much more complex than the simple FSM,
although many of the same principles apply. First, we need
read/write memory that can be easily accessed: this is easy
enough to do. Static or dynamic RAM chips do the job well,
and are inexpensive. Secondly, we need some form of logic
to process the data stored in memory. Because standard and
Boolean arithmetic functions are so useful, we can use an
Arithmetic Logic Unit (ALU) such as the look-up table ROM
example explored earlier. Finally, we need a device that
controls how and where data flows between the memory, the
ALU, and the outside world. This so-called Contro/ Unit is the
most mysterious piece of the puzzle yet, being comprised of
tri-state buffers (to direct data to and from buses) and
decoding logic which interprets certain binary codes as
instructions to carry out. Sample instructions might be
something like: "add the number stored at memory address
0010 with the number stored at memory address 1101," or,
"determine the parity of the data in memory address 0111."
The choice of which binary codes represent which
instructions for the Control Unit to decode is largely
arbitrary, just as the choice of which binary codes to use in
representing the letters of the alphabet in the ASCII
standard was largely arbitrary. ASCII, however, is now an
internationally recognized standard, whereas control unit
instruction codes are almost always manufacturer-specific.
Putting these components together (read/write memory,
ALU, and control unit) results in a digital device that is
typically called a processor. If minimal memory is used, and
all the necessary components are contained on a single
integrated circuit, it is called a microprocessor. When
combined with the necessary bus-control support circuitry, it
is Known as a Central Processing Unit, or CPU.
CPU operation is summed up in the so-called fetch/execute
cycle. Fetch means to read an instruction from memory for
the Control Unit to decode. A small binary counter in the
CPU (known as the program counter or instruction pointer)
holds the address value where the next instruction is stored
in main memory. The Control Unit sends this binary address
value to the main memory's address lines, and the memory's
data output is read by the Control Unit to send to another
holding register. If the fetched instruction requires reading
more data from memory (for example, in adding two
numbers together, we have to read both the numbers that
are to be added from main memory or from some other
source), the Control Unit appropriately addresses the
location of the requested data and directs the data output to
ALU registers. Next, the Control Unit would execute the
instruction by signaling the ALU to do whatever was
requested with the two numbers, and direct the result to
another register called the accumulator. The instruction has
now been "fetched" and "executed," so the Control Unit now
increments the program counter to step the next instruction,
and the cycle repeats itself.
Microprocessor (CPU)
** Program counter ** |
(increments address value sent to |
3
external memory chip(s) to fetch |==========> Address bus
the next instruction) | (to RAM
emory)
ae Control Unit ** |<=========> Control Bus
| (decodes instructions read from | (to all devices
sharing
| program in memory, enables flow | address and/or data
busses;
| of data to and from ALU, internal | arbitrates all bus
communi -
| registers, and external devices) | cations)
| ** Arithmetic Logic Unit (ALU) ** |
| (performs all mathematical |
| calculations and Boolean |
| functions) |
** Registers ** |
(small read/write memories for |<=========> Data Bus
holding instruction codes, | (from RAM memory and
error codes, ALU data, etc; | external devices)
includes the "accumulator" ) |
|
|
|
other
|
|
As one might guess, carrying out even simple instructions is
a tedious process. Several steps are necessary for the
Control Unit to complete the simplest of mathematical
procedures. This is especially true for arithmetic procedures
such as exponents, which involve repeated executions
("iterations") of simpler functions. Just imagine the sheer
quantity of steps necessary within the CPU to update the
bits of information for the graphic display on a flight
simulator game! The only thing which makes such a tedious
process practical is the fact that microprocessor circuits are
able to repeat the fetch/execute cycle with great speed.
In some microprocessor designs, there are minimal programs
stored within a special ROM memory internal to the device
(called microcode) which handle all the sub-steps necessary
to carry out more complex math operations. This way, only a
single instruction has to be read from the program RAM to
do the task, and the programmer doesn't have to deal with
trying to tell the microprocessor how to do every minute
step. In essence, its a processor inside of a processor; a
program running inside of a program.
Microprocessor programming
The "vocabulary" of instructions which any particular
microprocessor chip possesses is specific to that model of
chip. An Intel 80386, for example, uses a completely
different set of binary codes than a Motorola 68020, for
designating equivalent functions. Unfortunately, there are
no standards in place for microprocessor instructions. This
makes programming at the very lowest level very confusing
and specialized.
When a human programmer develops a set of instructions to
directly tell a microprocessor how to do something (like
automatically control the fuel injection rate to an engine),
they're programming in the CPU's own "language." This
language, which consists of the very same binary codes
which the Control Unit inside the CPU chip decodes to
perform tasks, is often referred to as machine language.
While machine language software can be "worded" in binary
notation, it is often written in hexadecimal form, because it
is easier for human beings to work with. For example, I'll
present just a few of the common instruction codes for the
Intel 8080 micro-processor chip:
Hexadecimal Binary Instruction description
| 7B 01111011 Move contents of register A to
register E
|
| 87 10000111 Add contents of register A to
register D
|
| 1C 00011100 Increment the contents of register E
by 1
|
| D3 11010011 Output byte of data to data bus
Even with hexadecimal notation, these instructions can be
easily confused and forgotten. For this purpose, another aid
for programmers exists called assembly language. With
assembly language, two to four letter mnemonic words are
used in place of the actual hex or binary code for describing
program steps. For example, the instruction 7B for the Intel
8080 would be "Mov A,E" in assembly language. The
mnemonics, of course, are useless to the microprocessor,
which can only understand binary codes, but it is an
expedient way for programmers to manage the writing of
their programs on paper or text editor (word processor).
There are even programs written for computers called
assemblers which understand these mnemonics, translating
them to the appropriate binary codes for a specified target
microprocessor, so that the programmer can write a program
in the computer's native language without ever having to
deal with strange hex or tedious binary code notation.
Once a program is developed by a person, it must be written
into memory before a microprocessor can execute it. If the
program is to be stored in ROM (which some are), this can be
done with a special machine called a ROM programmer, or
(if you're masochistic), by plugging the ROM chip into a
breadboard, powering it up with the appropriate voltages,
and writing data by making the right wire connections to the
address and data lines, one at a time, for each instruction. If
the program is to be stored in volatile memory, such as the
operating computer's RAM memory, there may be a way to
type it in by hand through that computer's keyboard (Some
computers have a mini-program stored in ROM which tells
the microprocessor how to accept keystrokes from a
keyboard and store them as commands in RAM), even if it is
too dumb to do anything else. Many "hobby" computer kits
work like this. If the computer to be programmed is a fully-
functional personal computer with an operating system, disk
drives, and the whole works, you can simply command the
assembler to store your finished program onto a disk for
later retrieval. To "run" your program, you would simply type
your program's filename at the prompt, press the Enter key,
and the microprocessor's Program Counter register would be
set to point to the location ("address") on the disk where the
first instruction is stored, and your program would run from
there.
Although programming in machine language or assembly
language makes for fast and highly efficient programs, it
takes a lot of time and skill to do so for anything but the
simplest tasks, because each machine language instruction
is So crude. The answer to this is to develop ways for
programmers to write in "high level" languages, which can
more efficiently express human thought. Instead of typing in
dozens of cryptic assembly language codes, a programmer
writing in a high-level language would be able to write
something like this...
Print "Hello, world! "
...and expect the computer to print "Hello, world!" with no
further instruction on how to do so. This is a great idea, but
how does a microprocessor understand such "human"
thinking when its vocabulary is so limited?
The answer comes in two different forms: /nterpretation, or
compilation. Just like two people speaking different
languages, there has to be some way to transcend the
language barrier in order for them to converse. A translator
is needed to translate each person's words to the other
person's language, one way at a time. For the
microprocessor, this means another program, written by
another programmer in machine language, which recognizes
the ASCII character patterns of high-level commands such as
Print (P-r-i-n-t) and can translate them into the necessary
bite-size steps that the microprocessor can directly
understand. If this translation is done during program
execution, just like a translator intervening between two
people in a live conversation, it is called "interpretation." On
the other hand, if the entire program is translated to
machine language in one fell swoop, like a translator
recording a monologue on paper and then translating all the
words at one sitting into a written document in the other
language, the process is called "compilation."
Interpretation is simple, but makes for a slow-running
program because the microprocessor has to continually
translate the program between steps, and that takes time.
Compilation takes time initially to translate the whole
program into machine code, but the resulting machine code
needs no translation after that and runs faster as a
consequence. Programming languages such as BASIC and
FORTH are interpreted. Languages such as C, C++,
FORTRAN, and PASCAL are compiled. Compiled languages
are generally considered to be the languages of choice for
professional programmers, because of the efficiency of the
final product.
Naturally, because machine language vocabularies vary
widely from microprocessor to microprocessor, and since
high-level languages are designed to be as universal as
possible, the interpreting and compiling programs necessary
for language translation must be microprocessor-specific.
Development of these interpreters and compilers is a most
impressive feat: the people who make these programs most
definitely earn their keep, especially when you consider the
work they must do to keep their software product current
with the rapidly-changing microprocessor models appearing
on the market!
To mitigate this difficulty, the trend-setting manufacturers of
microprocessor chips (most notably, Intel and Motorola) try
to design their new products to be backwardly compatible
with their older products. For example, the entire instruction
set for the Intel 80386 chip is contained within the latest
Pentium IV chips, although the Pentium chips have
additional instructions that the 80386 chips lack. What this
means is that machine-language programs (compilers, too)
written for 80386 computers will run on the latest and
greatest Intel Pentium IV CPU, but machine-language
programs written specifically to take advantage of the
Pentium's larger instruction set will not run on an 80386,
because the older CPU simply doesn't have some of those
instructions in its vocabulary: the Control Unit inside the
80386 cannot decode them.
Building on this theme, most compilers have settings that
allow the programmer to select which CPU type he or she
wants to compile machine-language code for. If they select
the 80386 setting, the compiler will perform the translation
using only instructions known to the 80386 chip; if they
select the Pentium setting, the compiler is free to make use
of all instructions known to Pentiums. This is analogous to
telling a translator what minimum reading level their
audience will be: a document translated for a child will be
understandable to an adult, but a document translated for
an adult may very well be gibberish to a child.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4/l—
— 4 —
Appendix 1
ABOUT THIS BOOK
Purpose
They say that necessity is the mother of invention. At least
in the case of this book, that adage is true. As an industrial
electronics instructor, | was forced to use a sub-standard
textbook during my first year of teaching. My students were
daily frustrated with the many typographical errors and
obscure explanations in this book, having spent much time
at home struggling to comprehend the material within.
Worse yet were the many incorrect answers in the back of
the book to selected problems. Adding insult to injury was
the $100+ price.
Contacting the publisher proved to be an exercise in futility.
Even though the particular text | was using had been in
print and in popular use for a couple of years, they claimed
my complaint was the first they'd ever heard. My request to
review the draft for the next edition of their book was met
with disinterest on their part, and | resolved to find an
alternative text.
Finding a Suitable alternative was more difficult than | had
imagined. Sure, there were plenty of texts in print, but the
really good books seemed a bit too heavy on the math and
the less intimidating books omitted a lot of information | felt
was important. Some of the best books were out of print, and
those that were still being printed were quite expensive.
It was out of frustration that | compiled Lessons in Electric
Circuits from notes and ideas | had been collecting for years.
My primary goal was to put readable, high-quality
information into the hands of my students, but a secondary
goal was to make the book as affordable as possible. Over
the years, | had experienced the benefit of receiving free
instruction and encouragement in my pursuit of learning
electronics from many people, including several teachers of
mine in elementary and high school. Their selfless
assistance played a key role in my own studies, paving the
way for a rewarding career and fascinating hobby. If only |
could extend the gift of their help by giving to other people
what they gavetome...
So, | decided to make the book freely available. More than
that, | decided to make it "open," following the same
development model used in the making of free software
(most notably the various UNIX utilities released by the Free
Software Foundation, and the Linux operating system,
whose fame Is growing even as | write). The goal was to
copyright the text -- so as to protect my authorship -- but
expressly allow anyone to distribute and/or modify the text
to suit their own needs with a minimum of legal
encumbrance. This willful and formal revoking of standard
distribution limitations under copyright is whimsically
termed copyleft. Anyone can "copyleft" their creative work
simply by appending a notice to that effect on their work,
but several Licenses already exist, covering the fine legal
points in great detail.
The first such License | applied to my work was the GPL --
General Public License -- of the Free Software Foundation
(GNU). The GPL, however, is intended to copyleft works of
computer software, and although its introductory language
is broad enough to cover works of text, its wording is not as
clear as it could be for that application. When other, less
specific copyleft Licenses began appearing within the free
software community, | chose one of them (the Design
Science License, or DSL) as the official notice for my project.
In "copylefting" this text, | guaranteed that no instructor
would be limited by a text insufficient for their needs, as |
had been with error-ridden textbooks from major publishers.
I'm sure this book in its initial form will not satisfy everyone,
but anyone has the freedom to change it, leveraging my
efforts to suit variant and individual requirements. For the
beginning student of electronics, learn what you can from
this book, editing it as you feel necessary if you come across
a useful piece of information. Then, if you pass it on to
someone else, you will be giving them something better
than what you received. For the instructor or electronics
professional, feel free to use this as a reference manual,
adding or editing to your heart's content. The only "catch" is
this: if you plan to distribute your modified version of this
text, you must give credit where credit is due (to me, the
Original author, and anyone else whose modifications are
contained in your version), and you must ensure that
whoever you give the text to is aware of their freedom to
similarly share and edit the text. The next chapter covers
this process in more detail.
It must be mentioned that although | strive to maintain
technical accuracy in all of this book's content, the subject
matter is broad and harbors many potential dangers.
Electricity maims and kills without provocation, and
deserves the utmost respect. | strongly encourage
experimentation on the part of the reader, but only with
circuits powered by small batteries where there is no risk of
electric shock, fire, explosion, etc. High-power electric
circuits should be left to the care of trained professionals!
The Design Science License clearly states that neither | nor
any contributors to this book bear any liability for what is
done with its contents.
The use of SPICE
One of the best ways to learn how things work is to follow
the inductive approach: to observe specific instances of
things working and derive general conclusions from those
observations. In science education, labwork is the
traditionally accepted venue for this type of learning,
although in many cases labs are designed by educators to
reinforce principles previously learned through lecture or
textbook reading, rather than to allow the student to learn
on their own through a truly exploratory process.
Having taught myself most of the electronics that | know, |
appreciate the sense of frustration students may have in
teaching themselves from books. Although electronic
components are typically inexpensive, not everyone has the
means or opportunity to set up a laboratory in their own
homes, and when things go wrong there's no one to ask for
help. Most textbooks seem to approach the task of education
from a deductive perspective: tell the student how things
are supposed to work, then apply those principles to specific
instances that the student may or may not be able to
explore by themselves. The inductive approach, as useful as
it is, is hard to find in the pages of a book.
However, textbooks don't have to be this way. | discovered
this when | started to learn a computer program called
SPICE. It is a text-based piece of software intended to model
circuits and provide analyses of voltage, current, frequency,
etc. Although nothing is quite as good as building real
circuits to gain knowledge in electronics, computer
simulation is an excellent alternative. In learning how to use
this powerful tool, | made a discovery: SPICE could be used
within a textbook to present circuit simulations to allow
students to "observe" the phenomena for themselves. This
way, the readers could learn the concepts inductively (by
interpreting SPICE's output) as well as deductively (by
interpreting my explanations). Furthermore, in seeing SPICE
used over and over again, they should be able to
understand how to use it themselves, providing a perfectly
safe means of experimentation on their own computers with
circuit simulations of their own design.
Another advantage to including computer analyses in a
textbook is the empirical verification it adds to the concepts
presented. Without demonstrations, the reader is left to take
the author's statements on faith, trusting that what has
been written is indeed accurate. The problem with faith, of
course, is that it is only as good as the authority in which it
is placed and the accuracy of interpretation through which it
is understood. Authors, like all human beings, are liable to
err and/or communicate poorly. With demonstrations,
however, the reader can immediately see for themselves
that what the author describes is indeed true.
Demonstrations also serve to clarify the meaning of the text
with concrete examples.
SPICE is introduced early in volume | (DC) of this book
series, and hopefully in a gentle enough way that it doesn't
create confusion. For those wishing to learn more, a chapter
in the Reference volume (volume V) contains an overview of
SPICE with many example circuits. There may be more flashy
(graphic) circuit simulation programs in existence, but SPICE
is free, a virtue complementing the charitable philosophy of
this book very nicely.
Acknowledgements
First, | wish to thank my wife, whose patience during those
many and long evenings (and weekends!) of typing has
been extraordinary.
| also wish to thank those whose open-source software
development efforts have made this endeavor all the more
affordable and pleasurable. The following is a list of various
free computer software used to make this book, and the
respective programmers:
e GNU/Linux Operating System -- Linus Torvalds, Richard
Stallman, and a host of others too numerous to mention.
e Vim text editor -- Bram Moolenaar and others.
Xcircuit drafting program -- Tim Edwards.
SPICE circuit simulation program -- too many
contributors to mention.
e T-X text processing system -- Donald Knuth and others.
e Texinfo document formatting system -- Free Software
Foundation.
¢ LATEX document formatting system -- Leslie Lamport and
others.
e Gimp image manipulation program -- too many
contributors to mention.
Appreciation is also extended to Robert L. Boylestad, whose
first edition of Introductory Circuit Analysis taught me more
about electric circuits than any other book. Other important
texts in my electronics studies include the 1939 edition of
The "Radio" Handbook, Bernard Grob's second edition of
Introduction to Electronics I, and Forrest Mims' original
Engineer's Notebook.
Thanks to the staff of the Bellingham Antique Radio
Museum, who were generous enough to let me terrorize their
establishment with my camera and flash unit. Thanks as well
to David Randolph of the Arlington Water Treatment facility
in Arlington, Washington, for allowing me to take
photographs of the equipment during a technical tour.
| wish to specifically thank Jeffrey Elkner and all those at
Yorktown High School for being willing to host my book as
part of their Open Book Project, and to make the first effort
in contributing to its form and content. Thanks also to David
Sweet (website: [*]) and Ben Crowell (website: [*]) for
providing encouragement, constructive criticism, and a
wider audience for the online version of this book.
Thanks to Michael Stutz for drafting his Design Science
License, and to Richard Stallman for pioneering the concept
of copyleft.
Last but certainly not least, many thanks to my parents and
those teachers of mine who saw in me a desire to learn
about electricity, and who kindled that flame into a passion
for discovery and intellectual adventure. | honor you by
helping others as you have helped me.
Tony Kuphaldt, July 2001
"A candle loses nothing of its light when lighting
another"
Kahlil Gibran
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] 4]\—
—| | +]
Appendix 2
George Zogopoulos Papaliakos
CONTRIBUTOR LIST
How to contribute to this book
As a copylefted work, this book is open to revision and expansion by
any interested parties. The only "catch" is that credit must be given
where credit is due. This /s a copyrighted work: it is notin the public
domain!
If you wish to cite portions of this book in a work of your own, you
must follow the same guidelines as for any other copyrighted work.
Here is a Sample from the Design Science License:
The Work is copyright the Author. All rights to the Work are reserved
by the Author, except as specifically described below. This License
describes the terms and conditions under which the Author permits you
to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright Law.
Your right to operate, perform, read or otherwise interpret and/or
execute the Work is unrestricted; however, you do so at your own risk,
because the Work comes WITHOUT ANY WARRANTY -- see Section 7 ("NO
WARRANTY") below.
If you wish to modify this book in any way, you must document the
nature of those modifications in the "Credits" section along with your
name, and ideally, information concerning how you may be
contacted. Again, the Design Science License:
Permission is granted to modify or sample from a copy of the Work,
producing a derivative work, and to distribute the derivative work
under the terms described in the section for distribution above,
provided that the following terms are met:
(a) The new, derivative work is published under the terms of this
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title can not be confused with the Work, or with a version of
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(c) Appropriate authorship credit is given: for the differences
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the Work remains attributed to the original Author; appropriate
notice must be included with the new work indicating the nature
and the dates of any modifications of the Work made by you.
Given the complexities and security issues surrounding the
maintenance of files comprising this book, it is recommended that
you submit any revisions or expansions to the original author (Tony R.
Kuphaldt). You are, of course, welcome to modify this book directly by
editing your own personal copy, but we would all stand to benefit
from your contributions if your ideas were incorporated into the
online “master copy” where all the world can see it.
Credits
All entries arranged in alphabetical order of surname. Major
contributions are listed by individual name with some detail on the
nature of the contribution(s), date, contact info, etc. Minor
contributions (typo corrections, etc.) are listed by name only for
reasons of brevity. Please understand that when | classify a
contribution as “minor,” it is in no way inferior to the effort or value of
a “major” contribution, just smaller in the sense of less text changed.
Any and all contributions are gratefully accepted. | am indebted to all
those who have given freely of their own knowledge, time, and
resources to make this a better book!
Tony R. Kuphaldt
« Date(s) of contribution(s): 1996 to present
¢ Nature of contribution: Original author.
e Contact at: liec0@lycos.com
Dennis Crunkilton
« Date(s) of contribution(s): July 2004 to present
e Nature of contribution:Original author: Karnaugh mapping
chapter; 04/2004; Shift registers chapter, June 2005.
¢ Nature of contribution: Mini table of contents, all chapters
except appendicies; html, latex, ps, pdf; See Devel/tutorial.html;
01/2006.
¢ Contact at: dcrunkilton(at)att(dot)net
George Zogopoulos Papaliakos
« Date(s) of contribution(s): November 2010
e Nature of contribution: Original author: “Author of Finite State
Machines” section, chapter 11.
e Contact at: Georacer@allaboutcircuits.com
David Zitzelsberger
Date(s) of contribution(s): November 2007
Nature of contribution: Original author: “Combinatorial Logic
Functions” chapter 9.
Contact at: davidzitzelsberger(at) yahoo(dot) com
Your name here
Date(s) of contribution(s): Month and year of contribution
Nature of contribution: Insert text here, describing how you
contributed to the book.
Contact at: my email@provider.net
Typo corrections and other “minor” contributions
line-allaboutcircuits.com (June 2005) Typographical error
correction in Volumes 1,2,3,5, various chapters ,(:s/visa-versa/vice
versa/).
Dennis Crunkilton (October 2005) Typographical capitlization
correction to sectiontitles, chapter 9.
Jeff DeFreitas (March 2006)Improve appearance: replace “/" and
”/" Chapters: Al, A2.
Paul Stokes, Program Chair, Computer and Electronics
Engineering Technology, ITT Technical Institute, Houston, Tx
(October 2004) Change (10015 = -849 + 749 = -1j9) to (1001, =
-819 + lio = -110), CH2, Binary Arithmetic
Paul Stokes, Program Chair Computer and Electronics
Engineering Technology, ITT Technical Institute, Houston, Tx
(October 2004) Near "Fold up the corners" change Out=B'C' to
Out=B'D', 14118.eps same change, Karnaugh Mapping
The students of Bellingham Technical College's Instrumentation
program, .
Roger Hollingsworth (May 2003) Suggested a way to make the
PLC motor control system fail-safe.
Jan-Willem Rensman (May 2002) Suggested the inclusion of
Schmitt triggers and gate hysteresis to the "Logic Gates" chapter.
Don Stalkowski (June 2002) Technical help with PostScript-to-
PDF file format conversion.
¢ Joseph Teichman (June 2002) Suggestion and technical help
regarding use of PNG images instead of JPEG.
MWalden@allaboutcircuits.com (June 2008) “Karnaugh
Mapping”, Larger Karnaugh Maps, error: s/A'B'D/A'B'D'/.
studiot@allaboutcircuits.com (March 2008) Ch 15, s/disk/disc/
in CDROM .
Keith@allaboutcircuits.com (April 2008) Ch 12, s/sat/stage ;
0437 3.eps correction to caption.
psomero@allaboutcircuits.com (April 2008) Ch 8, image
14122.eps replace 2nd instance A'B'C'D' with A'B'C'D.
Ron Harrison (March 2009) Ch 13, image 04256.png,
04257 .png Change text and images from 8-comparator to 7-
comparator, s/16/15 s/256/255 .
johndb@allaboutcircuits.com (June 2009) Ch 7, s/first on/first
one.
ruXx@allaboutcircuits.com (November 2009) Ch 7, s/if any
only/if and only/ .
tone_b@allaboutcircuits.com (January 2010) Ch 1, 9,
s/Lets/Let's/ ; ch 9 too/also.
manual@allaboutcircuits.com (January 2012) Ch 9, images:
04477.eps, 0447 8.eps, 0447 9.eps corrected.
Dcrunkilton@allaboutcircuits.com (January 2012) Ch 8,
image: 14159.eps corrected.
tshuck@allaboutcircuits.com (January 2014) Ch 11,
numerous: http://forum.allaboutcircuits.com/showthread.php?
t=80569
Schoen8 5@allaboutcircuits.com (February 2014) Ch 9, 7-
segment text, images: 14169.* 14174.* 14175.* 14176.* 14171.*
04464.* 04483.* 04489.* 04487 .*
jJetBlue@allaboutcircuits.com (August 2015) Ch 9, 7-segment
images: 14176.* 14171.* 04464.* 04483.*
kiroma@allaboutcircuits.com (August 2015) Ch 8, s/(A'+B') =
AB/(A'+B')' = AB/
djsfantasi@allaboutcircuits.com (August 2015) Ch 11,
s/Initial-Stan By/Initial-Stand By/
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt,
under the terms and conditions of the Design Science License.
—/ | 4]
Appendix 3
DESIGN SCIENCE LICENSE
Copyright © 1999-2000 Michael Stutz stutz@dsl.org
Verbatim copying of this document is permitted, in any
medium.
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The intent of this license is to be a general "copyleft" that
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distributed, and modified.
Whereas "design science" is a strategy for the development
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and subsequently improve the universal standard of living,
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END OF TERMS AND CONDITIONS
[ $Id: dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]
— 4 —
Lessons In Electrig@ircuits
Volume V - Rejeraae
an
ie
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or
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% £
si
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F a i
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* 5 =
ata .
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2 :
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ts
D io
» ¥ ~
”
Copyright (C) 2000-2020, Tony R.
Kuphaldt
See the Design Science License (Appendix 3)
for details regarding copying and distribution
Revised April 19, 2007
Master Index
Chapter 1: USEFUL EQUATIONS AND CONVERSION
FACTORS
Chapter 2: COLOR CODES
Chapter 3: CONDUCTOR AND INSULATOR TABLES
Chapter 4: ALGEBRA REFERENCE
Chapter 5: TRIGONOMETRY REFERENCE
Chapter 6: CALCULUS REFERENCE
Chapter 7: USING THE SPICE CIRCUIT SIMULATION
PROGRAM
Chapter 8: TROUBLESHOOTING -- THEORY AND PRACTICE
Chapter 9: CIRCUIT SCHEMATIC SYMBOLS
Chapter 10: PERIODIC TABLE OF THE ELEMENTS
Appendix 1: ABOUT THIS BOOK
Appendix 2: CONTRIBUTOR LIST
Appendix 3: DESIGN SCIENCE LICENSE
Download printable versions of this
volume
Adobe PDF format:
REF. pdf
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PostScript
1
"How do! view and/or print PostScript documents," you ask?
Easy! Just download some free software at:
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There you'll find GSview and Ghostscript, two progams
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computer system, you can get by with a little skill and a
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or word processor, and contains all the instructions
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e Sed (stands for Stream EDitor), a common UNIX utility
for performing search-and-replace commands on text
files. Required to convert SUbML source code into HTML,
TeX, LaTeX, and other formats. This is all you need for
generating HTML output!
LaTeX2e, a document formatting system designed as an
extension to TeX, Donald Knuth's outstanding text
processing system. You can also get by with just plain
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it will lack table-of-contents and index entries.
If you opt for the smaller of the two files (REFtiny.tar.gz),
you'll also need a set of graphic manipulation utilities
released as a package called ImageMagick. Specifically, the
utility you'll need is named Mogrify. The larger of the two
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This makes for a large file. The smaller source archive file
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conversion on your end. If you have access to other image
manipulation software capable of converting hundreds of
files with a batch command, you won't have to use
ImageMagick.
Back to Master Index
—/ | 4]
Lessons In Electric Circuits
-- Volume V
Chapter 1
USEFUL EQUATIONS AND
CONVERSION FACTORS
DC circuit equations and laws
o Ohm's and Joule's Laws
o Kirchhoff's Laws
Series circuit rules
Parallel circuit rules
Series and parallel component equivalent values
o Series and parallel resistances
o Series and parallel inductances
o Series and Parallel Capacitances
Capacitor sizing equation
Inductor sizing equation
Time constant equations
o Value of time constant in series RC and RL circuits
o Calculating voltage or current at specified time
o Calculating time at specified voltage or current
AC circuit equations
o Inductive reactance
o Capacitive reactance
Impedance in relation to R and X
Ohm's Law for AC
Series and Parallel Impedances
Resonance
o AC power
Decibels
Metric prefixes and unit conversions
Data
oO Oo 0 O
e Contributors
DC circuit equations and laws
Ohm's and Joule's Laws
Ohm’s Law
= sacle =
E=I1R l= R= i
Joule’s Law
P=1E p-E p-=(LR
R
Where,
E= Voltage in volts
1= Current in amperes (amps)
R= Resistance in ohms
P = Power in watts
NOTE: the symbol "V" ("U" in Europe) is sometimes used to
represent voltage instead of "E". In some cases, an author or
circuit designer may choose to exclusively use "V" for
voltage, never using the symbol "E." Other times the two
symbols are used interchangeably, or "E" is used to
represent voltage from a power source while "V" is used to
represent voltage across a load (voltage "drop").
Kirchhoff's Laws
"The algebraic sum of all voltages in a loop must equal
zero."
Kirchhoff's Voltage Law (KVL)
"The algebraic sum of all currents entering and exiting a
node must equal zero."
Kirchhoff's Current Law (KCL)
Series circuit rules
e Components in a series circuit share the same current.
total = 1a = lo =. -- Ip
e Total resistance in a series circuit is equal to the sum of
the individual resistances, making it greaterthan any of
the individual resistances. Riota) = Ry + Ro +... Ry
e Total voltage in a series circuit is equal to the sum of the
individual voltage drops. Eyota) = E, + Eo +... Ep,
Parallel circuit rules
e Components in a parallel circuit share the same voltage.
Etotal = FE, = Er =..- Ep
e Total resistance in a parallel circuit is /ess than any of
the individual resistances. Riota; = 1 /(1/R, + 1/Ro +...
1/R,,)
e Total current in a parallel circuit is equal to the sum of
the individual branch currents. liora) = 14 + lo +... Ip
Series and parallel component
equivalent values
Series and parallel resistances
Resistances
| | eae = R, + R, +... R,
1
1 1 oa
Ry Rot R
2 n
R
parallel —
Series and parallel inductances
Inductances
i = L, + L, +... L,,
l
Lara = ——_———
parallel cL 1 ES
ig i” *“*#e Le
Where,
L = Inductance in henrys
Series and Parallel Capacitances
Capacitances
l
l cg
© l
oe Cs Cc,
series
Coarallel > C; 7 C, +... C,,
Where,
C = Capacitance in farads
Capacitor sizing equation
C= EA
d
Where,
C= Capacitance in Farads
€= Permittivity of dielectric (absolute, not
relative)
A= Area of plate overlap in square meters
d= Distance between plates in meters
e&= Eo K
€)>= Permittivity of free space
&)= 8.8562 x 10'* F/m
K= Dielectric constant of material
between plates (see table)
Dielectric constants
Dielectric K Dielectric K
PTFE, Teflon
Mineral oil
Polypropylene
Polystyrene
Waxed paper
Transformer oil
Wood, oak
Hard Rubber
Silicones
Bakelite
Quartz, fused 8
Wood, maple 4
Glass 9-7.
Castor oil 0
Wood, birch =
Mica, muscovite .0-
3.
Glass-bonded mica 6.
Poreclain, steatite 6.5
Alumina Al,O, 8-10.0
Water, distilled 80
27.6
1200-1500
8.7
9.3
QDONWNN NYONNN?-
A formula for capacitance in picofarads using practical
dimensions:
Cc
Where,
or
K =
A=
A’ =
d=
d=
n=
_ 0.0885K(n-1) A _ 0.225K(n-1)A’
d i
ee
Capacitance in picofarads t
Dielectric constant
Area of one plate in square centimeters
Area of one plate in square inches
Thickness in centimeters
Thickness in inches
Number of plates
Inductor sizing equation
N7HA
|
LL = [Lo
L=
Where,
L = Inductance of coil in Henrys
N= Number of turns in wire coil (straight wire = 1)
i= Permeability of core material (absolute, not relative)
L;= Relative permeability, dimensionless (1,=1 for air)
lg = 1.26 x 10 *T-m/At permeability of free space
A = Area of coil in square meters = tr°
|= Average length of coil in meters
Wheeler's formulas for inductance of air core coils which
follow are useful for radio frequency inductors. The following
formula for the inductance of a single layer air core solenoid
coil is accurate to approximately 1% for 2r/l < 3. The thick
coil formula is 1% accurate when the denominator terms are
approximately equal. Wheeler's spiral formula is 1%
accurate for c>0.2r. While this is a "round wire" formula, it
may still be applicable to printed circuit spiral inductors at
reduced accuracy.
ad an y
: 18 c 45 Cc
a) r 7% r T
ae ey +
N-r-
oe Or + 10-1 ||}
0.8N7r Nr
L = 5 +914 10c = 8r+11c
Where,
= Inductance of coil in microhenrys
N= Number of turns of wire
Mean radius of coil in inches
Length of coil in inches
Thickness of coil in inches
‘
|
Cc
The inductance in henries of a square printed circuit
inductor is given by two formulas where p=q, and p#q.
L = 27-10'°(D**/p? (14R"'Y?
Where,
D = coil dimension in cm
N = number of turns
R= p/q
L=85-10°°DN*™?
Where.
D = dimension, cm
N = number turns
P=q
The wire table provides "turns per inch" for enamel magnet
wire for use with the inductance formulas for coils.
AWG turns/ |AWG turns/ | AWG turns/|AWG turns/
gauge inch gauge inch gauge inch gauge inch
r =
RP We rio ~ A
ON ~ NY ~ hd
ox
N~OO~AWWOr se
~6
Pa
<0
0
.8
9
6
4
Time constant equations
Value of time constant in series RC and RL
circuits
Time constant in seconds = RC
Time constant in seconds = L/R
Calculating voltage or current at specified
time
Universal Time Constant Formula
Change = Final ea ( ae
et
Where,
Final = Value of calculated variable after infinite time
(its ultimate value)
Start= Initial value of calculated variable
e= Euler's number (=2.7182818)
t= Timein seconds
t= Timeconstant for circuit in seconds
Calculating time at specified voltage or
current
t=—t {In/l - _ Change
Final - Start
AC circuit equations
Inductive reactance
X, = 2nfL
Where,
X, = Inductive reactance in ohms
f= Frequency in hertz
L =Inductance in henrys
Capacitive reactance
Where,
X, = Inductive reactance in ohms
f= Frequency in hertz
C = Capacitance in farads
Impedance in relation to R and X
Z, = R + JX,
Zc= R-jXc
Ohm's Law for AC
- _E _E
E=1Z l= > Za
Where,
E= Voltage in volts
1= Current in amperes (amps)
Z= Impedance in ohms
Series and Parallel Impedances
Lreries = Zi + Z, +... Zh
l
ee =
aay ie, 7
Zoarallel =
n
NOTE: All impedances must be calculated in complex
number form for these equations to work.
Resonance
£ l
resonant — >
2m \V LC
NOTE: This equation applies to a non-resistive LC circuit. In
circuits containing resistance as well as inductance and
Capacitance, this equation applies only to series
configurations and to parallel configurations where R is very
small.
AC power
P = true power Pore. pes
Measured in units of Watts
Q=reactive power Q=lVX Q= =
Measured in units of Volt-Amps-Reactive (VAR)
S=apparent power S=ITZ S= a ge
Zz
Measured in units of Volt-Amps
P = (1E)(power factor)
S= VP+Q
Power factor = cos (Z phase angle)
Decibels
Avian)
Ay ids) = 20 log Aviratio) Ay (ratioy = 10
Avan)
Aygpy = 29 log Ajgatic) rae a
Apap)
10
Apiap) = 10 log Ap iratioy Apiratioy = 10
Metric prefixes and unit conversions
e Metric prefixes
¢ Yotta = 1024 Symbol: Y
¢ Zetta = 102! Symbol: Z
¢ Exa = 10!8 Symbol: E
¢ Peta = 101° Symbol: P
¢ Tera = 1012 Symbol: T
¢ Giga = 102 Symbol: G
¢ Mega = 10°© Symbol: M
¢ Kilo = 103 Symbol: k
¢ Hecto = 102 Symbol: h
Deca = 10! Symbol: da
Deci = 10°! Symbol: d
Centi = 10° Symbol: c
Milli = 10°? Symbol: m
Micro = 10° Symbol: yu
Nano = 10°9 Symbol: n
Pico = 10°12 Symbol: p
Femto = 10°!° Symbol: f
Atto = 10°18 Symbol: a
Zepto = 102! Symbol: z
Yocto = 104 Symbol: y
METRIC PREFIX SCALE
T G M kK m Mu n p
tera giga mega kilo (none) milli micro nano pico
Lo ge: get’ ae ie ee ae ier. Lor
Fl bg be
io 10° 10°: 20°
hecto deca deci centi
a da d Cc
e Conversion factors for temperature
e OF = (°C)(9/5) + 32
e °C = (°F - 32)(5/9)
e OR = °F + 459.67
e 0% = °C + 273.15
Conversion equivalencies for volume
1 US gallon (gal) = 231.0 cubic inches (in?) = 4 quarts
(qt) = 8 pints (pt) = 128 fluid ounces (fl. oz.) = 3.7854
liters (1)
1 Imperial gallon (gal) = 160 fluid ounces (fl. oz.) =
4.546 liters (1)
Conversion equivalencies for distance
1 inch (in) = 2.540000 centimeter (cm)
Conversion equivalencies for velocity
1 mile per hour (mi/h) = 88 feet per minute (ft/m) =
1.46667 feet per second (ft/s) = 1.60934 kilometer per
hour (km/h) = 0.44704 meter per second (m/s) =
0.868976 knot (knot -- international)
Conversion equivalencies for weight
1 pound (lb) = 16 ounces (0z) = 0.45359 kilogram (kg)
Conversion equivalencies for force
1 pound-force (lbf) = 4.44822 newton (N)
Acceleration of gravity (free fall), Earth standard
9.806650 meters per second per second (m/s?) =
32.1740 feet per second per second (ft/s?)
Conversion equivalencies for area
1 acre = 43560 square feet (ft?) = 4840 square yards
(yd?) = 4046.86 square meters (m2)
Conversion equivalencies for pressure
1 pound per square inch (psi) = 2.03603 inches of
mercury (in. Hg) = 27.6807 inches of water (in. W.C.) =
6894.757 pascals (Pa) = 0.0680460 atmospheres (Atm)
= 0.0689476 bar (bar)
Conversion equivalencies for energy or work
1 british thermal unit (BTU -- "International Table") =
251.996 calories (cal -- "International Table") = 1055.06
joules J) = 1055.06 watt-seconds (W-s) = 0.293071
watt-hour (W-hr) = 1.05506 x 102° ergs (erg) = 778.169
foot-pound-force (ft-Ibf)
Conversion equivalencies for power
1 horsepower (hp -- 550 ft-lbf/s) = 745.7 watts (W) =
2544.43 british thermal units per hour (BTU/hr) =
0.0760181 boiler horsepower (hp -- boiler)
Conversion equivalencies for motor torque
Newton-meter Gram-centimeter Pound-inch Pound-ftoot Ounce-inch
(n-m) (g-cm) (lb-in) (1b-ft) (oz-in)
l 8.85 0.738
981 x 10° 8.68x10° 723x 10°
0.113 l 0.0833
1.36 12 l
7.062 x 10° 0.0625 5.21x 10°
Locate the row corresponding to known unit of torque along
the left of the table. Multiply by the factor under the column
for the desired units. For example, to convert 2 oz-in torque
to n-m, locate oz-in row at table left. Locate 7.062 x 10-3 at
intersection of desired n-m units column. Multiply 2 oz-in x
(7.062 x 103 ) = 14.12 x 103 n-m.
Converting between units is easy if you have a set of
equivalencies to work with. Suppose we wanted to convert
an energy quantity of 2500 calories into watt-hours. What
we would need to do is find a set of equivalent figures for
those units. In our reference here, we see that 251.996
calories is physically equal to 0.293071 watt hour. To
convert from calories into watt-hours, we must form a "unity
fraction" with these physically equal figures (a fraction
composed of different figures and different units, the
numerator and denominator being physically equal to one
another), placing the desired unit in the numerator and the
initial unit in the denominator, and then multiply our initial
value of calories by that fraction.
Since both terms of the "unity fraction" are physically equal
to one another, the fraction as a whole has a physical value
of 1, and so does not change the true value of any figure
when multiplied by it. When units are canceled, however,
there will be a change in units. For example, 2500 calories
multiplied by the unity fraction of (0.293071 w-hr/ 251.996
cal) = 2.9075 watt-hours.
Original figure
"Unity fraction”
... Cancelling units .. .
2500 caloeies 0.293071 watt-hour
l
25 1.996 caloriés
Converted figure | 2.9075 watt-hours
The "unity fraction" approach to unit conversion may be
extended beyond single steps. Suppose we wanted to
convert a fluid flow measurement of 175 gallons per hour
into liters per day. We have two units to convert here:
gallons into liters, and hours into days. Remember that the
word "per" in mathematics means "divided by," so our initial
figure of 175 gallons perhour means 175 gallons divided by
hours. Expressing our original figure as such a fraction, we
multiply it by the necessary unity fractions to convert
gallons to liters (3.7854 liters = 1 gallon), and hours to days
(1 day = 24 hours). The units must be arranged in the unity
fraction in such a way that undesired units cancel each
other out above and below fraction bars. For this problem it
means using a gallons-to-liters unity fraction of (3.7854
liters / 1 gallon) and a hours-to-days unity fraction of (24
hours / 1 day):
Original figure 175 gallons/hour
: . 3.7854 liters
"Unity fraction" | 2S?" eS
y 1 gallon
"Unity fraction" 24 hours
1 day
.. cancelling units. . .
175 gallefis 3.7854 liters 24 hours
1 hout 1 galton l day
Converted figure | 15,898.68 liters/day
Our final (converted) answer is 15898.68 liters per day.
Data
Conversion factors were found in the 78" edition of the CRC
Handbook of Chemistry and Physics, and the 3" edition of
Bela Liptak's /nstrument Engineers’ Handbook -- Process
Measurement and Analysis.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Gerald Gardner (January 2003): Addition of Imperial
gallons conversion.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=|] 4]\—
—/ | 4]
Lessons In Electric Circuits
-- Volume V
Chapter 2
COLOR CODES
Resistor Color Codes
o Example #1
Example #2
Example #3
Example #4
Example #5
Example #6
Wiring Color Codes
Bibliography
e
o Oo 0 0 O
Resistor Color Codes
Components and wires are coded with colors to identify their
value and function.
Tolerance (%
orange] 3 | 10° |
Yetow [4 | 10° |
Toren | 5 | 10° | 05
Paue [6 | 10 | 025
voit [7 | 107 [01
Torey |e | 1]
Pwnte [9 [10 |
reoa || 107 [5
rsiver || 107 | 10
a
The colors brown, red, green, blue, and violet are used as
tolerance codes on 5-band resistors only. All 5-band resistors
use a colored tolerance band. The blank (20%) "band" is
only used with the "4-band" code (3 colored bands + a blank
"band").
Digit Digit Multiplier Tolerance
ne eee
— TTL +-—
4-band code
Digit Digit Digit Multiplier Tolerance
— TL —
5-band code
Example #1
—iL -—-
A resistor colored Yellow-Violet-Orange-Gold would be 47 kQ
with a tolerance of +/- 5%.
Example #2
—h F—
A resistor colored Green-Red-Gold-Si/ver would be 5.2 Q with
a tolerance of +/- 10%.
Example #3
—lL +
A resistor colored White-Violet-Black would be 97 QO witha
tolerance of +/- 20%. When you see only three color bands
on a resistor, you know that it is actually a 4-band code with
a blank (20%) tolerance band.
Example #4
—Ihh—
A resistor colored Orange-Orange-Black-Brown-Violet would
be 3.3 kQ with a tolerance of +/- 0.1%.
Example #5
—HLt—
A resistor colored Brown-Green-Grey-Si/ver-Red would be
1.58 QO with a tolerance of +/- 2%.
Example #6
—HLt—
A resistor colored B/ue-Brown-Green-Silver-Blue would be
6.15 QO with a tolerance of +/- 0.25%.
Wiring Color Codes
Wiring for AC and DC power distribution branch circuits are
color coded for identification of individual wires. In some
jurisdictions all wire colors are specified in legal documents.
In other jurisdictions, only a few conductor colors are so
codified. In that case, local custom dictates the “optional”
wire colors.
IEC, AC: Most of Europe abides by IEC (International
Electrotechnical Commission) wiring color codes for AC
branch circuits. These are listed in Table below. The older
color codes in the table reflect the previous style which did
not account for proper phase rotation. The protective ground
wire (listed as green-yellow) is green with yellow stripe.
IEC (most of Europe) AC power circuit wiring color codes.
[Function |label| Color, 1EC [Color, old 1EC
Protective earth PE _[green-yellow/green-yellow _
Neutral iN plue blue
Line, single phase__[orown __[brown or black
Line, 3-phase brown or black
Line, 3-phase brown or black
Line, 3-phase brown or black
UK, AC: The United Kingdom now follows the IEC AC wiring
color codes. Table below lists these along with the obsolete
domestic color codes. For adding new colored wiring to
existing old colored wiring see Cook. [PCk]
UK AC power circuit wiring color codes.
Neutral _N blue black
Line, single phase. brown [red
ee ee ee
Line, 3-phase L1 brown
ine spss 42 lek yw
US, AC:The US National Electrical Code only mandates
white (or grey) for the neutral power conductor and bare
copper, green, or green with yellow stripe for the protective
ground. In principle any other colors except these may be
used for the power conductors. The colors adopted as local
practice are shown in Table below. Black, red, and blue are
used for 208 VAC three-phase; brown, orange and yellow are
used for 480 VAC. Conductors larger than #6 AWG are only
available in black and are color taped at the ends.
US AC power circuit wiring color codes.
label Color, common Color,
alternative
Protective PG bare, green, or green- green
ground yellow
Neutral |N_white grey
aaa | black or red (2nd hot) |
phase
Line, 3-phase |L1_ black =—_|brown
Line, 3-phase |L2 red ==~—_jorange
Line, 3-phase |L3 |blue —_—_—ifyellow
Canada: Canadian wiring is governed by the CEC (Canadian
Electric Code). See Table below. The protective ground is
green or green with yellow stripe. The neutral is white, the
hot (live or active) single phase wires are black , and red in
the case of a second active. Three-phase lines are red, black,
and blue.
Canada AC power circuit wiring color codes.
Protective ground IPG green or green-yellow
Neutral IN white
IEC, DC: DC power installations, for example, solar power
and computer data centers, use color coding which follows
the AC standards. The IEC color standard for DC power
cables is listed in Table below, adapted from Table 2, Cook.
[PCk]
IEC DC power circuit wiring color codes.
label] Color
green-
Protective earth
yellow
2-wire unearthed DC Power
System
L
ositive
egative
2-wire earthed DC Power System
ositive (of a negative earthed) circuit brown
egative (of a negative earthed) circuit |M b|
ositive (of a positive earthed) circuit M
own
=
28
S
a)
label
grey
__2-wire earthed DC Power System | __
Positive (of a negative earthed) circuit _|L+ _|
Sa
|
Negative (of a positive earthed) circuit |/L-
3-wire earthed DC Power System
grey
blue
grey
US DC power: The US National Electrical Code (for both AC
and DC) mandates that the grounded neutral conductor of a
power system be white or grey. The protective ground must
be bare, green or green-yellow striped. Hot (active) wires
may be any other colors except these. However, common
practice (per local electrical inspectors) is for the first hot
(live or active) wire to be black and the second hot to be red.
The recommendations in Table below are by Wiles. [JWi] He
makes no recommendation for ungrounded power system
colors. Usage of the ungrounded system is discouraged for
safety. However, red (+) and black (-) follows the coloring of
the grounded systems in the table.
:
Le
ae
Positive brown
US recommended DC power circuit wiring color codes.
Protective ground
2-wire ungrounded DC
Power System
2-wire grounded DC Power
System
Positive (of a negative
Color
bare, green, or
green-yellow
no recommendation
(red)
no recommendation
(black)
Hh
red
“Pe
rounded) circuit
Negative (of a negative
grounded) circuit
Positive (of a positive grounded)
circuit
N
_
Negative (of a positive L- black
grounded) circuit
L+
N
3-wire grounded DC Power
System
ositive
Mid-wire (center tap)
|
Negative SSCS Clack
Bibliography
1. [PCk]Paul Cook, “Harmonised colours and alphanumeric
marking”, IEE Wiring Matters, Spring 2004 at
http://www.iee.org/Publish/WireRegs/IEE_ Harmonized _co
lours.pdf
2. JWiljohn Wiles, “Photovoltaic Power Systems and the
National Electrical Code: Suggested Practices”,
Southwest Technology Development Institute, New
Mexico State University, March 2001 at
http://www.re.sandia.gov/en/ti/tu/Copy%200f%20NEC20
00.pdf
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
— 4 —>
—/ | 4]
Lessons In Electric Circuits
-- Volume V
Chapter 3
CONDUCTOR AND
INSULATOR TABLES
Copper wire gage table
Coefficients of specific resistance
Temperature coefficients of resistance
Critical temperatures for superconductors
Dielectric strengths for insulators
Data
Copper wire gage table
Soild copper wire table: below
Soild copper wire table:
Diameter soley area Weight
sectional
inches cir. mils sq. inches nee
/O0 |0.4600 211,600 0.1662 640.5
/O |0.4096 167,800 0.1318 07.9
/O0 0.3648 133,100 0.1045
1/0 0.3249 105,500 0.08289
83,690 0.06573
0.2576 66,370 0.05213 200.9
SSS 2S eSSS_S_aai
rT
=
N
00
Ye)
WW
2
Heed
|
2294
.2043
.1819
.1620
.1443
1285
.1144
.1019
.09074
.08081
.07196
.06408
.05707
.05082
04526
.04030
.03589
.03196
.02846
02535
02257
02010
.01790
.01594
.01420
.01264
.01126
.01003
oO
oO
Socios cosci
52,630
41,740
33,100
26,250
20,820
16,510
13,090
10,380
8,234
6,530
3,178
4,107
3,257
2,983
2,048
1,624
1,288
1,022
810.1
642.5
509.5
404.0
320.4
254.1
201500
1598
126.7
2005 0
GJ) NOT NOT NOT INO TINO TRO TRO TRO JRO YEN fT tt | ~S U1 WWJ
|} CO})/ NM |} Oy }} O1]) BI} GUI] NO] Ee |} O |] MO] CO]] N |} Oy |} O1]]) BI] GUN eR |] oO
|
|
0.008155
0.006467
0.005129
0.004067
0.003225
0.002558
0.002028
0.001609
0.001276
0.001012
0.0008023
0.0006363
0.0005046
0.0004001
0.0003173
0.0002517
0.0001996
0.0001583
0.0001255
0.00009954
0.00007 894
——
0.04134
0.03278
0.02600
0.02062
0.01635
0.01297
0.01028
3) fe)
26.4
00.2
79.46
3.02
o:07
9.63
1.43
4.92
oy
5.68
2.43
858
.818
.200
917
899
.092
A452
1.945
542
253
wooo
1692
.6100
4837
3836
3042
4
0
0
ERR nt
31 0.008928
[32 0.007950
33 0.007080
4 (0.006305
35 0.005615
36 0.005000
rf
79.70
63:21
50.13
39.75
31.52
25.00
2983 0
15.72
12.47
9.888
7.842
6.219
4.932
3.911
0.00006260
0.00004964
0.00003937
0.00003122
0.00002476
0.00001963
0.00001557
0.00001235_
0.000009793
0.000007766
0.000006159
0.000004884
0.00000387 3
0.000003072
0.2413
1905
1517
.1203
09542
.07567
.06001
.04759
.03774
02993
.02374
.01882
.01493
.01184
0
0
0
0
0
0
FEE EEESSE:
Ampacities of copper wire: below
Ampacities of copper wire, in free air at 30° C:
INSULATION
TYPE:
RUW, T
Current
Rating
@ 60 degrees
C
9
ae
THW, THWN FEP, FEPB
THHN, XHHW
Current
Rating
@ 90 degrees
Current Rating
@ 75 degrees C
ri2S
is
ee
jas F13 18
hes ——‘ifiss_——i20
qo 9530S
20225. —~*iaes——~—S—iSOO
Bi0_260 ‘(glo ——~—S—i5O
ao_g00——*igeo. SOS
* = estimated values; normally, these small wire sizes are
not manufactured with these insulation types, above.
Coefficients of specific resistance
Specific resistance table: below
Specific resistance at 20° C:
lea al
Nichrome Alloy —(675——=SSS2.2
NichromeV Alloy 650. ——«a108..—SS
Manganin Alloy (290. 48.21
Constantan Alloy 272.97 45.38
a nn a
Steel* Alloy 100 16.62
Platinum [Element (63.16
ron Element 57.81
Nickel Element (41.69
35.49
Molybdenum|Element (32.12
Mungsten Element (31.76
Aluminum Element [15.94 [2.650
13.32 214
Copper [Element [10.09 —_—i1.678
Silver Element (9.546 (1.587
TM
. = Steel alloy at 99.5% iron, 0.5%
carbon
Temperature coefficients of resistance
Temperature coefficient table: below
Temperature coefficient (a) per degree C:
| Material |[Element/Alloy|Temp. coefficient
Nickel ___|Element__|0.005866
ron Element (0.005671
MolybdenumElement 0.004579
Tungsten [Element __|0.004403
Aluminum _|Element___|0.004308
Copper _|Element__—(0.004041
Silver Element 0.003819
Platinum [Element 0.003729
Gold __—(Element__|0.003715
Zinc Element 0.003847
Stee* [Alloy (0.003
Nichrome Alloy (0.00017
NichromeV Alloy 0.00013
Manganin [Alloy _|0.000015
Constantan [Alloy __|+0.000074
iron, 0.5%
* = Steel alloy at 99.5%
carbon
Critical temperatures for
superconductors
Critical temperature, superconductors below
Critical temperatures given in Kelvins
Material
Aluminum
Cadmium
lead
Mercury
Niobium
Thorium
Tin
Titanium
Uranium
Niobium/Tin
Cupric
sulphide
Element or
Alloy
Element
Element
Element
Element
Element
Element
Element
Element
ELement
Element
Alloy
Compound
Critical
temperature(K)
1.20
0.56
72
4.16
8.70
1.37
Bae
0.39
1.0
0.91
18.1
Critical temperatures, high temperature
superconuctors below
Critical temperatures, high temperature superconuctors in
Kelvins
Note: all critical temperatures given at zero magnetic field
strength, above.
Dielectric strengths for insulators
Dielectric strength: below
Dielectric strength in kilovolts per inch (kV/in):
Material* Dielectric strength
Vacuum 20
Air 20to75
|
|
|
Porcelain
araffin Wax
Transformer Oil
akelite
ubber
hellac
aper
Teflon
Glass
Mica
Porcelain _|
Paraffin Wax _|
Transformer Oil
Bakelite
Rubber
Shellac
Paper
Teflon
Glass
Mica
40 to 200
200 to 300
300 to 550
50 to 700
250
500
000 to 3000
000
* = Materials listed are specially prepared for electrical use,
above.
Data
Tables of specific resistance and temperature coefficient of
resistance for elemental materials (not alloys) were derived
from figures found in the 78th edition of the CRC Handbook
of Chemistry and Physics. Superconductivity data from
Collier's Encyclopedia (volume 21, 1968, page 640).
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
the terms and conditions of the Design
Kuphaldt, under
Science License.
|| 4] l_—
—/ | 4]
Lessons In Electric Circuits
-- Volume V
Chapter 4
ALGEBRA REFERENCE
Basic identities
Arithmetic properties
o The associative property
o The commutative property
o The distributive property
Properties of exponents
Radicals
o Definition of a radical
o Properties of radicals
Important constants
o Euler's number
o Pi
Logarithms
o Definition of a logarithm
o Properties of logarithms
Factoring equivalencies
The quadratic formula
Sequences
o Arithmetic sequences
o Geometric sequences
Factorials
o Definition of a factorial
o Strange factorials
Solving_simultaneous equations
o Substitution method
o Addition method
Contributors
Basic identities
a+O=a la=a Oa= 0
a _ O _ a _
r= 4 = 0 z= 1
5 = undefined
Note: while division by zero is popularly thought to be equal
to infinity, this is not technically true. In some practical
applications it may be helpful to think the result of such a
fraction approaching positive infinity as a positive
denominator approaches zero (imagine calculating current
I=E/R in a circuit with resistance approaching zero -- current
would approach infinity), but the actual fraction of anything
divided by zero is undefined in the scope of either real or
complex numbers.
Arithmetic properties
The associative property
In addition and multiplication, terms may be arbitrarily
associated with each other through the use of parentheses:
a+(b+c)=(a+b)+c a(bc) = (ab)c
The commutative property
In addition and multiplication, terms may be arbitrarily
interchanged, or commutated:
a+b=b+a ab=ba
The distributive property
a(b + c)= ab+ ac
Properties of exponents
mon + nm nym
a a! _ al n (ab) = al b‘
myn _ mn a mn
(a a
Radicals
Definition of a radical
When people talk of a "Square root," they're referring toa
radical with a root of 2. This is mathematically equivalent to
a number raised to the power of 1/2. This equivalence is
useful to know when using a calculator to determine a
strange root. Suppose for example you needed to find the
fourth root of anumber, but your calculator lacks a "4th
root" button or function. If it has a y* function (which any
scientific calculator should have), you can find the fourth
root by raising that number to the 1/4 power, or x9-2°.
It is important to remember that when solving for an even
root (Square root, fourth root, etc.) of any number, there are
two valid answers. For example, most people know that the
square root of nine is three, but negative three is also a valid
answer, since (-3)? = 9 just as 32 = 9.
Properties of radicals
\/ ab = \/ a \/ b
on fa _Va_ =
b Vb
Important constants
Euler's number
Euler's constant is an important value for exponential
functions, especially scientific applications involving decay
(such as the decay of a radioactive substance). It is
especially important in calculus due to its uniquely self-
similar properties of integration and differentiation.
e€ approximately equals:
2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69996
=3 l
=) ah
k=0
cs ae Sone Sree ai.
a 1 oe ee
n!
Pi
Pi (11) is defined as the ratio of a circle's circumference to its
diameter.
Pi approximately equals:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37511
Note: For both Euler's constant (e) and pi (m1), the spaces
shown between each set of five digits have no mathematical
significance. They are placed there just to make it easier for
your eyes to "piece" the number into five-digit groups when
manually copying.
Logarithms
Definition of a logarithm
b* =x
Then:
log, X= y
Where,
b = "Base" of the logarithm
"log" denotes a common logarithm (base = 10), while "In"
denotes a natural logarithm (base = e).
Properties of logarithms
(log a) + (log b) = log ab
(log a) - (log b) = log >
log a" = (m)(log a)
(l )
a og m) _ m
These properties of logarithms come in handy for performing
complex multiplication and division operations. They are an
example of something called a transform function, whereby
one type of mathematical operation is transformed into
another type of mathematical operation that is simpler to
solve. Using a table of logarithm figures, one can multiply or
divide numbers by adding or subtracting their logarithms,
respectively. then looking up that logarithm figure in the
table and seeing what the final product or quotient is.
Slide rules work on this principle of logarithms by
performing multiplication and division through addition and
subtraction of distances on the slide.
Slide rule
Cursor
Slide
Numerical quantities are represented by
the positioning of the slide.
Marks on a slide rule's scales are spaced in a logarithmic
fashion, so that a linear positioning of the scale or cursor
results in a nonlinear indication as read on the scale(s).
Adding or subtracting lengths on these logarithmic scales
results in an indication equivalent to the product or
quotient, respectively, of those lengths.
Most slide rules were also equipped with special scales for
trigonometric functions, powers, roots, and other useful
arithmetic functions.
Factoring equivalencies
x -y = (x+y)(x-y)
x -y =(x-y\(x° +xy+y)
The quadratic formula
For a polynomial expression in
the form of: ax” + bx +c =0
-b + Vb’ -4ac
2a
A =
Sequences
Arithmetic sequences
An arithmetic sequence is a series of numbers obtained by
adding (or subtracting) the same value with each step. A
child's counting sequence (1, 2,3,4,...) is a simple
arithmetic sequence, where the common difference is 1:
that is, each adjacent number in the sequence differs by a
value of one. An arithmetic sequence counting only even
numbers (2, 4, 6, 8, ...) or only odd numbers (1, 3, 5,7,9,..
.) would have a common difference of 2.
In the standard notation of sequences, a lower-case letter "a"
represents an element (a single number) in the sequence.
The term "a," refers to the element at the n" step in the
sequence. For example, "a3" in an even-counting (common
difference = 2) arithmetic sequence starting at 2 would be
the number 6, "a" representing 4 and "a," representing the
starting point of the sequence (given in this example as 2).
A capital letter "A" represents the sum of an arithmetic
sequence. For instance, in the same even-counting
sequence starting at 2, A, is equal to the sum of all
elements from a, through ay, which of course would be 2 +
4+6+8,or 20.
a, =a,.,+d a, =a, +d(n-1)
Where:
d= The "common difference"
Example of an arithmetic sequence:
Fy cy hy Dp Oy bg Ay eyed aca
A, =a, +a,+... 4,
n
Geometric sequences
A geometric sequence, on the other hand, is a series of
numbers obtained by multiplying (or dividing) by the same
value with each step. A binary place-weight sequence (1, 2,
4,8,16, 32, 64,...) is a simple geometric sequence, where
the common ratio is 2: that is, each adjacent number in the
sequence differs by a factor of two.
n-1l
a, = r(a,_;) a, = a,(r )
Where:
r= The "common ratio"
Example of a geometric sequence:
3, 12, 48, 192, 768, 3072...
A,=a,+a,+...a
rt
a,(1 - r")
a
Factorials
Definition of a factorial
Denoted by the symbol "!" after an integer; the product of
that integer and all integers in descent to 1.
Example of a factorial:
S!=5x4x3x2x1
5! = 120
Strange factorials
O!=1 I!=1
Solving simultaneous equations
The terms simultaneous equations and systems of equations
refer to conditions where two or more unknown variables are
related to each other through an equal number of equations.
Consider the following example:
x+y=24
2x-y=-6
For this set of equations, there is but a single combination of
values for x and y that will satisfy both. Either equation,
considered separately, has an infinitude of valid (x,y)
solutions, but together there is only one. Plotted on a graph,
this condition becomes obvious:
Each line is actually a continuum of points representing
possible x and y solution pairs for each equation. Each
equation, separately, has an infinite number of ordered pair
(x,y) solutions. There is only one point where the two linear
functions x + y = 24 and 2x - y = -6 intersect (where one of
their many independent solutions happen to work for both
equations), and that is where x is equal to a value of 6 and y
is equal to a value of 18.
Usually, though, graphing is not a very efficient way to
determine the simultaneous solution set for two or more
equations. It is especially impractical for systems of three or
more variables. In a three-variable system, for example, the
solution would be found by the point intersection of three
planes in a three-dimensional coordinate space -- not an
easy scenario to visualize.
Substitution method
Several algebraic techniques exist to solve simultaneous
equations. Perhaps the easiest to comprehend is the
substitution method. Take, for instance, our two-variable
example problem:
x+y= 24
2x- y=-6
In the substitution method, we manipulate one of the
equations such that one variable is defined in terms of the
other:
x+y =24
y=24-x
Defining y in terms ofx
Then, we take this new definition of one variable and
substitute it for the same variable in the other equation. In
this case, we take the definition of y, which is 24 - x and
substitute this for the y term found in the other equation:
y=24-x
aaa
2x-y=-6
Vv
2x - (24-x)=-6
Now that we have an equation with just a single variable (x),
we can solve it using "normal" algebraic techniques:
2x - (24-x)=-6
Lb Distributive property
2x -244+x= -6
A} Combining like terms
3x -24 = -6
Lb Adding 24 to each side
3x = 18
A) Dividing both sides by 3
t=
Now that x is Known, we can plug this value into any of the
original equations and obtain a value for y. Or, to save us
some work, we can plug this value (6) into the equation we
just generated to define y in terms of x, being that it is
already in a form to solve for y:
1=b
| substitute
4-x
NM
y=
ie
II
NM
=
'
oO
y= 18
Applying the substitution method to systems of three or
more variables involves a similar pattern, only with more
work involved. This is generally true for any method of
solution: the number of steps required for obtaining
solutions increases rapidly with each additional variable in
the system.
To solve for three unknown variables, we need at least three
equations. Consider this example:
x-y+z=10
3x+y+2z=34
-3x + 2y-z=-14
Being that the first equation has the simplest coefficients (1,
-1, and 1, for x, y, and z, respectively), it seems logical to use
it to develop a definition of one variable in terms of the
other two. In this example, I'll solve for x in terms of y and z:
x-y+z=10
Adding y and subtracting z
from both sides
x=y-z+10
Now, we can substitute this definition of x where x appears
in the other two equations:
x=y-z+10 x=y-z+10
| substitute | substitute
3x+ y+ 2z=34 -5x+2y -z=-14
3(y -z+ 10)+ y +2z=34 -S(y-z+10)+2y-z=-l4
Reducing these two equations to their simplest forms:
3(y -z+ 10)+ y +2z=34 -S(y-z+10)+2y-z=-14
Lt Distributive property LL
3y -3z+ 30+ y + 2z2= 34 -Sy +5z-50+2y-z=-l4
Lt Combining like terms LL
4y -z+30=34 -3y +4z-50=-14
Lt Moving constant values to right LL
of the "=" sign
4y-z=4 -3y + 4z = 36
So far, our efforts have reduced the system from three
variables in three equations to two variables in two
equations. Now, we can apply the substitution technique
again to the two equations 4y - z = 4and -3y + 4z = 36to
solve for either y or z. First, I'll manipulate the first equation
to define z in terms of y:
4y -z=¢
4
LL Adding z to both sides;
subtracting 4 from both sides
z=4y-4
Next, we'll substitute this definition of z in terms of y where
we see z in the other equation:
z=4y-4
| substitute
-3y + 4z = 36
V
-3y + 4(4y - 4) =36
Lb Distributive property
-3y + l6y - 16=36
LL Combining like terms
13y - 16 =36
Lt Adding 16 to both sides
13y =52
<4 Dividing both sides by 13
y=4
Now that y is a Known value, we can plug it into the
equation defining z in terms of y and obtain a figure for z:
y=4
substitute
z=4y-4
Se
z= 16-4
we
Z= 12
Now, with values for y and z known, we can plug these into
the equation where we defined x in terms of y and z, to
obtain a value for x:
~=9
substitute | z= 12
| substitute
x=y-z+10
Pd
II
i
<<
+
r=)
NM
x
In closing, we've found values for x, y, and z of 2,4, and 12,
respectively, that satisfy all three equations.
Addition method
While the substitution method may be the easiest to grasp
on a conceptual level, there are other methods of solution
available to us. One such method is the so-called addition
method, whereby equations are added to one another for
the purpose of canceling variable terms.
Let's take our two-variable system used to demonstrate the
substitution method:
One of the most-used rules of algebra is that you may
perform any arithmetic operation you wish to an equation so
long as you do it equally to both sides. With reference to
addition, this means we may add any quantity we wish to
both sides of an equation -- so long as its the same quantity
-- without altering the truth of the equation.
An option we have, then, is to add the corresponding sides
of the equations together to form a new equation. Since
each equation is an expression of equality (the same
quantity on either side of the = sign), adding the left-hand
side of one equation to the left-hand side of the other
equation is valid so long as we add the two equations' right-
hand sides together as well. In our example equation set, for
instance, we may add x + y to 2x - y, and add 24 and -6
together as well to form a new equation. What benefit does
this hold for us? Examine what happens when we do this to
our example equation set:
x+y=24
+2x-y=-6
3x+0=18
Because the top equation happened to contain a positive y
term while the bottom equation happened to contain a
negative y term, these two terms canceled each other in the
process of addition, leaving no y term in the sum. What we
have left is a new equation, but one with only a single
unknown variable, x! This allows us to easily solve for the
value of x:
3x+0= 18
LL Dropping the 0 term
3x = 18
<1) Dividing both sides by 3
x=6
Once we have a known value for x, of course, determining y's
value is a simply matter of substitution (replacing x with the
number 6) into one of the original equations. In this
example, the technique of adding the equations together
worked well to produce an equation with a single unknown
variable. What about an example where things aren't so
simple? Consider the following equation set:
2x + 2y = 14
3x+ y= 13
We could add these two equations together -- this being a
completely valid algebraic operation -- but it would not
profit us in the goal of obtaining values for x and y:
2x + 2y = 14
+ 3x+y=13
5x + 3y = 27
The resulting equation still contains two unknown variables,
just like the original equations do, and so we're no further
along in obtaining a solution. However, what if we could
manipulate one of the equations so as to have a negative
term that wou/d cancel the respective term in the other
equation when added? Then, the system would reduce to a
single equation with a single unknown variable just as with
the last (fortuitous) example.
If we could only turn the y term in the lower equation into a -
2y term, so that when the two equations were added
together, both y terms in the equations would cancel,
leaving us with only an x term, this would bring us closer to
a solution. Fortunately, this is not difficult to do. If we
multiply each and every term of the lower equation by a -2,
it will produce the result we seek:
-2(3x + y) = -2(13)
Lt Distributive property
-6x - 2y = -26
Now, we may add this new equation to the original, upper
equation:
2x + 2y = 14
+ -6x - 2y = -26
-4x + Oy =-12
Solving for x, we obtain a value of 3:
-4x + Oy =-12
Lb Dropping the 0 term
-4x = -12
< | Dividing both sides by -4
L=3
Substituting this new-found value for x into one of the
Original equations, the value of y is easily determined:
x=3
| substitute
2x + 2y = 14
V7
6+2y=14
.u; Subtracting 6 from both sides
2y=8
<4 Dividing both sides by 2
¥=4
Using this solution technique on a three-variable system is a
bit more complex. As with substitution, you must use this
technique to reduce the three-equation system of three
variables down to two equations with two variables, then
apply it again to obtain a single equation with one unknown
variable. To demonstrate, I'll use the three-variable equation
system from the substitution section:
x-y+z=10
3x+y+2z=34
-Sx + 2y-z=-14
Being that the top equation has coefficient values of 1 for
each variable, it will be an easy equation to manipulate and
use as a cancellation tool. For instance, if we wish to cancel
the 3x term from the middle equation, all we need to do is
take the top equation, multiply each of its terms by -3, then
add it to the middle equation like this:
x-y+z=10
<4 Multiply both sides by -3
-3(x - y +z) =-3(10)
Lb Distributive property
-3x + 3y - 3z = -30
-3x + 3y - 3z= -30
+3x+y+2z=34
Ox+4y-z=4
or
4y-z=4
(Adding)
We can rid the bottom equation of its -5x term in the same
manner: take the original top equation, multiply each of its
terms by 5, then add that modified equation to the bottom
equation, leaving a new equation with only y and z terms:
x-y+z=10
{+ Multiply both sides by 5
5(x - y +z) = 5(10)
Lb Distributive property
5x - Sy +5z=50
; 5x - Sy + 5z=50
(Adding) :
+-5x+ 2y-z=-l4
Ox - 3y + 4z= 36
or
-3y + 4z = 36
At this point, we have two equations with the same two
unknown variables, y and z:
4y-z=4
-3y + 4z= 36
By inspection, it should be evident that the -z term of the
upper equation could be leveraged to cancel the 4z term in
the lower equation if only we multiply each term of the
upper equation by 4 and add the two equations together:
4y-z=4
{Multiply both sides by 4
4(4y - z) = 4(4)
Lb Distributive property
l6y - 4z= 16
loy - 4z= 16
+ -3y +4z= 36
L3y + Oz =52
or
13y = 52
(Adding)
Taking the new equation 13y = 52 and solving for y (by
dividing both sides by 13), we get a value of 4 for y.
Substituting this value of 4 for y in either of the two-variable
equations allows us to solve for z. Substituting both values
of y and z into any one of the original, three-variable
equations allows us to solve for x. The final result (I'll spare
you the algebraic steps, since you should be familiar with
them by now!) is that x = 2, y = 4,andz = 12.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Chirvasuta Constantin (April 2, 2003): Pointed out error
in quadratic equation formula.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | +4]
—/ | 4]
Lessons In Electric Circuits
-- Volume V
Chapter 5
TRIGONOMETRY
REFERENCE
o Trigonometric identities
o The Pythagorean theorem
e
=
Oo
ES
4
ie)
—
or
cr
ae
ted}
ie
a
OD
cr
=.
ie)
oO
ie |
oO
=
OD
or
iam |
<
Trigonometric equivalencies
Hyperbolic functions
Contributors
Hypotenuse (H)
Opposite (O)
Adjacent (A)
A right triangle is defined as having one angle precisely
equal to 90° (a right angle).
Trigonometric identities
_ _O _ A __ O oh , (RIX
sin X= cosSxX= + tan x= a tanx= 250
_. H __ H _ A -_ COsx
csc x= Oo ee t= cotx= > cotx= =>
H is the Hypotenuse, always being opposite the right angle.
Relative to angle x, O is the Opposite and A is the Adjacent.
"Arc" functions such as "arcsin", "arccos", and "arctan" are
the complements of normal trigonometric functions. These
functions return an angle for a ratio input. For example, if
the tangent of 45° is equal to 1, then the "arctangent"
(arctan) of 1 is 45°. "Arc" functions are useful for finding
angles in a right triangle if the side lengths are known.
The Pythagorean theorem
H = A*+ 0°
_ sinb _ sinc
B C
sina
A
A’ =B*+C’- (2BC)\(cos a)
B’ = A’ +C° - (2AC)(cos b)
C? = A?+B?- (2AB)(cos c)
Trigonometric equivalencies
sin -x = -sin x COS -X = COS X tan -t= -tan t
csc -t= -csc ft sec -t=sect cot -t= -cot t
sin 2x = 2(sin x)(cos x) cos 2x = (cos x) - (sin’ x)
2(tan x)
tan 2t= ——__
l - tan” x
ne en | cos 2x , a | cos 2x
sin eer, —=3. con: x= a 5
Hyperbolic functions
sinh x =
x
e+e
-X
cosh x =
sinh x
tanhx= ————
cosh x
Note: all angles (x) must be expressed in units of radians for
these hyperbolic functions. There are 2m radians in a circle
(360°).
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Harvey Lew (??? 2003): Corrected typographical error:
"tangent" should have been "cotangent".
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
—/ | 4]
—/ | 4]
Lessons In Electric Circuits
-- Volume V
Chapter 6
CALCULUS REFERENCE
Rules for limits
Derivative of a constant
Common derivatives
Derivatives of power functions of e
Trigonometric derivatives
Rules for derivatives
o Constant rule
o Rule of sums
o Rule of differences
o Product rule
o Quotient rule
o Power rule
o Functions of other functions
The antiderivative (Indefinite integral)
Common antiderivatives
Antiderivatives of power functions of e
Rules for antiderivatives
o Constant rule
o Rule of sums
o Rule of differences
Definite integrals and the fundamental theorem of
calculus
Differential equations
Rules for limits
lim [f(x) + g(x)] = lim fix) + lim g(x)
xa xa xa
lim [f(x) - g(x)] = lim f(x) - lim g(x)
xa x—a xa
lim [f(x) g(x)] = Dim f(x)] Dim g(x)]
xa xa xa
Derivative of a constant
If:
fixy=Hc
Then:
d Fe
ao
("c" being a constant)
Common derivatives
d x2 - nxt!
dx
d ee |
— In AF =>
dx x
a‘ = (In a)(a*)
d
dx
Derivatives of power functions of e
If:
If:
fix)=e fixy=2”
Then: Then:
d _ ox d _ isis) do
d= Fe ae te
Example:
fix) = e + 2x)
ax) = eft +2x) d
(x7 42)
dx X x _
fix) = (e* + ™)(2x + 2)
Trigonometric derivatives
d
— sinx =cosx de cos X = -sin x
dx dx
d 2 d 2
tan x = sec” X cotx =-csc xX
dx dx
4. sec x = (sec x (tan x) d
— csc x = (-csc x)(cot x)
dx
Rules for derivatives
Constant rule
d ae |
‘dx: I= 6)
Rule of sums
d ee ee ee ee:
Gr Led + sCOl= 4 feo + (x)
ax?
Rule of differences
d L)- G(x = de - A o
Gr Le g(x)] = Te Fix) dx g(x)
Product rule
d A oietl — 5 > _d_
Sr Led 8001 = OL FG 800] + sCol G— Al
Quotient rule
fix) so fx)] - fix) a g(x)]
de
Power rule
d \a yal d
jr SHY) = affix)] Gr AH)
Functions of other functions
Ds geo
Break the function into two functions:
u=g(x) and y=fiu)
Solve:
dy foxy = FY fy) WU,
dx fis) dat) dx 8 (x)
The antiderivative (Indefinite
integral)
If:
2 fix) = g(x)
Then:
g(x) is the derivative of fix)
fix) is the antiderivative of g(x)
Je(x) dx =f(x)+c
Notice something important here: taking the derivative of
f(x) may precisely give you g(x), but taking the
antiderivative of g(x) does not necessarily give you f(x) in its
Original form. Example:
Ax) = 3x45
2. fix) = 6x
lox dx = 3x*+c
Note that the constant c is unknown! The original function
f(x) could have been 3x2 + 5, 3x? + 10, 3x? + anything, and
the derivative of f(x) would have still been 6x. Determining
the antiderivative of a function, then, is a bit less certain
than determining the derivative of a function.
Common antiderivatives
n+l
Ix" dx = +c
+1
| °% dx = (In Ixl) +c
Where,
c = aconstant
Antiderivatives of power functions of
e
le“ dx =e* 4c
Note: this is a very unique and useful property of e. As in the
case of derivatives, the antiderivative of such a function is
that same function. In the case of the antiderivative, a
constant term "c" is added to the end as well.
Rules for antiderivatives
Constant rule
lcfix) dx = c [Aix) dx
Rule of sums
[Ufo + 200] dx = [[fx) dx 1+ [eGo dx J
Rule of differences
[Ex - g(x] dx = (Ax) dx ] - Pe(x) dx J
Definite integrals and the
fundamental theorem of calculus
If:
lAx) dx=g(x) or * g(x) = fix)
Then:
b
Jfix) dx = g(b) - g(a)
Where,
a and b are constants
If:
lAx) dx = g(x) and a=0
Then:
x
Ax) dx = g(x)
Differential equations
As opposed to normal equations where the solution is a
number, a differential equation is one where the solution is
actually a function, and which at least one derivative of that
unknown function is part of the equation.
As with finding antiderivatives of a function, we are often
left with a solution that encompasses more than one
possibility (consider the many possible values of the
constant "c" typically found in antiderivatives). The set of
functions which answer any differential equation is called
the "general solution" for that differential equation. Any one
function out of that set is referred to as a "particular
solution" for that differential equation. The variable of
reference for differentiation and integration within the
differential equation is known as the "independent variable."
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
Next
—
nts
E¢
—_
Previous Contents Next
— 4 —>
Lessons In Electric Circuits --
Volume V
Chapter 7
USING THE SPICE CIRCUIT
SIMULATION PROGRAM
Introduction
History of SPICE
Fundamentals of SPICE programming
The command-line interface
Circuit components
o Passive components
» CAPACITORS
» INDUCTORS
» INDUCTOR COUPLING (transformers),
» RESISTORS
o Active components
» DIODES
« JFET, junction field-effect transistor
« MOSFET, transistor
o Sources
Analysis options
Quirks
A good beginning
A good ending
Must have a node 0
Avoid open circuits
Avoid certain component loops
Current measurement
Fourier analysis
Example circuits and netlists
Multiple-source DC resistor network, part 1
Multiple-source DC resistor network, part 2
RC time-constant circuit
°
o Oo 0 0 0 90
Simple AC resistor-capacitor circuit
Low-pass filter
Multiple-source AC network
AC phase shift demonstration
Transformer circuit
Full-wave bridge rectifier
Common-base BJT transistor amplifier
o 0 0 0 00 00 0 0 0
Common-source JFET amplifier with self-bias
Inverting_op-amp circuit
Noninverting_op-amp circuit
Instrumentation amplifier
o Oo 0 0 0 0
Introduction
"With Electronics Workbench, you can create circuit schematics that look
Just the same as those you're already familiar with on paper -- plus you can
flip the power switch so the schematic behaves like a real circuit. With
other electronics simulators, you may have to type in SPICE node lists as
text files -- an abstract representation of a circuit beyond the capabilities of
all but advanced electronics engineers."
(Electronics Workbench User's guide -- version 4, page 7)
This introduction comes from the operating manual for a circuit simulation
program called Electronics Workbench. Using a graphic interface, it allows the
user to draw a circuit schematic and then have the computer analyze that
circuit, displaying the results in graphic form. It is a very valuable analysis tool,
but it has its shortcomings. For one, it and other graphic programs like it tend to
be unreliable when analyzing complex circuits, as the translation from picture
to computer code is not quite the exact science we would want it to be (yet).
Secondly, due to its graphics requirements, it tends to need a significant
amount of computational "horsepower" to run, and a computer operating
system that supports graphics. Thirdly, these graphic programs can be costly.
However, underneath the graphics skin of Electronics Workbench lies a robust
(and free!) program called SPICE, which analyzes a circuit based on a text-file
description of the circuit's components and connections. What the user pays for
with Electronics Workbench and other graphic circuit analysis programs is the
convenient "point and click" interface, while SPICE does the actual
mathematical analysis.
By itself, SPICE does not require a graphic interface and demands little in
system resources. It is also very reliable. The makers of Electronic Workbench
would like you to think that using SPICE in its native text mode is a task suited
for rocket scientists, but I'm writing this to prove them wrong. SPICE is fairly
easy to use for simple circuits, and its non-graphic interface actually lends itself
toward the analysis of circuits that can be difficult to draw. | think it was the
programming expert Donald Knuth who quipped, "What you See is all you get"
when it comes to computer applications. Graphics may look more attractive, but
abstracted interfaces (text) are actually more efficient.
This document is not intended to be an exhaustive tutorial on how to use SPICE.
I'm merely trying to show the interested user how to apply it to the analysis of
simple circuits, as an alternative to proprietary ($$$) and buggy programs.
Once you learn the basics, there are other tutorials better suited to take you
further. Using SPICE -- a program originally intended to develop integrated
circuits -- to analyze some of the really simple circuits showcased here may
seem a bit like cutting butter with a chain saw, but it works!
All options and examples have been tested on SPICE version 2g6 on both MS-
DOS and Linux operating systems. As far as | know, I'm not using features
specific to version 2g6, so these simple functions should work on most versions
of SPICE.
History of SPICE
SPICE is a computer program designed to simulate analog electronic circuits. It
original intent was for the development of integrated circuits, from which it
derived its name: Simulation Program with Integrated Circuit Emphasis.
The origin of SPICE traces back to another circuit simulation program called
CANCER. Developed by professor Ronald Rohrer of U.C. Berkeley along with
some of his students in the late 1960's, CANCER continued to be improved
through the early 1970's. When Rohrer left Berkeley, CANCER was re-written
and re-named to SPICE, released as version 1 to the public domain in May of
1972. Version 2 of SPICE was released in 1975 (version 2g6 -- the version used
in this book -- is a minor revision of this 1975 release). Instrumental in the
decision to release SPICE as a public-domain computer program was professor
Donald Pederson of Berkeley, who believed that all significant technical
progress happens when information is freely shared. | for one thank him for his
vision.
A major improvement came about in March of 1985 with version 3 of SPICE
(also released under public domain). Written in the C language rather than
FORTRAN, version 3 incorporated additional transistor types (the MESFET, for
example), and switch elements. Version 3 also allowed the use of alphabetical
node labels rather than only numbers. Instructions written for version 2 of
SPICE should still run in version 3, though.
Despite the additional power of version 3, | have chosen to use version 2g6
throughout this book because it seems to be the easiest version to acquire and
run on different computer systems.
Fundamentals of SPICE programming
Programming a circuit simulation with SPICE is much like programming in any
other computer language: you must type the commands as text in a file, save
that file to the computer's hard drive, and then process the contents of that file
with a program (compiler or interpreter) that understands such commands.
In an interpreted computer language, the computer holds a special program
called an interpreter that translates the program you wrote (the so-called
source file) into the computer's own language, on the fly, as its being executed:
Computer
software
Source Interpreter
File | >
In a compiled computer language, the program you wrote is translated all at
once into the computer's own language by a special program called a compiler.
After the program you've written has been "compiled," the resulting executable
file needs no further translation to be understood directly by the computer. It
can now be "run" on a computer whether or not compiler software has been
installed on that computer:
Computer
Source Compiler
ans >
Computer
SPICE is an interpreted language. In order for a computer to be able to
understand the SPICE instructions you type, it must have the SPICE program
(interpreter) installed:
Computer
Source SPICE
"netlist"
SPICE source files are commonly referred to as "netlists," although they are
sometimes known as "decks" with each line in the file being called a "card."
Cute, don't you think? Netlists are created by a person like yourself typing
instructions line-by-line using a word processor or text editor. Text editors are
much preferred over word processors for any type of computer programming, as
they produce pure ASCII text with no special embedded codes for text
highlighting (like /ta/ic or boldface fonts), which are uninterpretable by
interpreter and compiler software.
As in general programming, the source file you create for SPICE must follow
certain conventions of programming. It is a computer language in itself, albeit a
simple one. Having programmed in BASIC and C/C++, and having some
experience reading PASCAL and FORTRAN programs, it is my opinion that the
language of SPICE is much simpler than any of these. It is about the same
complexity as a markup language such as HTML, perhaps less so.
There is a cycle of steps to be followed in using SPICE to analyze a circuit. The
cycle starts when you first invoke the text editing program and make your first
draft of the netlist. The next step is to run SPICE on that new netlist and see
what the results are. If you are a novice user of SPICE, your first attempts at
creating a good netlist will be fraught with small errors of syntax. Don't worry --
as every computer programmer knows, proficiency comes with lots of practice.
If your trial run produces error messages or results that are obviously incorrect,
you need to re-invoke the text editing program and modify the netlist. After
modifying the netlist, you need to run SPICE again and check the results. The
sequence, then, looks something like this:
e Compose a new netlist with a text editing program. Save that netlist to a
file with a name of your choice.
e Run SPICE on that netlist and observe the results.
e If the results contain errors, start up the text editing program again and
modify the netlist.
e Run SPICE again and observe the new results.
e If there are still errors in the output of SPICE, re-edit the netlist again with
the text editing program. Repeat this cycle of edit/run as many times as
necessary until you are getting the desired results.
e Once you've "debugged" your netlist and are getting good results, run
SPICE again, only this time redirecting the output to a new file instead of
just observing it on the computer screen.
e Start up a text editing program ora word processor program and open the
SPICE output file you just created. Modify that file to suit your formatting
needs and either save those changes to disk and/or print them out on
paper.
To "run" a SPICE "program," you need to type in a command at a terminal
prompt interface, such as that found in MS-DOS, UNIX, or the MS-Windows DOS
prompt option:
Spice < example.cir
The word "spice" invokes the SPICE interpreting program (providing that the
SPICE software has been installed on the computer!), the "<" symbol redirects
the contents of the source file to the SPICE interpreter, and example.cir is the
name of the source file for this circuit example. The file extension ".cir" is not
mandatory; | have seen ".inp" (for "input") and just plain ".txt" work well, too. It
will even work when the netlist file has no extension. SPICE doesn't care what
you name it, so long as it has a name compatible with the filesystem of your
computer (for old MS-DOS machines, for example, the filename must be no
more than 8 characters in length, with a 3 character extension, and no spaces
or other non-alphanumerical characters).
When this command is typed in, SPICE will read the contents of the example.cir
file, analyze the circuit specified by that file, and send a text report to the
computer terminal's standard output (usually the screen, where you can see it
scroll by). A typical SPICE output is several screens worth of information, so you
might want to look it over with a slight modification of the command:
spice < example.cir | more
This alternative "pipes" the text output of SPICE to the "more" utility, which
allows only one page to be displayed at a time. What this means (in English) is
that the text output of SPICE is halted after one screen-full, and waits until the
user presses a keyboard key to display the next screen-full of text. If you're just
testing your example circuit file and want to check for any errors, this is a good
way to do it.
Spice < example.cir > example.txt
This second alternative (above) redirects the text output of SPICE to another
file, called example.txt, where it can be viewed or printed. This option
corresponds to the last step in the development cycle listed earlier. It is
recommended by this author that you use this technique of "redirection" to a
text file only after you've proven your example circuit netlist to work well, so
that you don't waste time invoking a text editor just to see the output during
the stages of "debugging."
Once you have a SPICE output stored in a .txt file, you can use a text editor or
(better yet!) a word processor to edit the output, deleting any unnecessary
banners and messages, even specifying alternative fonts to highlight the
headings and/or data for a more polished appearance. Then, of course, you can
print the output to paper if you so desire. Being that the direct SPICE output is
plain ASCII text, such a file will be universally interpretable on any computer
whether SPICE is installed on it or not. Also, the plain text format ensures that
the file will be very small compared to the graphic screen-shot files generated
by "point-and-click" simulators.
The netlist file format required by SPICE is quite simple. A netlist file is nothing
more than a plain ASCII text file containing multiple lines of text, each line
describing either a circuit component or special SPICE command. Circuit
architecture is specified by assigning numbers to each component's connection
points in each line, connections between components designated by common
numbers. Examine the following example circuit diagram and its corresponding
SPICE file. Please bear in mind that the circuit diagram exists only to make the
simulation easier for human beings to understand. SPICE only understands
netlists:
Example netlist
vl 10dc 15
rl 10 2.2k
r2 1 2 3.3k
r3 2 @ 150
.end
Each line of the source file shown above is explained here:
e vl represents the battery (voltage source 1), positive terminal numbered 1,
negative terminal numbered 0, with a DC voltage output of 15 volts.
¢ rl represents resistor R; in the diagram, connected between points 1 and 0,
with a value of 2.2 kQ.
¢ r2 represents resistor R> in the diagram, connected between points 1 and 2,
with a value of 3.3 kQ.
* r3 represents resistor R3 in the diagram, connected between points 2 and 0,
with a value of 150 kQ.
Electrically common points (or "nodes") in a SPICE circuit description share
common numbers, much in the same way that wires connecting common points
in a large circuit typically share common wire labels.
To simulate this circuit, the user would type those six lines of text on a text
editor and save them as a file with a unique name (Such as example.cir). Once
the netlist is composed and saved to a file, the user then processes that file
with one of the command-line statements shown earlier (spice < example.cir),
and will receive this text output on their computer's screen:
1******* 10/10/99 2K OK OK OK OK OK OK OK Spice 29.6 3/15/83 KKAKKKKKOQT 37: GDKKKKK
Oexample netlist
Q**** input listing
vl 10dc 15
rl 1 0 2.2k
r2 1 2 3.3k
r3 2 0 150
.end
**KEKKITQ/10/99 2K KK OK OK OK OK KOK Spice 29.6
Oexample netlist
QtEK* small signal bias solution
node voltage node voltage
( 1) 15.0000 ( 2) 0.6522
voltage source currents
name current
temperature = 27.000 deg c
3/15/83 ****EK O71 32:4 QKKKKK
temperature = 27.000 deg c
vl -1.117E-02
total power dissipation 1.67E-01 watts
job concluded
0 total job time 0.02
]****EEEITQ/10/99 FEKK**** Spice 2g.6 3/15/83 ******Q7:32:42*****
O**** input listing temperature = 27.000 deg c
SPICE begins by printing the time, date, and version used at the top of the
output. It then lists the input parameters (the lines of the source file), followed
by a display of DC voltage readings from each node (reference number) to
ground (always reference number 0). This is followed by a list of current
readings through each voltage source (in this case there's only one, v1). Finally,
the total power dissipation and computation time in seconds is printed.
All output values provided by SPICE are displayed in scientific notation.
The SPICE output listing shown above is a little verbose for most peoples’ taste.
For a final presentation, it might be nice to trim all the unnecessary text and
leave only what matters. Here is a sample of that same output, redirected to a
text file (spice < example.cir > example.txt), then trimmed down judiciously with
a text editor for final presentation and printed:
example netlist
vl 10 dc 15
rl 10 2.2k
r2 1 2 3.3k
r3 2 0 150
.end
node voltage node voltage
( 1) 15.0000 ( 2) 0.6522
voltage source currents
name current
v1 -1.117E-02
total power dissipation 1.67E-01 watts
One of the very nice things about SPICE is that both input and output formats
are plain-text, which is the most universal and easy-to-edit electronic format
around. Practically any computer will be able to edit and display this format,
even if the SPICE program itself is not resident on that computer. If the user
desires, he or she is free to use the advanced capabilities of word processing
programs to make the output look fancier. Comments can even be inserted
between lines of the output for further clarity to the reader.
The command-line interface
If you've used DOS or UNIX operating systems before in a command-line shell
environment, you may wonder why we have to use the "<" symbol between the
word "spice" and the name of the netlist file to be interpreted. Why not just
enter the file name as the first argument to the command "spice" as we do
when we invoke the text editor? The answer is that SPICE has the option of an
interactive mode, whereby each line of the netlist can be interpreted as it is
entered through the computer's Standard Input (stdin). If you simple type
"spice" at the prompt and press [Enter], SPICE will begin to interpret anything
you type in to it (live).
For most applications, its nice to save your netlist work in a separate file and
then let SPICE interpret that file when you're ready. This is the way | encourage
SPICE to be used, and so this is the way its presented in this lesson. In order to
use SPICE this way in a command-line environment, we need to use the "<"
redirection symbol to direct the contents of your netlist file to Standard Input
(stdin), which SPICE can then process.
Circuit components
Remember that this tutorial is not exhaustive by any means, and that all
descriptions for elements in the SPICE language are documented here in
condensed form. SPICE is a very capable piece of software with lots of options,
and I'm only going to document a few of them.
All components in a SPICE source file are primarily identified by the first letter
in each respective line. Characters following the identifying letter are used to
distinguish one component of a certain type from another of the same type (rl,
r2, r3, rload, rpullup, etc.), and need not follow any particular naming
convention, so long as no more than eight characters are used in both the
component identifying letter and the distinguishing name.
For example, suppose you were simulating a digital circuit with "pullup" and
"pulldown" resistors. The name rpullup would be valid because it is seven
characters long. The name rpulltdown, however, is nine characters long. This may
cause problems when SPICE interprets the netlist.
You can actually get away with component names in excess of eight total
characters if there are no other similarly-named components in the source file.
SPICE only pays attention to the first eight characters of the first field in each
line, SO rpulldown is actually interpreted as rpulldow with the "n" at the end
being ignored. Therefore, any other resistor having the first eight characters in
its first field will be seen by SPICE as the same resistor, defined twice, which will
Cause an error (i.e. rpulldownl and rpulldown2 would be interpreted as the same
name, rpulldow).
It should also be noted that SPICE ignores character case, so r1 and R1 are
interpreted by SPICE as one and the same.
SPICE allows the use of metric prefixes in specifying component values, which is
a very handy feature. However, the prefix convention used by SPICE differs
somewhat from standard metric symbols, primarily due to the fact that netlists
are restricted to standard ASCII characters (ruling out Greek letters such as u
for the prefix "micro") and that SPICE is case-insensitive, so "m" (which is the
standard symbol for "milli") and "M" (which is the standard symbol for "Mega")
are interpreted identically. Here are a few examples of prefixes used in SPICE
netlists:
rl 1 @ 2t (Resistor Ry, 2t = 2 Tera-ohms = 2 TQ)
r2 1 0 4g (Resistor R>, 4g = 4 Giga-ohms = 4 GQ)
r3 1 0 47meg (Resistor R3, 47 meg = 47 Mega-ohms = 47 MQ)
r4 1 @ 3.3k (Resistor Ry, 3.3k = 3.3 kilo-ohms = 3.3 kQ)
r5 1 @ 55m (Resistor Rs, 55m = 55 milli-ohms = 55 mQ)
r6é 1 @ 10u (Resistor Re, LOU = 10 micro-ohms 10 yO)
r7 1 © 30n (Resistor R7, 30n = 30 nano-ohms = 30 nQ)
r8 1 0 5p (Resistor Rg, 5p = 5 pico-ohms = 5 pQ)
r9 1 @ 250f (Resistor Rg, 250f = 250 femto-ohms = 250 fQ)
Scientific notation is also allowed in specifying component values. For example:
r10 1 0 4.7e3 (Resistor Ry9, 4.7e3 = 4.7 x 103 ohms = 4.7 kilo-ohms = 4.7 kQ)
r11 1 @ 1e-12 (Resistor R,3, 1e-12 = 1 x 10°! ohms = 1 pico-ohm = 1 pQ)
The unit (ohms, volts, farads, henrys, etc.) is automatically determined by the
type of component being specified. SPICE "knows" that all of the above
examples are "ohms" because they are all resistors (rl, r2, r3,...). If they were
Capacitors, the values would be interpreted as "farads," if inductors, then
"henrys," etc.
Passive components
CAPACITORS
General form: c[ name] [nodel] [ node2] [ value] ic=[ initial voltage]
Example 1: cl 12 33 10u
Example 2: cl 12 33 10u ic=3.5
Comments: The “initial condition" (ic=) variable is the capacitor's voltage in
units of vo/ts at the start of DC analysis. It is an optional value, with the starting
voltage assumed to be zero if unspecified. Starting current values for capacitors
are interpreted by SPICE only if the .tran analysis option is invoked (with the
"uic" option).
INDUCTORS
General form: tlU[ name] [nodel] [node2] [value] ic=[ initial current]
Example 1: l1 12 33 133m
Example 2: ll 12 33 133m ic=12.7m
Comments: The “initial condition" (ic=) variable is the inductor's current in
units of amps at the start of DC analysis. It is an optional value, with the
starting current assumed to be zero if unspecified. Starting current values for
inductors are interpreted by SPICE only if the .tran analysis option is invoked.
INDUCTOR COUPLING (transformers)
General form: k[ name] l[ name] l[ name] [ coupling factor]
Example 1: k1 11 12 0.999
Comments: SPICE will only allow coupling factor values between 0 and 1 (non-
inclusive), with 0 representing no coupling and 1 representing perfect coupling.
The order of specifying coupled inductors (11, 12 or 12, 11) is irrelevant.
RESISTORS
General form: r[{ name] [nodel] [ node2] [ value]
Example: rload 23 15 3.3k
Comments: In case you were wondering, there is no declaration of resistor
power dissipation rating in SPICE. All components are assumed to be
indestructible. If only real life were this forgiving!
Active components
All semiconductor components must have their electrical characteristics
described in a line starting with the word ".modet", which tells SPICE exactly how
the device will behave. Whatever parameters are not explicitly defined in the
.model card will default to values pre-programmed in SPICE. However, the .model
card must be included, and at least specify the model name and device type (d,
npn, pnp, njf, pjf, nmos, or pmos).
DIODES
General form: d[{name] [ anode] [ cathode] [ model]
Example: dl 1 2 modl
DIODE MODELS:
General form: .model [modelname] d [ parmtri=x] [ parmtr2=x]
Example: .model modi d
Example: .model mod2 d vj=0.65 rs=1.3
diodeparameter
Parameter definitions:
is = saturation current in amps
rs = Junction resistance in ohms
n = emission coefficient (unitless)
tt = transit time in seconds
cjo = zero-bias junction capacitance in farads
vj = junction potential in volts
m = grading coefficient (unitless)
eg = activation energy in electron-volts
xti = saturation-current temperature exponent (unitless)
kf = flicker noise coefficient (unitless)
af = flicker noise exponent (unitless)
fc = forward-bias depletion capacitance coefficient (unitless)
bv = reverse breakdown voltage in volts
ibv = current at breakdown voltage in amps
Comments: The model name must begin with a letter, not a number. If you
plan to specify a model for a 1N4003 rectifying diode, for instance, you cannot
use "1n4003" for the model name. An alternative might be "m1n4003" instead.
TRANSISTORS, bipolar junction -- BJT
General form: q[ name] [collector] [base] [ emitter] [ model]
Example: ql 2 3 0 modi
BJT TRANSISTOR MODELS:
General form: .model [modelname] [npn or pnp] [ parmtri1=x]
Example: .model modi pnp
Example: .model mod2 npn bf=75 is=le-14
The model examples shown above are very nonspecific. To accurately model
real-life transistors, more parameters are necessary. Take these two examples,
for the popular 2N2222 and 2N2907 transistors (the "+") characters represent
line-continuation marks in SPICE, when you wish to break a single line (card)
into two or more separate lines on your text editor:
Example: -model m2n2222 npn is=19f bf=150 vaf=100 ikf=.18
+ ise=50p ne=2.5 br=7.5 var=6.4 ikr=12m
+ isc=8.7p nc=1.2 rb=50 re=0.4 rc=0.4 cje=26p
+ tf=0.5n cjc=llp tr=7n xtb=1.5 kf=0.032f af=1
Example: .model m2n2907 pnp is=1.1p bf=200 nf=1.2 vaf=50
+ ikf=0.1 ise=13p ne=1.9 br=6 rc=0.6 cje=23p
+ vje=0.85 mje=1.25 tf=0.5n cjc=19p vjc=0.5
+ mjc=0.2 tr=34n xtb=1.5
Parameter definitions:
is = transport saturation current in amps
bf = ideal maximum forward Beta (unitless)
nf
forward current emission coefficient (unitless)
vaf = forward Early voltage in volts
ikf
corner for forward Beta high-current rolloff in amps
ise = B-E leakage saturation current in amps
ne = B-E leakage emission coefficient (unitless)
br = ideal maximum reverse Beta (unitless)
nr = reverse current emission coefficient (unitless)
bar = reverse Early voltage in volts
ikrikr = corner for reverse Beta high-current rolloff in amps
iscisc = B-C leakage saturation current in amps
nc = B-C leakage emission coefficient (unitless)
rb = zero bias base resistance in ohms
irb = current for base resistance halfway value in amps
rbm = minimum base resistance at high currents in ohms
emitter resistance in ohms
re
collector resistance in ohms
rc
cje = B-E zero-bias depletion capacitance in farads
vje = B-E built-in potential in volts
mje = B-E junction exponential factor (unitless)
tf = ideal forward transit time (Seconds)
xtf = coefficient for bias dependence of transit time (unitless)
vtf = B-C voltage dependence on transit time, in volts
itf = high-current parameter effect on transit time, in amps
ptf = excess phase at f=1/(transit time)(2)(pi) Hz, in degrees
cjc = B-C zero-bias depletion capacitance in farads
vjc = B-C built-in potential in volts
mjc = B-C junction exponential factor (unitless)
xjcj = B-C depletion capacitance fraction connected in base node (unitless)
tr = ideal reverse transit time in seconds
cjs = zero-bias collector-substrate capacitance in farads
vjs = substrate junction built-in potential in volts
mjs = substrate junction exponential factor (unitless)
xtb = forward/reverse Beta temperature exponent
eg = energy gap for temperature effect on transport saturation current in
electron-volts
xti = temperature exponent for effect on transport saturation current (unitless)
kf = flicker noise coefficient (unitless)
af = flicker noise exponent (unitless)
fc = forward-bias depletion capacitance formula coefficient (unitless)
Comments: Just as with diodes, the model name given for a particular
transistor type must begin with a letter, not a number. That's why the examples
given above for the 2N2222 and 2N2907 types of BJTs are named "m2n2222"
and "q2n2907" respectively.
As you can see, SPICE allows for very detailed specification of transistor
properties. Many of the properties listed above are well beyond the scope and
interest of the beginning electronics student, and aren't even useful apart from
knowing the equations SPICE uses to model BJT transistors. For those interested
in learning more about transistor modeling in SPICE, consult other books, such
as Andrei Vladimirescu's The Spice Book (ISBN 0-47 1-60926-9).
JFET, junction field-effect transistor
General form: j[ name] [drain] [ gate] [ source] [ model]
Example: jl 2 3 © modi
JFET TRANSISTOR MODELS:
General form: .model [modelname] [njf or pjf] [ parmtr1=x]
Example: .model modl pjf
Example: .model mod2 njf lLambda=1le-5 pb=0.75
Parameter definitions:
vto = threshold voltage in volts
beta = transconductance parameter in amps/volts?
Lambda = channel length modulation parameter in units of 1/volts
rd = drain resistance in ohms
rs = source resistance in ohms
cgs = zero-bias G-S junction capacitance in farads
cgd = zero-bias G-D junction capacitance in farads
pb = gate junction potential in volts
is = gate junction saturation current in amps
kf = flicker noise coefficient (unitless)
af = flicker noise exponent (unitless)
fc = forward-bias depletion capacitance coefficient (unitless)
MOSFET, transistor
General form: m[ name] [drain] [ gate] [source] [substrate] [model]
Example: ml 2 3 0 © modl
MOSFET TRANSISTOR MODELS:
General form: .model [modelname] [nmos or pmos] [ parmtr1=x]
Example: .model mod1 pmos
Example: .model mod2 nmos level=2 phi=0.65 rd=1.5
Example: .model mod3 nmos vto=-1 (depletion)
Example: .model mod4 nmos vto=1 (enhancement)
Example: .model mod5 pmos vto=1 (depletion)
Example: .model mod6 pmos vto=-1 (enhancement)
Comments: In order to distinguish between enhancement mode and
depletion-mode (also known as depletion-enhancement mode) transistors, the
model parameter "vto" (zero-bias threshold voltage) must be specified. Its
default value is zero, but a positive value (+1 volts, for example) on a P-channel
transistor or a negative value (-1 volts) on an N-channel transistor will specify
that transistor to be a dep/etion (otherwise known as dep/etion-enhancement)
mode device. Conversely, a negative value on a P-channel transistor or a
positive value on an N-channel transistor will specify that transistor to be an
enhancement mode device.
Remember that enhancement mode transistors are normally-off devices, and
must be turned on by the application of gate voltage. Depletion-mode
transistors are normally "on," but can be "pinched off" as well as enhanced to
greater levels of drain current by applied gate voltage, hence the alternate
designation of "depletion-enhancement" MOSFETs. The "vto" parameter
specifies the threshold gate voltage for MOSFET conduction.
Sources
AC SINEWAVE VOLTAGE SOURCES (when using .ac card to specify
frequency):
General form: v[ name] [+node] [ -node] ac [voltage] [ phase] sin
Example 1: vl 10 ac 12 sin
Example 2: vl 10 ac 12 240 sin (12 V Z 240°)
Comments: This method of specifying AC voltage sources works well if you're
using multiple sources at different phase angles from each other, but all at the
same frequency. If you need to specify sources at different frequencies in the
Same circuit, you must use the next method!
AC SINEWAVE VOLTAGE SOURCES (when NOT using .ac card to specify
frequency):
General form: v[ name] [+node] [ -node] sin([ offset] [ voltage]
+ [ freq] [delay] [damping factor] )
Example 1: v1 10 sin(0 12 60 0 0)
Parameter definitions:
offset = DC bias voltage, offsetting the AC waveform by a specified voltage.
voltage = peak, or crest, AC voltage value for the waveform.
freq = frequency in Hertz.
delay = time delay, or phase offset for the waveform, in seconds.
damping factor = a figure used to create waveforms of decaying amplitude.
Comments: This method of specifying AC voltage sources works well if you're
using multiple sources at different frequencies from each other. Representing
phase shift is tricky, though, necessitating the use of the delay factor.
DC VOLTAGE SOURCES (when using .dc card to specify voltage):
General form: v[ name] [+node] [ -node] dc
Example 1: vl 10 dc
Comments: If you wish to have SPICE output voltages notin reference to node
0, you must use the .dc analysis option, and to use this option you must specify
at least one of your DC sources in this manner.
DC VOLTAGE SOURCES (when NOT using .dc card to specify voltage):
General form: v[ name] [+node] [ -node] dc [ voltage]
Example 1: vl 1 0dc 12
Comments: Nothing noteworthy here!
PULSE VOLTAGE SOURCES
General form: v[ name] [+node] [ -node] pulse ([i] [p] [td] [tr
+ [tf] [ pw] [ pd] )
Parameter definitions:
initial value
i
p = pulse value
td = delay time (all time parameters in units of seconds)
rise time
tr
tf = fall time
pw = pulse width
pd = period
Example 1: vl 1 0 pulse (-3 3 0 0 O 10m 20m)
Comments: Example 1 is a perfect square wave oscillating between -3 and +3
volts, with zero rise and fall times, a 20 millisecond period, and a 50 percent
duty cycle (+3 volts for 10 ms, then -3 volts for 10 ms).
AC SINEWAVE CURRENT SOURCES (when using .ac card to specify
frequency):
General form: if name] [+node] [ -node] ac [current] [ phase] sin
Example 1: il 1 0 ac 3 sin (3 amps)
Example 2: il 1 0 ac 1m 240 sin (1 mA 2 240°)
Comments: The same comments apply here (and in the next example) as for
AC voltage sources.
AC SINEWAVE CURRENT SOURCES (when NOT using .ac card to specify
frequency):
General form: if name] [+node] [ -node] sin([ offset]
+ [ current] [ freq] 0 0)
Example 1: il 1 0 sin(@ 1.5 60 0 0)
DC CURRENT SOURCES (when using .dc card to specify current):
General form: if name] [+node] [ -node] dc
Example 1: il 10dc
DC CURRENT SOURCES (when NOT using .dc card to specify current):
General form: if name] [+node] [ -node] dc [ current]
Example 1: il 10 dc 12
Comments: Even though the books all say that the first node given for the DC
Current source is the positive node, that's not what I've found to be in practice.
In actuality, a DC current source in SPICE pushes current in the same direction
as a voltage source (battery) would with its negative node specified first.
PULSE CURRENT SOURCES
General form: if[ name] [+node] [ -node] pulse ([i] [p] [td] [tr
+ [tf] [ pw] [ pd] )
Parameter definitions:
initial value
i
p = pulse value
td = delay time
tr = rise time
tf = fall time
pw = pulse width
pd = period
Example 1: il 1 0 pulse (-3m 3m 0 © O 17m 34m)
Comments: Example 1 is a perfect square wave oscillating between -3 mA and
+3 mA, with zero rise and fall times, a 34 millisecond period, and a 50 percent
duty cycle (+3 mA for 17 ms, then -3 mA for 17 ms).
VOLTAGE SOURCES (dependent):
General form: ef name] [ out+node] [ out-node] [ in+node] [ in-node]
+ [ gain]
Example 1: el 201 2 999k
Comments: Dependent voltage sources are great to use for simulating
operational amplifiers. Example 1 shows how such a source would be
configured for use as a voltage follower, inverting input connected to output
(node 2) for negative feedback, and the noninverting input coming in on node
1. The gain has been set to an arbitrarily high value of 999,000. One word of
caution, though: SPICE does not recognize the input of a dependent source as
being a load, so a voltage source tied only to the input of an independent
voltage source will be interpreted as "open." See op-amp circuit examples for
more details on this.
CURRENT SOURCES (dependent):
Analysis options
AC ANALYSIS:
General form: .ac [curve] [points] [start] [ final]
Example 1: .ac Lin 1 1000 1000
Comments: The [curve] field can be "lin" (linear), "dec" (decade), or "oct"
(octave), specifying the (non)linearity of the frequency sweep. specifies how
many points within the frequency sweep to perform analyses at (for decade
sweep, the number of points per decade; for octave, the number of points per
octave). The [start] and [final] fields specify the starting and ending
frequencies of the sweep, respectively. One final note: the "start" value cannot
be zero!
DC ANALYSIS:
General form: .dc [source] [start] [ final] [ increment]
Example 1: .dc vin 1.5 15 0.5
Comments: The .dc card is necessary if you want to print or plot any voltage
between two nonzero nodes. Otherwise, the default "small-signal" analysis only
prints out the voltage between each nonzero node and node zero.
TRANSIENT ANALYSIS:
General form: .tran [increment] [stop time] [ start_time]
+ [ comp interval]
Example 1: .tran 1m 50m uic
Example 2: .tran .5m 32m 0 .O01m
Comments: Example 1 has an increment time of 1 millisecond and a stop time
of 50 milliseconds (when only two parameters are specified, they are increment
time and stop time, respectively). Example 2 has an increment time of 0.5
milliseconds, a stop time of 32 milliseconds, a start time of 0 milliseconds (no
delay on start), and a computation interval of 0.01 milliseconds.
Default value for start time is zero. Transient analysis a/ways beings at time
zero, but storage of data only takes place between start time and stop time.
Data output interval is increment time, or (Stop time - start time)/50, which ever
is smallest. However, the computing interval variable can be used to force a
computational interval smaller than either. For large total interval counts, the
it15 variable in the .options card may be set to a higher number. The "“uic"
option tells SPICE to "use initial conditions."
PLOT OUTPUT:
General form: .plot [type] [ outputl1] [output2] . . . [output nj
Example 1: .plot dc v(1,2) i(v2)
Example 2: .plot ac v(3,4) vp(3,4) i(vl) ip(v1)
Example 3: .plot tran v(4,5) i(v2)
Comments: SPICE can't handle more than eight data point requests on a
single .plot or .print card. If requesting more than eight data points, use
multiple cards!
Also, here's a major caveat when using SPICE version 3: if you're performing AC
analysis and you ask SPICE to plot an AC voltage as in example #2, the v(3,4)
command will only output the rea/ component of a rectangular-form complex
number! SPICE version 2 outputs the po/ar magnitude of a complex number: a
much more meaningful quantity if only a single quantity is asked for. To coerce
SPICE3 to give you polar magnitude, you will have to re-write the .print or .plot
argument as such: vm(3,4).
PRINT OUTPUT:
General form: .print [type] [output1] [ output2] . . . [output nl
Example 1: -print dc v(1,2) i(v2)
Example 2: .print ac v(2,4) i(vinput) vp(2,3)
Example 3: .print tran v(4,5) i(v2)
Comments: SPICE can't handle more than eight data point requests on a
single .plot or .print card. If requesting more than eight data points, use
multiple cards!
FOURIER ANALYSIS:
General form: .four [freq] [ outputi1] [output2] . . . [ output n]
Example 1: .four 60 v(1,2)
Comments: The . four card relies on the .tran card being present somewhere in
the deck, with the proper time periods for analysis of adequate cycles. Also,
SPICE may "crash" if a .plot analysis isn't done along with the .four analysis,
even if all .tran parameters are technically correct. Finally, the .four analysis
option only works when the frequency of the AC source is specified in that
source's card line, and notin an .ac analysis option line.
It helps to include a computation interval variable in the .tran card for better
analysis precision. A Fourier analysis of the voltage or current specified is
performed up to the 9th harmonic, with the [freq] specification being the
fundamental, or starting frequency of the analysis spectrum.
MISCELLANEOUS:
General form: .options [option1] [ option2]
Example 1: .options Limpts=500
Example 2: options itl5=0
Example 3: .options method=gear
Example 4: .options list
Example 5 .options nopage
Example 6 .options numdgt=6
Comments: There are lots of options that can be specified using this card.
Perhaps the one most needed by beginning users of SPICE is the "limpts"
setting. When running a simulation that requires more than 201 points to be
printed or plotted, this calculation point limit must be increased or else SPICE
will terminate analysis. The example given above (limpts=500) tells SPICE to
allocate enough memory to handle at least 500 calculation points in whatever
type of analysis is specified (DC, AC, or transient).
In example 2, we see an /teration variable (it15) being set to a value of 0. There
are actually six different iteration variables available for user manipulation.
They control the iteration cycle limits for solution of nonlinear equations. The
variable it15 sets the maximum number of iterations for a transient analysis.
Similar to the Limpts variable, itl5 usually needs to be set when a small
computation interval has been specified on a .tran card. Setting itl5 to a value
of 0 turns off the limit entirely, allowing the computer infinite iteration cycles
(infinite time) to compute the analysis. Warning: this may result in long
simulation times!
Example 3 with "method=gear" sets the numerical integration method used by
SPICE. The default is "trapezoid" rather than "gear," trapezoid being a simple
geometric approximation of area under a curve found by slicing up the curve
into trapezoids to approximate the shape. The "gear" method is based on
second-order or better polynomial equations and is named after C.W. Gear
(Numerical Integration of Stiff Ordinary Equations, Report 221, Department of
Computer Science, University of Illinois, Urbana). The Gear method of
integration is more demanding of the computer (computationally "expensive")
and will sometimes give slightly different results from the trapezoid method.
The "list" option shown in example 4 gives a verbose summary of all circuit
components and their respective values in the final output.
By default, SPICE will insert ASCII page-break control codes in the output to
separate different sections of the analysis. Specifying the "nopage" option
(example 5) will prevent such pagination.
The "numdgt" option shown in example 6 specifies the number of significant
digits output when using one of the ". print" data output options. SPICE defaults
at a precision of 4 significant digits.
WIDTH CONTROL:
General form: .width in=[ columns] out=[ columns]
Example 1: .width out=80
Comments: The .width card can be used to control the width of text output
lines upon analysis. This is especially handy when plotting graphs with the
.plot card. The default value is 120, which can cause problems on 80-character
terminal displays unless set to 80 with this command.
Quirks
“Garbage in, garbage out."
Anonymous
SPICE is a very reliable piece of software, but it does have its little quirks that
take some getting used to. By "quirk" | mean a demand placed upon the user to
write the source file in a particular way in order for it to work without giving
error messages. | do not mean any kind of fault with SPICE which would produce
erroneous or misleading results: that would be more properly referred to asa
"bug." Speaking of bugs, SPICE has a few of them as well.
Some (or all) of these quirks may be unique to SPICE version 2g6, which is the
only version I've used extensively. They may have been fixed in later versions.
A good beginning
SPICE demands that the source file begin with something other than the first
"card" in the circuit description "deck." This first character in the source file can
be a linefeed, title line, or a comment: there just has to be something there
before the first component-specifying line of the file. If not, SPICE will refuse to
do an analysis at all, claiming that there is a serious error (Such as improper
node connections) in the "deck."
A good ending
SPICE demands that the .end line at the end of the source file not be terminated
with a linefeed or carriage return character. In other words, when you finish
typing ".end" you should not hit the [Enter] key on your keyboard. The cursor
on your text editor should stop immediately to the right of the "d" after the
".end" and go no further. Failure to heed this quirk will result in a "missing .end
cara" error message at the end of the analysis output. The actual circuit
analysis is not affected by this error, so | normally ignore the message.
However, if you're looking to receive a "perfect" output, you must pay heed to
this idiosyncrasy.
Must have a node 0
You are given much freedom in numbering circuit nodes, but you must have a
node 0 somewhere in your netlist in order for SPICE to work. Node 0 is the
default node for circuit ground, and it is the point of reference for all voltages
specified at single node locations.
When simple DC analysis is performed by SPICE, the output will contain a listing
of voltages at all non-zero nodes in the circuit. The point of reference (ground)
for all these voltage readings is node O. For example:
node voltage node voltage
( 1) 15.0000 ( 2) 0.6522
In this analysis, there is a DC voltage of 15 volts between node 1 and ground
(node 0), and a DC voltage of 0.6522 volts between node 2 and ground (node
0). In both these cases, the voltage polarity is negative at node 0 with reference
to the other node (in other words, both nodes 1 and 2 are positive with respect
to node 0).
Avoid open circuits
SPICE cannot handle open circuits of any kind. If your netlist specifies a circuit
with an open voltage source, for example, SPICE will refuse to perform an
analysis. A prime example of this type of error is found when "connecting" a
voltage source to the input of a voltage-dependent source (used to simulate an
operational amplifier). SPICE needs to see a complete path for current, so |
usually tie a high-value resistor (call it rbogus!) across the voltage source to act
as a minimal load.
Avoid certain component loops
SPICE cannot handle certain uninterrupted loops of components in a circuit,
namely voltage sources and inductors. The following loops will cause SPICE to
abort analysis:
Parallel inductors
= 10 mH
netlist
11 2 4 10m
12 2 4 50m
13 2 4 25m
Voltage source / inductor loop
150 mH
netlist
vl 10 dc 12
11 1 0 150m
Series capacitors
5
Cc, | 33 [LF
|
rol T 47 \\F
7
netlist
cl 5 6 33u
c2 6 7 47u
The reason SPICE can't handle these conditions stems from the way it performs
DC analysis: by treating all inductors as shorts and all capacitors as opens.
Since short-circuits (0 Q) and open circuits (infinite resistance) either contain or
generate mathematical infinitudes, a computer simply cannot deal with them,
and so SPICE will discontinue analysis if any of these conditions occur.
In order to make these component configurations acceptable to SPICE, you
must insert resistors of appropriate values into the appropriate places,
eliminating the respective short-circuits and open-circuits. If a series resistor is
required, choose a very low resistance value. Conversely, if a parallel resistor is
required, choose a very high resistance value. For example:
To fix the parallel inductor problem, insert a very low-value resistor in series
with each offending inductor.
Original circuit
"Fixea” circuit
3 Ryogus 1 2 Roogu s2 5
Original netlist
ll 2 4 10m
12 2 4 50m
13 2 4 25m
fixed netlist
rbogusl 2 3 le-12
rbogus2 2 5 le-12
11 3 4 10m
12 2 4 50m
13 5 4 25m
The extremely low-resistance resistors Rpogusi ANd Rpogus2 (each one with a
mere 1 pico-ohm of resistance) "break up" the direct parallel connections that
existed between Lj, Lz, and L3. It is important to choose very low resistances
here so that circuit operation is not substantially impacted by the "fix."
To fix the voltage source / inductor loop, insert a very low-value resistor in series
with the two components.
Original circuit
150 mH
"Fixed" circuit
Riogus
V, — 12V 150 mH
Original netlist
vl 10 dc 12
l1 1 0 150m
fixed netlist
vl 10 dc 12
l1 2 0 150m
rbogus 1 2 le-12
As in the previous example with parallel inductors, it is important to make the
correction resistor (Rpogus) very low in resistance, so as to not substantially
impact circuit operation.
To fix the series capacitor circuit, one of the capacitors must have a resistor
shunting across it. SPICE requires a DC current path to each capacitor for
analysis.
Original circuit "Fixed" circuit
5 5
ale LF bn
6 6 6
C, 47 LF C, 47 UF : Ee
| 7
7 7
Original netlist
cl 5 6 33u
c2 6 7 47u
fixed netlist
cl 5 6 33u
c2 6 7 47u
rbogus 6 7 9e12
The Rpyogus Value of 9 Tera-ohms provides a DC current path to C, (and around
C,) without substantially impacting the circuit's operation.
Current measurement
Although printing or plotting of voltage is quite easy in SPICE, the output of
current values is a bit more difficult. Voltage measurements are specified by
declaring the appropriate circuit nodes. For example, if we desire to know the
voltage across a capacitor whose leads connect between nodes 4 and 7, we
might make out .print statement look like this:
4 7
a
22 LF
cl 4 7 22u
.print ac v(4,7)
However, if we wanted to have SPICE measure the current through that
Capacitor, it wouldn't be quite so easy. Currents in SPICE must be specified in
relation to a voltage source, not any arbitrary component. For example:
6 Vinput 4 2
«—-] 22 WF
cl 4 7 22u
vinput 6 4 ac 1 sin
print ac i(vinput)
This .print card instructs SPICE to print the current through voltage source
Vinput- Which happens to be the same as the current through our capacitor
between nodes 4 and 7. But what if there is no such voltage source in our
circuit to reference for current measurement? One solution is to insert a shunt
resistor into the circuit and measure voltage across it. In this case, | have
chosen a shunt resistance value of 1 QO to produce 1 volt per amp of current
through Cy:
Cc
6 Rehunt 4 ie
12 22 UF
=— 1
cl 4 7 22u
rshunt 6 4 1
.print ac v(6,4)
However, the insertion of an extra resistance into our circuit large enough to
drop a meaningful voltage for the intended range of current might adversely
affect things. A better solution for SPICE is this, although one would never seek
such a current measurement solution in real life:
Vbogns Cc;
6 | 4 Ff
=
Ov 22 WF
=— 1
cl 4 7 22u
vbogus 6 4 dc 0
print ac i(vbogus)
Inserting a "bogus" DC voltage source of zero volts doesn't affect circuit
operation at all, yet it provides a convenient place for SPICE to take a current
measurement. Interestingly enough, it doesn't matter that Vpogus is a DC source
when we're looking to measure AC current! The fact that SPICE will output an
AC current reading is determined by the "ac" specification in the .print card and
nothing more.
It should also be noted that the way SPICE assigns a polarity to current
measurements is a bit odd. Take the following circuit as an example:
example
v1 10
rl 1 2 5k
r2 2 0 5k
.dc v1 10 10 1
print dc i(vl)
.end
With 10 volts total voltage and 10 kQ total resistance, you might expect SPICE
to tell you there's going to be 1 mA (1e-03) of current through voltage source
V,, but in actuality SPICE will output a figure of negative 1 mA (-1e-03)! SPICE
regards current out of the negative end of a DC voltage source (the normal
direction) to be a negative value of current rather than a positive value of
current. There are times I'Il throw in a "bogus" voltage source in a DC circuit like
this simply to get SPICE to output a positive current value:
example
v1 10
rl 1 2 5k
r2 2 3 5k
vbogus 3 0 dc 0
.dc v1 10 10 1
print dc i(vbogus)
.end
Notice how Vpogus is positioned so that the circuit current will enter its positive
side (node 3) and exit its negative side (node 0). This orientation will ensure a
positive output figure for circuit current.
Fourier analysis
When performing a Fourier (frequency-domain) analysis on a waveform, | have
found it necessary to either print or plot the waveform using the .print or .plot
cards, respectively. If you don't print or plot it, SPICE will pause for a moment
during analysis and then abort the job after outputting the "initial transient
solution."
Also, when analyzing a square wave produced by the "pulse" source function,
you must give the waveform some finite rise and fall time, or else the Fourier
analysis results will be incorrect. For some reason, a perfect square wave with
zero rise/fall time produces significant levels of even harmonics according to
SPICE's Fourier analysis option, which is not true for real square waves.
Example circuits and netlists
The following circuits are pre-tested netlists for SPICE 2g6, complete with short
descriptions when necessary. Feel free to "copy" and "paste" any of the netlists
to your own SPICE source file for analysis and/or modification. My goal here is
twofold: to give practical examples of SPICE netlist design to further
understanding of SPICE netlist syntax, and to show how simple and compact
SPICE netlists can be in analyzing simple circuits.
All output listings for these examples have been "trimmed" of extraneous
information, giving you the most succinct presentation of the SPICE output as
possible. | do this primarily to save space on this document. Typical SPICE
outputs contain lots of headers and summary information not necessarily
germane to the task at hand. So don't be surprised when you run a simulation
on your own and find that the output doesn't exactly look like what | have
shown here!
Multiple-source DC resistor network, part 1
Without a .dc card and a .print or .plot card, the output for this netlist will only
display voltages for nodes 1, 2, and 3 (with reference to node 0, of course).
Netlist:
Multiple dc sources
vl 10 dc 24
v2 3 @ de 15
rl 1 2 10k
r2 2 3 8.1k
r3 2 @ 4.7k
.end
Output:
node voltage node voltage node voltage
( 1) 24.0000 ( 2) 9.7470 ( 3) 15.0000
voltage source currents
name current
vl -1.425E-03
v2 -6.485E-04
total power dissipation 4.39E-02 watts
Multiple-source DC resistor network, part 2
By adding a .dc analysis card and specifying source V, from 24 volts to 24 volts
in 1 step (in other words, 24 volts steady), we can use the .print card analysis
to print out voltages between any two points we desire. Oddly enough, when
the .dc analysis option is invoked, the default voltage printouts for each node
(to ground) disappears, so we end up having to explicitly specify them in the
.print card to see them at all.
Netlist:
Multiple dc sources
vl 10
v2 3 0 15
rl 12 10k
r2 2 3 8.1k
r3 2 0 4.7k
.dc vl 24 24 1
.print dc v(1) v(2) v(3) v(1,2) v(2,3)
.end
Output:
vl v(1) v(2) v(3) v(1,2)
2.400E+01 2.400E+01 9.747E+00 1.500E+01 1.425E+01
RC time-constant circuit
1 1 1
Wy 10-9 al _L
47 [LF 22 WF
R,
0 2
3.3kQ 2
v(2,3)
-5.253E+00
For DC analysis, the initial conditions of any reactive component (C or L) must
be specified (voltage for capacitors, current for inductors). This is provided by
the last data field of each capacitor card (ic=0). To perform a DC analysis, the
.tran ("transient") analysis option must be specified, with the first data field
specifying time increment in seconds, the second specifying total analysis
timespan in seconds, and the "uic" telling it to "use initial conditions" when
analyzing.
Netlist:
RC time delay circuit
v1 10 dc 10
cl 1 2 47u ic=0
c2 1 2 22u ic=0
rl 2 0 3.3k
.tran .05 1 uic
.print tran v(1,2)
.end
Output:
time v(1,2)
0.000E+00 7.701E-06
5.000E-02 1.967E+00
1.000E-01 3.551E+00
1.500E-01 4.824E+00
2.000E-01 5.844E+00
2.500E-01 6.664E+00
3.000E-01 7.322E+00
3.500E-01 7.851E+00
4.000E-01 8.274E+00
4.500E-01 8.615E+00
5.000E-01 8.888E+00
5.500E-01 9.107E+00
6.000E-01 9,283E+00
6.500E-01 9.425E+00
7.000E-01 9.538E+00
7.500E-01 9.629E+00
8.000E-01 9.702E+00
8.500E-01 9.761E+00
9.000E-01 9.808E+00
9.500E-01 9.846E+00
1.000E+00 9.877E+00
1 1
Vi
Is Vv (\) Rioad 10 kQ
0 0
This exercise does show the proper setup for plotting instantaneous values of a
sine-wave voltage source with the .plot function (as a transient analysis). Not
surprisingly, the Fourier analysis in this deck also requires the .tran (transient)
analysis option to be specified over a suitable time range. The time range in
this particular deck allows for a Fourier analysis with rather poor accuracy. The
more cycles of the fundamental frequency that the transient analysis is
performed over, the more precise the Fourier analysis will be. This is not a quirk
of SPICE, but rather a basic principle of waveforms.
Netlist:
v1 1 0 sin(0 15 60 0 0)
rload 1 0 10k
* change tran card to the following for better Fourier precision
* .tran 1m 30m .01m and include .options card:
* .options itl5=30000
.tran 1m 30m
.plot tran v(1)
.four 60 v(1)
.end
Output:
time v(1) -2.000E+01 -1.000E+01 0. 000E+00 1.000E+01
©.000E+00 ©.000E+00 . : *
1.000E-03 5.487E+00 . : : * :
2.000E-03 1.025E+01 . ‘ : *
3.000E-03 1.350E+01 . : : : ky
4.000E-03 1.488E+01 . : : : ae
5.000E-03 1.425E+01 . : : : +
6.000E-03 1.150E+01 . : : joe
7.000E-03 7.184E+00 . : ; *
8.000E-03 1.879E+00 . ‘ sue
9.000E-03 -3.714E+00 . : *
1.Q0Q00E-02 -8.762E+00 . ae
1.100E-02 -1.265E+01 . .
1.200E-02 -1.466E+01 . *
1.300E-02 -1.465E+01 . *
1.4Q00E-02 -1.265E+01 . og
1.500E-02 -8.769E+00 . 5
.600E-02 -3.709E+00 . ; * :
. 700E-02 .876E+00 . : 2
. 800E-02 .191E+00 . : , ss ‘
. 900E-02 .149E+01 . ; : am
.QQ00E-02 -425E+01 . . . : ae
. 100E-02 .489E+01 . ‘ : ‘ ee
. 200E-02 .349E+01 . ; : : *
. 300E-02 .Q26E+01 . : . a
-400E-02 .491E+00 . : : mn
. 00E-02 .553E-03 . ; a
.600E-02 -5.514E+00 . ; ee
.700E-02 -1.022E+01 . =
.800E-02 -1.349E+01 . as
.900E-02 -1.495E+01 . *
.QO0E-02 -1.427E+01 . .
PUP RPRPPRPYNPH
'WNONNNNNNNNNPFPRR RF
fourier components of transient response v(1)
dc component = -1.885E-03
harmonic frequency fourier normalized phase normalized
no (hz) component component (deg) phase (deg)
1 6.000E+01 1.494E+01 1.000000 -71.998 0.000
1.200E+02 1.886E-02 0.001262 -50.162 21.836
3 1.800E+02 1.346E-03 0.000090 102.674 174.671
4 2.400E+02 1.799E-02 0.001204 -10.866 61.132
5 3.000E+02 3.604E-03 0.000241 160.923 232.921
6 3.600E+02 5 .642E-03 0.000378 -176.247 -104.250
7 4.200E+02 2.095E-03 0.000140 122.661 194.658
8 4.800E+02 4.574E-03 0.000306 -143.754 -71.757
9 5.400E+02 4.896E-03 0.000328 -129.418 -57.420
total harmonic distortion = 0.186350 percent
Simple AC resistor-capacitor circuit
The .ac card specifies the points of ac analysis from 60Hz to 60Hz, at a single
point. This card, of course, is a bit more useful for multi-frequency analysis,
where a range of frequencies can be analyzed in steps. The .print card outputs
the AC voltage between nodes 1 and 2, and the AC voltage between node 2 and
ground.
Netlist:
Demo of a simple AC circuit
vl 10 ac 12 sin
rl 1 2 30
cl 2 0 100u
-ac lin 1 60 60
-print ac v(1,2) v(2)
.end
Output:
freq v(1,2) v(2)
6.000E+01 8.990E+00 7.949E+00
Low-pass filter
250 mH
100 LF Rioad > 1 kQ
This low-pass filter blocks AC and passes DC to the Rjgag resistor. Typical of a
filter used to suppress ripple from a rectifier circuit, it actually has a resonant
frequency, technically making it a band-pass filter. However, it works well
anyway to pass DC and block the high-frequency harmonics generated by the
AC-to-DC rectification process. Its performance is measured with an AC source
sweeping from 500 Hz to 15 kHz. If desired, the .print card can be substituted
or supplemented with a .plot card to show AC voltage at node 4 graphically.
Netlist:
Lowpass filter
vl 2 1 ac 24 sin
v2 10 dc 24
rload 4 0 1k
l1 2 3 100m
12 3 4 250m
cl 3 0 100u
ac Lin 30 500 15k
print ac v(4)
.plot ac v(4)
.end
f
PRPRP RP RPP PRPPPPOUWUMADHNNDDUUBRWWNNPRU
OUUBBRWWNNPR OU!
req
-000E+02
-000E+03
-500E+03
-000E+03
-500E+03
.000E+03
-500E+03
- 000E+03
-500E+03
-000E+03
-500E+03
-000E+03
-500E+03
-000E+03
-500E+03
.000E+03
-500E+03
.000E+03
-500E+03
.000E+04
-050E+04
. L1OOE+04
. 150E+04
. 200E+04
-250E+04
. 300E+04
.350E+04
-400E+04
-450E+04
-500E+04
req
. QQ0E+O02
. Q00E+03
. 500E+03
. Q00E+03
. 500E+03
. Q00E+03
. 500E+03
. 000E+03
. 500E+03
. Q00E+03
. 500E+03
. 000E+03
PNWHRUOrRNABRrPWeH
&
.935E-01
.275E-02
.057E-02
.614E-03
-402E-03
.403E-03
.884E-04
.973E-04
. 206E-04
.072E-04
.311E-04
.782E-04
.403E-04
.124E-04
.141E-05
.536E-05
.285E-05
.296E-05
.504E-05
.863E-05
.337E-05
.903E-05
.541E-05
.237E-05
.979E-05
.760E-05
.571E-05
-409E-05
.268E-05
. 146E-05
PREP RPRPENNNWWAUDYNOREPHENWAUDHEPNARPWH CS
v(4) 1.000E-06
.935E-01
.275E-02
.Q57E-02
.614E-03
.402E-03
-403E-03 .
.884E-04 .
.973E-04 .
.206E-04 .
.O72E-04 .
.311E-04 .
.782E-04 .
1.000E-04
1.000E-02
1.000E+00
*
6.500E+03 1.403E-04 . as
7.Q000E+03 1.124E-04 . a
7.500E+03 9.141E-05 . *
8.Q000E+03 7.536E-05 . oy
8.500E+03 6.285E-05 . hee
9.Q000E+03 5.296E-05 . =
9.500E+03 4.504E-05 . =
1.000E+04 3.863E-05 . ba
1.050E+04 3.337E-05 . =
1.100E+04 2.903E-05 . *
1.150E+04 2.541E-05 . :
1.200E+04 2.237E-05 . ‘
1.250E+04 1.979E-05 . x
1.300E+04 1.760E-05 . bs
1.350E+04 1.571E-05 . 7
1.400E+04 1.409E-05 . a
1.450E+04 1.268E-05 . *
1.500E+04 1.146E-05 . 2
Multiple-source AC network
0 0 0
One of the idiosyncrasies of SPICE is its inability to handle any loop in a circuit
exclusively composed of series voltage sources and inductors. Therefore, the
"loop" of Vj-Ly-Lo-V2-V, is unacceptable. To get around this, | had to insert a
low-resistance resistor somewhere in that loop to break it up. Thus, we have
Rpogus between 3 and 4 (with 1 pico-ohm of resistance), and V2 between 4 and
0. The circuit above is the original design, while the circuit below has Rpogus
inserted to avoid the SPICE error.
L; L,
450 mH 150 mH
Riogus
330 [LF
Netlist:
Multiple ac source
vl 10 ac 55 © sin
v2 4 0 ac 43 25 sin
11 1 2 450m
cl 2 0 330u
12 2 3 150m
rbogus 3 4 le-12
-ac lin 1 30 30
.print ac v(2)
.end
Output:
freq v(2)
3.000E+01 1.413E+02
AC phase shift demonstration
1 1 1
The currents through each leg are indicated by the voltage drops across each
respective shunt resistor (1 amp = 1 volt through 1 Q), output by the v(1,2) and
v(1,3) terms of the .print card. The phase of the currents through each leg are
indicated by the phase of the voltage drops across each respective shunt
resistor, output by the vp(1,2) and vp(1,3) terms in the .print card.
Netlist:
phase shift
vl 10 ac 4 sin
rshuntl 12 1
rshunt2 13 1
11201
rl 3 0 6.3k
-ac lin 1 1000 1000
-print ac v(1,2) v(1,3) vp(1,2) vp(1,3)
.end
Output:
freq v(1,2) v(1,3) vp(1,2) vp(1,3)
1.000E+03 6.366E-04 6.349E-04 -9.000E+01 0©.000E+00
Transformer circuit
SPICE understands transformers as a set of mutually coupled inductors. Thus, to
simulate a transformer in SPICE, you must specify the primary and secondary
windings as separate inductors, then instruct SPICE to link them together with a
"k" card specifying the coupling constant. For ideal transformer simulation, the
coupling constant would be unity (1). However, SPICE can't handle this value,
sO we use something like 0.999 as the coupling factor.
Note that a// winding inductor pairs must be coupled with their own k cards in
order for the simulation to work properly. For a two-winding transformer, a
single k card will suffice. For a three-winding transformer, three k cards must be
specified (to link L; with Lj, L, with L3, and L, with L3).
The L,/Ly inductance ratio of 100:1 provides a 10:1 step-down voltage
transformation ratio. With 120 volts in we should see 12 volts out of the L,
winding. The L;/L3 inductance ratio of 100:25 (4:1) provides a 2:1 step-down
voltage transformation ratio, which should give us 60 volts out of the L3
winding with 120 volts in.
Netlist:
transformer
v1 10 ac 120 sin
rbogus® 1 6 le-3
11 6 0 100
12241
13 3 5 25
k1 11 12 0.999
k2 12 13 0.999
k3 11 13 0.999
rl 2 4 1000
r2 3 5 1000
rbogusl 5 0 1lel0
rbogus2 4 0 1lel0@
.ac lin 1 60 60
-print ac v(1,0) v(2,0) v(3,0)
.end
Output:
freq v(1) v(2) v(3)
6.000E+01 1.200E+02 1.199E+01 5.993E+01
In this example, Rpoguso IS a Very low-value resistor, serving to break up the
source/inductor loop of Vy/Ly. Rpogus1 ANd Rpogus2 are very high-value resistors
necessary to provide DC paths to ground on each of the isolated circuits. Note
as well that one side of the primary circuit is directly grounded. Without these
ground references, SPICE will produce errors!
Full-wave bridge rectifier
Diodes, like all semiconductor components in SPICE, must be modeled so that
SPICE knows all the nitty-gritty details of how they're supposed to work.
Fortunately, SPICE comes with a few generic models, and the diode is the most
basic. Notice the .model card which simply specifies "d" as the generic diode
model for mod1. Again, since we're plotting the waveforms here, we need to
specify all parameters of the AC source in a single card and print/plot all values
using the .tran option.
Netlist:
fullwave bridge rectifier
v1 10 sin(@ 15 60 0 0)
rload 1 0 10k
d
d
d
d
(
(
0
5
1
1
112 modl
2 0 2 modl
3 3 1 modl
4 3 0 modl
model modl d
tran .5m 25m
.plot tran v(1,0) v(2,3)
end
*
+)-------
. Q00E+00
.QQ00E-04
.QQ00E-03
. 500E-03
v(1)
-- -2.000E+01
-- -5.000E+00
0.Q00E+00 .
2.806E+00 .
5.483E+00 .
7.929E+00 .
-1.000E+01
0.000E+00
0. 000E+00
5.000E+00
1.000E+01
1.000E+01
2.000E+01
1.500E+01
. QQ00E-03
. 500E-03
.QQ00E-03
. 500E-03
.QQ00E-03
. 500E-03
.QOQ00E-03
. 500E-03
.QQ00E-03
. 500E-03
.QQ00E-03
. 500E-03
.QQ00E-03
. 500E-03
.QQ00E-03
. 500E-03
.QQ00E-02
.Q050E-02
. LOOE-02
. 150E-02
. 200E-02
.250E-02
. 300E-02
.350E-02
-400E-02
-450E-02
. 500E-02
.550E-02
. 600E-02
.650E-02
. 700E-02
.750E-02
. 800E-02
.850E-02
.900E-02
.950E-02
.QQ00E-02
.Q50E-02
. LOOE-02
. 150E-02
. 200E-02
.250E-02
. 300E-02
.350E-02
-400E-02
-450E-02
TNNNNNNNNNNNRPRPRPRPRPRPRPRPRPRPRPRPRPRPRPRPRPRPRPRFPOUOAONNODOUUBBHRWWNN
Common-base BJT transistor amplifier
ONDWOrRrABANORRPRPRPRPRPRPHEH
NUNP RP RPP RPRPRPRPRPONAPH
.013E+01
. 198E+01
.338E+01
-435E+01
-476E+01
-470E+01
-406E+01
.299E+01
.139E+01
.455E+00
. 113E+00
.591E+00 .
.841E+00
.177E-01
.689E+00 .
. 380E+00
. 784E+00
1.
.255E+01
.372E+01
-460E+01
-476E+01
-460E+01
.373E+01
.254E+01
.077E+01
.726E+00 .
075E+01
. 293E+00
.684E+00
.361E-01
.875E+00
.552E+00
.170E+00 .
.401E+00
. 146E+01
.293E+01
.414E+01
.464E+01
.483E+01
-430E+01
.344E+01
. 195E+01
.016E+01
.917E+00
-460E+00
. 809E+00
-500E-02 -8.
297E-04 .
+
Oto5 V 24V
This analysis sweeps the input voltage (Vin) from 0 to 5 volts in 0.1 volt
increments, then prints out the voltage between the collector and emitter leads
of the transistor v(2,3). The transistor (Q1) is an NPN with a forward Beta of 50.
Netlist:
Common-base BJT amplifier
vsupply 1 0 dc 24
vin 0 4 dc
rc 1 2 800
re 3 4 100
ql 2 0 3 modl
.model modl npn bf=50
.dc vin 05 0.1
.print dc v(2,3)
.plot dc v(2,3)
.end
Output:
vin v(2,3)
0.000E+00 2.400E+01
1.000E-01 2.410E+01
2.000E-01 2.420E+01
3.000E-01 2.430E+01
4.000E-01 2.440E+01
5.000E-01 2.450E+01
6.000E-01 2.460E+01
7.000E-01 2.466E+01
8.000E-01 2.439E+01
9.000E-01 2.383E+01
1.000E+00 2.317E+01
1.100E+00 2.246E+01
1.200E+00 2.174E+01
1.300E+00 2.101E+01
1.400E+00 2.026E+01
1.500E+00 1.951E+01
1.600E+00 1.876E+01
1.700E+00 1.800E+01
1.800E+00 1.724E+01
1.900E+00 1.648E+01
2. 000E+00 1.572E+01
2.100E+00 1.495E+01
2.200E+00 1.418E+01
2.300E+00 1.342E+01
2.400E+00 1.265E+01
2.500E+00 1.188E+01
2.600E+00 1.110E+01
2.700E+00 1.033E+01
2.800E+00 9.560E+00
2. 900E+00 8.787E+00
3. 000E+00 8.014E+00
3.100E+00 7.240E+00
3.200E+00 6.465E+00
3.300E+00 5.691E+00
3.400E+00 4.915E+00
3.500E+00 4.140E+00
3.600E+00 3. 364E+00
3. 700E+00 2.588E+00
3.800E+00 1.811E+00
3. 900E+00 1.034E+00
4.000E+00 2.587E-01
4.100E+00 9.744E-02
4.200E+00 7.815E-02
4.300E+00 6.806E-02
4.400E+00 6.141E-62
4.500E+00 5.657E-02
4.600E+00 5. 281E-02
4.700E+00 4.981E-02
4.800E+00 4.734E-02
4.900E+00 4.525E-02
5 .000E+00 4.346E-02
vin v(2,3) 0.000E+00 1.000E+01 2.000E+01 3.000E+01
0.000E+00 2.400E+01 *
1.000E-01 2.410E+01 *
2.000E-01 2.420E+01 *
3.000E-01 2.430E+01 *
4.000E-01 2.440E+01 *
5.000E-01 2.450E+01 *
6.000E-01 2.460E+01 . ; ; *
7.000E-01 2.466E+01 . *
8.000E-01 2.439E+01 . ; ; *
9.000E-01 2.383E+01 . .
1.000E+00 2.317E+01 . é . *
1.100E+00 2.246E+01 . ; . *
1.200E+00 2.174E+01 . _*
1.300E+00 2.101E+01 . cs
1.400E+00 2.026E+01 . *
1.500E+00 1.951E+01 . *,
1.600E+00 1.876E+01 . *
1.700E+00 1.800E+01 . *
1.800E+00 1.724E+01 . ; *
1.900E+00 1.648E+01 . *
2.000E+00 1.572E+01 .
2.100E+00 1.495E+01 . : *
2.200E+00 1.418E+01 . : i
2.300E+00 1.342E+01 . ‘ *
2.400E+00 1.265E+01 . ; *
2.500E+00 1.188E+01 . ae
2.600E+00 1.110E+01 . ra
2.700E+00 1.033E+01 . *
2.800E+00 9.560E+00 . ey
2.900E+00 8.787E+00 . i
3.000E+00 8.014E+00 . *
3.100E+00 7.240E+00 . :
3.200E+00 6.465E+00 .
3.300E+00 5.691E+00 . i
3.400E+00 4.915E+00 . *
3.500E+00 4.140E+00 . *
3.600E+00 3.364E+00 . as
3.700E+00 2.588E+00 . -
3.800E+00 1.811E+00 . *
3.900E+00 1.034E+00 .*
4.000E+00 2.587E-01 *
4.100E+00 9.744E-02 *
4.200E+00 7.815E-02 *
4.300E+00 6.806E-02 *
4.400E+00 6.141E-02 *
4.500E+00 5.657E-02 *
4.600E+00 5.281E-02 *
4.700E+00 4.981E-02 *
4.800E+00 4.734E-02 *
4.900E+00 4.525E-02 *
5.Q00E+00 4 *
. 346E-02
Common-source JFET amplifier with self-bias
3 3
Netlist:
common source jfet amplifier
vin 1 0 sin(0 1 60 © 0)
vdd 3 0 dc 20
rdrain 3 2 10k
rsource 4 0 lk
jl 2 1 4 modi
.model mod1 njf
.tran im 30m
.plot tran v(2,0) v(1,0)
.end
Output:
legend:
*: v(2)
+: v(1)
time v(2)
(*)--------- 1.400E+01 1.600E+01 1.800E+01 2.000E+01 2.200E+01
(#).sSssesese -1.000E+00 -5.000E-01 @.Q000E+00 5.000E-01 1.000E+00
Q@.Q000E+00 1.708E+01 . ; * + ‘
1.000E-03 1.609E+01 . oe ‘ + ,
2.Q000E-03 1.516E+01 . * : ‘ . o¢+
3.000E-03 1.448E+01 . * : : ‘ Sa
4.000E-03 1.419E+01 .* : F P +
5.000E-03 1.432E+01 . * ‘ ‘ F +,
6.Q000E-03 1.490E+01 . * 5 : : +
7.Q000E-03 1.577E+01 . F +,
8.Q000E-03 1.676E+01 . a a
9.Q000E-03 1.768E+01 . : + *.
1.000E-02 1.841E+01 . ae 2 wo
1.100E-02 1.890E+01 . + ‘ ‘ *
1.200E-02 1.912E+01 .+ ‘ : *
1.300E-02 1.912E+01 .+ ; ‘ *
1.400E-02 1.890E+01 . + : : *
1.500E-02 1.842E+01 . + , ok
1.600E-02 1.768E+01 . : + uae
1.700E-02 1.676E+01 . a . + ,
1.800E-02 1.577E+01 . aa ‘ +,
1.900E-02 1.491E+01 . * : ¢ P + :
2.000E-02 1.432E+01 . * : : ; +,
2.100E-02 1.419E+01 .* : : ‘ +
2.200E-02 1.449E+01 . * ; F P +
2.300E-02 1.516E+01 . * : ; . +
2.400E-02 1.609E+01 . J* . + ,
2.500E-02 1.708E+01 . . * +
2.600E-02 1.796E+01 . . + *
2.700E-02 1.861E+01 . fs fs |
2.800E-02 1.900E+O1 . + : F *
2.900E-02 1.916E+01 + : , *
3.000E-02 1.908E+01 .+ : . *
Inverting op-amp Circuit
To simulate an ideal operational amplifier in SPICE, we use a voltage-dependent
voltage source as a differential amplifier with extremely high gain. The "e" card
sets up the dependent voltage source with four nodes, 3 and 0 for voltage
output, and 1 and O for voltage input. No power supply is needed for the
dependent voltage source, unlike a real operational amplifier. The voltage gain
is set at 999,000 in this case. The input voltage source (V;) sweeps from 0 to
3.5 volts in 0.05 volt steps.
Netlist:
Inverting opamp
vl 2 0 dc
e300 1 999k
rl 3 1 3.29k
r2 12 1.18k
.dc vl 0 3.5 0.05
.print dc v(3,0)
.end
Output:
ry
. 000E+00
.QQ00E-02
.QQ00E-01
.500E-01
.QQ00E-01
.500E-01
.QQ00E-01
.500E-01
.QQ00E-01
.500E-01
.QQ00E-01
.500E-01
.QQ00E-01
.500E-01
.QQ00E-01
.500E-01
NNODODUUNBRWWNNPRUOK
v(3)
. 900E+00
.394E-01
.788E-01
.182E-01
.576E-01
.970E-01
.364E-01
.758E-01
. 115E+00
.255E+00
. 394E+00
.533E+00
.673E+00
.812E+00
.952E+00
.091E+00
8.000E-01 -2.231E+00
8.500E-01 -2.370E+00
9.000E-01 -2.509E+00
9.500E-01 -2.649E+00
1. 000E+00 -2.788E+00
1.050E+00 -2.928E+00
1.100E+00 -3.067E+00
1.150E+00 -3.206E+00
1.200E+00 -3.346E+00
1.250E+00 -3.485E+00
1.300E+00 -3.625E+00
1.350E+00 -3.764E+00
1.400E+00 -3.903E+00
1.450E+00 -4.043E+00
1.500E+00 -4.182E+00
1.550E+00 -4.322E+00
1.600E+00 -4.461E+00
1.650E+00 -4.600E+00
1.700E+00 -4.740E+00
1.750E+00 -4.879E+00
1.800E+00 -5.019E+00
1.850E+00 -5.158E+00
1.900E+00 -5.297E+00
1.950E+00 -5.437E+00
2.000E+00 -5.576E+00
2.050E+00 -5.716E+00
2.100E+00 -5.855E+00
2.150E+00 -5.994E+00
2.200E+00 -6.134E+00
2.250E+00 -6.273E+00
2.300E+00 -6.413E+00
2.350E+00 -6.552E+00
2.400E+00 -6.692E+00
2.450E+00 -6.831E+00
2.500E+00 -6.970E+00
2.550E+00 -7.110E+00
2.600E+00 -7.249E+00
2.650E+00 -7.389E+00
2.700E+00 -7.528E+00
2.750E+00 -7.667E+00
2.800E+00 -7.807E+00
2.850E+00 -7.946E+00
2.900E+00 -8.086E+00
2.950E+00 -8.225E+00
3.000E+00 -8.364E+00
3.050E+00 -8.504E+00
3.100E+00 -8.643E+00
3.150E+00 -8.783E+00
3.200E+00 -8.922E+00
3.250E+00 -9.061E+00
3.300E+00 -9.201E+00
3.350E+00 -9.340E+00
3.400E+00 -9.480E+00
3.450E+00 -9.619E+00
3.500E+00 -9.758E+00
Noninverting op-amp circuit
5 V
v=
JE
0-=
Another example of a SPICE quirk: since the dependent voltage source "e" isn't
considered a load to voltage source Vj, SPICE interprets V, to be open-circuited
and will refuse to analyze it. The fix is to connect Rpogus in parallel with V; to
act as a DC load. Being directly connected across Vj, the resistance of Rpogus IS
not crucial to the operation of the circuit, so 10 kQ will work fine. | decided not
to sweep the V, input voltage at all in this circuit for the sake of keeping the
netlist and output listing simple.
Netlist:
noninverting opamp
vl 2 @dc 5
rbogus 2 0 10k
e302 1 999k
rl 3 1 20k
r2 1 0 10k
.end
Output:
node voltage node voltage node
( 1) 5.0000 ( 2) 5.0000 ( 3)
Instrumentation amplifier
voltage
15.0000
Es)
Pg
+ R,
. (el) T VV
= 10 kQ
Roogust Vv;
Oto 1OV
als R,> 10kQ
o- o>
42
Rens 1010
-———————45
B, $ 1040
s*- Rs
(e2) + vv“
2
3
5
Note the very high-resistance Rpogus1 2Nd Rpogus2 resistors in the netlist (not
shown in schematic for brevity) across each input voltage source, to keep SPICE
from thinking V, and V> were open-circuited, just like the other op-amp circuit
examples.
Netlist:
Instrumentation amplifier
vl 10
rbogusl 1 0 9el12
v2 4 @dc5
rbogus2 4 0 9el12
el 3 0 1 2 999k
e2 6 0 4 5 999k
e3 9 0 8 7 999k
rload 9 0 10k
rl 2 3 10k
rgain 2 5 10k
r2 5 6 10k
r3 3 7 10k
r4 7 9 10k
r5 6 8 10k
r6 8 0 10k
.dc vl 0 10 1
print dc v(9) v(3,6)
.end
Output:
vl v(9) v(3,6)
0.000E+00 1.500E+01 -1.500E+01
1. 000E+00 1.200E+01 -1.200E+01
2.000E+00 9.000E+00 -9.000E+00
3.000E+00 6.000E+00 -6.000E+00
4.000E+00 3.000E+00 -3.000E+00
5. Q000E+00 9.955E-11 -9.956E-11
6.000E+00 -3.000E+00 3.000E+00
7.000E+00 -6.000E+00 6.000E+00
8. 000E+00 -9.000E+00 9.000E+00
9.000E+00 -1.200E+01 1.200E+01
1.000E+01 -1.500E+01 1.500E+01
Netlist:
Integrator with sinewave input
vin 1 0 sin (0 15 60 0 @)
rl 1 2 10k
cl 2 3 150u ic=0
e300 2 999k
.tran 1m 30m uic
.plot tran v(1,0) v(3,0)
.end
Output:
legend
*: v(1)
+: v(3)
time v(1)
(F)-44-222% -2.000E+01 -1.000E+01 0.Q00E+00 1.000E+01
(+)-------- -6.000E-02 -4.000E-02 -2.000E-02 0.000E+00
0.000E+00 6.536E-08 . ; * +
1.000E-03 5.516E+00 . ; : 7% +.
. QQ00E-03
.QQ00E-03
.QQ00E-03
. QQ00E-03
.QQ00E-03
.QQ00E-03
. QQ00E-03
.QQ00E-03
.QQ00E-02
. LOOE-02
. 200E-02
. 300E-02
-400E-02
. 500E-02
. 600E-02
. 700E-02
. 800E-02
.900E-02
.QQ00E-02
. LOOE-02
. 200E-02
. 300E-02
-400E-02
. 00E-02
.600E-02
. 700E-02
. 800E-02
.900E-02
.QOQ00E-02
2
3
4
5
6
7
8
9
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
3
Netlist:
Integrator with squarewave input
vin 1 0 pulse (-1 10 0 0 10m 20m)
rl 12 1k
.021E+01
.350E+01
-495E+01
-418E+01
.150E+01 .
.214E+00 .
.867E+00 .
.709E+00 .
.805E+00 .
.259E+01
-466E+01
.471E+01
.259E+01 .
.774E+00 .
.723E+00 .
.870E+00 .
.188E+00 .
.154E+01
.418E+01
-490E+01
.355E+01
.Q20E+01 .
.496E+00 .
-486E-03 .
.489E+00 .
.021E+01
.355E+01
.488E+01
.427E+01
cl 2 3 150u ic=0
e300 2 999k
.tran 1m 50m uic
.plot tran v(1,0) v(3,0)
.end
Output:
legend:
*: v(1)
+: v(3)
time v(1)
(*)-------- -1.000E+00
(+)-------- -1.000E-01
0.Q000E+00 -1.000E+00 *
1.000E-03 1.Q00E+00 .
2.000E-03 1.000E+00 .
3.000E-03 1.000E+00 .
4.000E-03 1.000E+00
5.000E-03 1.000E+00 .
6.000E-03 1.000E+00 .
7.000E-03 1.000E+00 .
8.000E-03 1.000E+00 .
9.000E-03 1.000E+00
1.000E-02 1.Q000E+00 .
1.100E-02 1.Q00E+00 .
1.200E-02 -1.Q000E+00 *
1.300E-02 -1.Q000E+00 *
1.400E-02 -1.Q000E+00 *
1.500E-02 -1.Q000E+00 *
1.600E-02 -1.Q000E+00 *
1.700E-02 -1.Q00E+00 *
1.800E-02 -1.Q000E+00 *
1.900E-02 -1.Q000E+00 *
2.Q000E-02 -1.000E+00 *
2.100E-02 1.000E+00 .
2.200E-02 1.000E+00
2.300E-02 1.000E+00 .
2.400E-02 1.000E+00 .
2.500E-02 1.000E+00
2.600E-02 1.000E+00 .
2.700E-02 1.000E+00 .
2.800E-02 1.000E+00
2.900E-02 1.000E+00 .
3.000E-02 1.000E+00 .
3.100E-02 1.000E+00 . +
3.200E-02 -1.000E+00 * +
3.300E-02 -1.000E+00 *
3.400E-02 -1.000E+00 *
3.500E-02 -1.000E+00 *
3.600E-02 -1.000E+00 *
3.700E-02 -1.000E+00 *
3.800E-02 -1.000E+00 *
3.900E-02 -1.000E+00 *
4.000E-02 -1.000E+00 *
-5.000E-01
-5.000E-02
0.000E+00 5.000E-01
0.000E+00 5.000E-02
1. 000E+00
1.000E-01
* *¥ ¥ ¥ *¥ ¥ ¥ ¥ ¥ *¥
* *¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ KK:
4.100E-02 1.000E+00 . i + *
4.200E-02 1.000E+00 . : + *
4.300E-02 1.000E+00 . 2 OF *
4.400E-02 1.000E+00 . + *
4.500E-02 1.000E+00 . +, in
4.600E-02 1.000E+00 . + *
4.700E-02 1.000E+00 . + *
4.800E-02 1.000E+00 . + *
4.900E-02 1.000E+00 . + *
5.Q00E-02 1.000E+00 + *
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt, under
the terms and conditions of the Design Science License.
Previous Contents
=|i4 1 |=
—/ | 4]
Lessons In Electric Circuits
-- Volume V
Chapter 8
TROUBLESHOOTING --
THEORY AND PRACTICE
Questions to ask before proceeding
General troubleshooting tips
o Prior occurrence
o Recent alterations
o Function vs. non-function
o Hypothesize
Specific troubleshooting techniques
Swap identical components
o Remove parallel components
o Divide system into sections and test those sections
o Simplify and rebuild
o Trap a signal
Likely failures in proven systems
o Operator error
o Bad wire connections
o Power supply _ problems
o Active components
o Passive components
Likely failures in unproven systems
o Wiring_problems
Power supply problems
Defective components
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Potential pitfalls
e Contributors
Perhaps the most valuable but difficult-to-learn skill any
technical person could have is the ability to troubleshoot a
system. For those unfamiliar with the term, troubleshooting
means the act of pinpointing and correcting problems in any
kind of system. For an auto mechanic, this means
determining and fixing problems in cars based on the car's
behavior. For a doctor, this means correctly diagnosing a
patient's malady and prescribing a cure. For a business
expert, this means identifying the source(s) of inefficiency in
a corporation and recommending corrective measures.
Troubleshooters must be able to determine the cause or
causes of a problem simply by examining its effects. Rarely
does the source of a problem directly present itself for all to
see. Cause/effect relationships are often complex, even for
seemingly simple systems, and often the proficient
troubleshooter is regarded by others as something of a
miracle-worker for their ability to quickly discern the root
cause of a problem. While some people are gifted with a
natural talent for troubleshooting, it is a skill that can be
learned like any other.
Sometimes the system to be analyzed is in so bad a state of
affairs that there is no hope of ever getting it working again.
When investigators sift through the wreckage of a crashed
airplane, or when a doctor performs an autopsy, they must
do their best to determine the cause of massive failure after
the fact. Fortunately, the task of the troubleshooter is
usually not this grim. Typically, a misbehaving system is still
functioning to some degree and may be stimulated and
adjusted by the troubleshooter as part of the diagnostic
procedure. In this sense, troubleshooting is a lot like
scientific method: determining cause/effect relationships by
means of live experimentation.
Like science, troubleshooting is a mixture of standard
procedure and personal creativity. There are certain
procedures employed as tools to discern cause(s) from
effects, but they are impotent if not coupled with a creative
and inquisitive mind. In the course of troubleshooting, the
troubleshooter may have to invent their own specific
technique -- adapted to the particular system they're
working on -- and/or modify tools to perform a special task.
Creativity is necessary in examining a problem from
different perspectives: learning to ask different questions
when the "standard" questions don't lead to fruitful answers.
If there is one personality trait I've seen positively
associated with excellent troubleshooting more than any
other, its technical curiosity. People fascinated by learning
how things work, and who aren't discouraged by a
challenging problem, tend to be better at troubleshooting
than others. Richard Feynman, the late physicist who taught
at Caltech for many years, illustrates to me the ultimate
troubleshooting personality. Reading any of his
(auto)biographical books is both educating and
entertaining, and | recommend them to anyone seeking to
develop their own scientific reasoning/troubleshooting skills.
Questions to ask before proceeding
e Has the system ever worked before? If yes, has anything
happened to it since then that could cause the problem?
e Has this system proven itself to be prone to certain
types of failure?
e How urgent is the need for repair?
What are the safety concerns, before | start
troubleshooting?
What are the process quality concerns, before | start
troubleshooting (what can | do without causing
interruptions in production)?
These preliminary questions are not trivial. Indeed, they are
essential to expedient and safe troubleshooting. They are
especially important when the system to be trouble-shot is
large, dangerous, and/or expensive.
Sometimes the troubleshooter will be required to work ona
system that is still in full operation (perhaps the ultimate
example of this is a doctor diagnosing a live patient). Once
the cause or causes are determined to a high degree of
certainty, there is the step of corrective action. Correcting a
system fault without significantly interrupting the operation
of the system can be very challenging, and it deserves
thorough planning.
When there is high risk involved in taking corrective action,
such as is the case with performing surgery on a patient or
making repairs to an operating process in a chemical plant,
it is essential for the worker(s) to plan ahead for possible
trouble. One question to ask before proceeding with repairs
is, "how and at what point(s) can | abort the repairs if
something goes wrong?" In risky situations, it is vital to have
planned "escape routes" in your corrective action, just in
case things do not go as planned. A surgeon operating ona
patient knows if there are any "points of no return" in sucha
procedure, and stops to re-check the patient before
proceeding past those points. He or she also knows how to
"back out" of a surgical procedure at those points if needed.
General troubleshooting tips
When first approaching a failed or otherwise misbehaving
system, the new troubleshooter often doesn't know where to
begin. The following strategies are not exhaustive by any
means, but provide the troubleshooter with a simple
checklist of questions to ask in order to start isolating the
problem.
As tips, these troubleshooting suggestions are not
comprehensive procedures: they serve as starting points
only for the troubleshooting process. An essential part of
expedient troubleshooting is probability assessment, and
these tips help the troubleshooter determine which possible
points of failure are more or less likely than others. Final
isolation of the system failure is usually determined through
more specific techniques (outlined in the next section --
Specific Troubleshooting Techniques).
Prior occurrence
If this device or process has been historically known to fail in
a certain particular way, and the conditions leading to this
common failure have not changed, check for this "way" first.
A corollary to this troubleshooting tip is the directive to keep
detailed records of failure. Ideally, a computer-based failure
log is optimal, so that failures may be referenced by and
correlated to a number of factors such as time, date, and
environmental conditions.
Example: 7he car's engine is overheating. The last two
times this happened, the cause was low coolant level in the
radiator.
What to do: Check the coolant level first. Of course, past
history by no means guarantees the present symptoms are
caused by the same problem, but since this is more likely, it
makes sense to check this first.
If, however, the cause of routine failure in a system has been
corrected (i.e. the leak causing low coolant level in the past
has been repaired), then this may not be a probable cause of
trouble this time.
Recent alterations
If a system has been having problems immediately after
some kind of maintenance or other change, the problems
might be linked to those changes.
Example: The mechanic recently tuned my car's engine,
and now | hear a rattling noise that | didn't hear before |
took the car in for repair.
What to do: Check for something that may have been left
loose by the mechanic after his or her tune-up work.
Function vs. non-function
If a system isn't producing the desired end result, look for
what it /s doing correctly; in other words, identify where the
problem is not, and focus your efforts elsewhere. Whatever
components or subsystems necessary for the properly
working parts to function are probably okay. The degree of
fault can often tell you what part of it is to blame.
Example: 7he radio works fine on the AM band, but not on
the FM band.
What to do: Eliminate from the list of possible causes,
anything in the radio necessary for the AM band's function.
Whatever the source of the problem is, it is specific to the
FM band and not to the AM band. This eliminates the audio
amplifier, soeakers, fuse, power supply, and almost all
external wiring. Being able to eliminate sections of the
system as possible failures reduces the scope of the problem
and makes the rest of the troubleshooting procedure more
efficient.
Hypothesize
Based on your knowledge of how a system works, think of
various kinds of failures that would cause this problem (or
these phenomena) to occur, and check for those failures
(starting with the most likely based on circumstances,
history, or knowledge of component weaknesses).
Example: 7he car's engine is overheating.
What to do: Consider possible causes for overheating, based
on what you know of engine operation. Either the engine is
generating too much heat, or not getting rid of the heat well
enough (most likely the latter). Brainstorm some possible
causes: a loose fan belt, clogged radiator, bad water pump,
low coolant level, etc. Investigate each one of those
possibilities before investigating alternatives.
Specific troubleshooting techniques
After applying some of the general troubleshooting tips to
narrow the scope of a problem's location, there are
techniques useful in further isolating it. Here are a few:
Swap identical components
In a system with identical or parallel subsystems, swap
components between those subsystems and see whether or
not the problem moves with the swapped component. If it
does, you've just swapped the faulty component; if it
doesn't, keep searching!
This is a powerful troubleshooting method, because it gives
you both a positive and a negative indication of the
Swapped component's fault: when the bad part is
exchanged between identical systems, the formerly broken
subsystem will start working again and the formerly good
subsystem will fail.
| was once able to troubleshoot an elusive problem with an
automotive engine ignition system using this method: |
happened to have a friend with an automobile sharing the
exact same model of ignition system. We swapped parts
between the engines (distributor, spark plug wires, ignition
coil -- one at a time) until the problem moved to the other
vehicle. The problem happened to be a "weak" ignition coil,
and it only manifested itself under heavy load (a condition
that could not be simulated in my garage). Normally, this
type of problem could only be pinpointed using an ignition
system analyzer (or oscilloscope) and a dynamometer to
simulate loaded driving conditions. This technique, however,
confirmed the source of the problem with 100% accuracy,
using no diagnostic equipment whatsoever.
Occasionally you may swap a component and find that the
problem still exists, but has changed in some way. This tells
you that the components you just swapped are somehow
different (different calibration, different function), and
nothing more. However, don't dismiss this information just
because it doesn't lead you straight to the problem -- look
for other changes in the system as a whole as a result of the
swap, and try to figure out what these changes tell you
about the source of the problem.
An important caveat to this technique is the possibility of
causing further damage. Suppose a component has failed
because of another, less conspicuous failure in the system.
Swapping the failed component with a good component will
cause the good component to fail as well. For example,
suppose that a circuit develops a short, which "blows" the
protective fuse for that circuit. The blown fuse is not evident
by inspection, and you don't have a meter to electrically test
the fuse, so you decide to swap the suspect fuse with one of
the same rating from a working circuit. As a result of this,
the good fuse that you move to the shorted circuit blows as
well, leaving you with two blown fuses and two non-working
circuits. At least you know for certain that the original fuse
was blown, because the circuit it was moved to stopped
working after the swap, but this knowledge was gained only
through the loss of a good fuse and the additional "down
time" of the second circuit.
Another example to illustrate this caveat is the ignition
system problem previously mentioned. Suppose that the
"weak" ignition coil had caused the engine to backfire,
damaging the muffler. If swapping ignition system
components with another vehicle causes the problem to
move to the other vehicle, damage may be done to the other
vehicle's muffler as well. As a general rule, the technique of
swapping identical components should be used only when
there is minimal chance of causing additional damage. It is
an excellent technique for isolating non-destructive
problems.
Example 1: You're working on a CNC machine tool with x,
Y, and Z-axis drives. The Y axis is not working, but the X and
Z axes are working. All three axes share identical
components (feedback encoders, servo motor drives, servo
motors).
What to do: Exchange these identical components, one ata
time, Y axis and either one of the working axes (X or Z), and
see after each swap whether or not the problem has moved
with the swap.
Example 2: A stereo system produces no sound on the left
speaker, but the right speaker works just fine.
What to do: Try swapping respective components between
the two channels and see if the problem changes sides, from
left to right. When it does, you've found the defective
component. For instance, you could swap the speakers
between channels: if the problem moves to the other side
(i.e. the same speaker that was dead before is still dead, now
that its connected to the right channel cable) then you know
that speaker is bad. If the problem stays on the same side
(i.e. the speaker formerly silent is now producing sound after
having been moved to the other side of the room and
connected to the other cable), then you know the speakers
are fine, and the problem must lie somewhere else (perhaps
in the cable connecting the silent speaker to the amplifier,
or in the amplifier itself).
If the soeakers have been verified as good, then you could
check the cables using the same method. Swap the cables
so that each one now connects to the other channel of the
amplifier and to the other speaker. Again, if the problem
changes sides (i.e. now the right speaker is now "dead" and
the left soeaker now produces sound), then the cable now
connected to the right speaker must be defective. If neither
swap (the speakers nor the cables) causes the problem to
change sides from left to right, then the problem must lie
within the amplifier (i.e. the left channel output must be
"dead").
Remove parallel components
If a system is composed of several parallel or redundant
components which can be removed without crippling the
whole system, start removing these components (one at a
time) and see if things start to work again.
Example 1: A "star" topology communications network
between several computers has failed. None of the
computers are able to communicate with each other.
What to do: Try unplugging the computers, one at atime
from the network, and see if the network starts working
again after one of them is unplugged. If it does, then that
last unplugged computer may be the one at fault (it may
have been "jamming" the network by constantly outputting
data or noise).
Example 2: A household fuse keeps blowing (or the breaker
keeps tripping open) after a short amount of time.
What to do: Unplug appliances from that circuit until the
fuse or breaker quits interrupting the circuit. If you can
eliminate the problem by unplugging a single appliance,
then that appliance might be defective. If you find that
unplugging almost any appliance solves the problem, then
the circuit may simply be overloaded by too many
appliances, neither of them defective.
Divide system into sections and test those
sections
In a system with multiple sections or stages, carefully
measure the variables going in and out of each stage until
you find a stage where things don't look right.
Example 1: A radio is not working (producing no sound at
the speaker))
What to do: Divide the circuitry into stages: tuning stage,
mixing stages, amplifier stage, all the way through to the
speaker(s). Measure signals at test points between these
stages and tell whether or not a stage is working properly.
Example 2: An analog summer circuit is not functioning
properly.
Analog summer circuit
R 2R
Vout
V inl
Vin
R
Vv
in3
What to do: | would test the passive averager network (the
three resistors at the lower-left corner of the schematic) to
see that the proper (averaged) voltage was seen at the
noninverting input of the op-amp. | would then measure the
voltage at the inverting input to see if it was the same as at
the noninverting input (or, alternatively, measure the
voltage difference between the two inputs of the op-amp, as
it should be zero). Continue testing sections of the circuit (or
just test points within the circuit) to see if you measure the
expected voltages and currents.
Simplify_ and rebuild
Closely related to the strategy of dividing a system into
sections, this is actually a design and fabrication technique
useful for new circuits, machines, or systems. It's always
easier begin the design and construction process in little
steps, leading to larger and larger steps, rather than to build
the whole thing at once and try to troubleshoot it as a whole.
Suppose that someone were building a custom automobile.
He or she would be foolish to bolt all the parts together
without checking and testing components and subsystems
as they went along, expecting everything to work perfectly
after its all assembled. Ideally, the builder would check the
proper operation of components along the way through the
construction process: start and tune the engine before its
connected to the drivetrain, check for wiring problems
before all the cover panels are put in place, check the brake
system in the driveway before taking it out on the road, etc.
Countless times I've witnessed students build a complex
experimental circuit and have trouble getting it to work
because they didn't stop to check things along the way: test
all resistors before plugging them into place, make sure the
power supply is regulating voltage adequately before trying
to power anything with it, etc. It is human nature to rush to
completion of a project, thinking that such checks area
waste of valuable time. However, more time will be wasted
in troubleshooting a malfunctioning circuit than would be
spent checking the operation of subsystems throughout the
process of construction.
Take the example of the analog summer circuit in the
previous section for example: what if it wasn't working
properly? How would you simplify it and test it in stages?
Well, you could reconnect the op-amp as a basic comparator
and see if its responsive to differential input voltages, and/or
connect it as a voltage follower (buffer) and see if it outputs
the same analog voltage as what is input. If it doesn't
perform these simple functions, it will never perform its
function in the summer circuit! By stripping away the
complexity of the summer circuit, paring it down to an
(almost) bare op-amp, you can test that component's
functionality and then build from there (add resistor
feedback and check for voltage amplification, then add
input resistors and check for voltage summing), checking for
expected results along the way.
Trap a signal
Set up instrumentation (such as a datalogger, chart
recorder, or multimeter set on "record" mode) to monitor a
signal over a period of time. This is especially helpful when
tracking down intermittent problems, which have a way of
showing up the moment you've turned your back and
walked away.
This may be essential for proving what happens first in a
fast-acting system. Many fast systems (especially shutdown
"trip" systems) have a "first out" monitoring capability to
provide this kind of data.
Example #1: A turbine contro! system shuts automatically
in response to an abnormal condition. By the time a
technician arrives at the scene to survey the turbine's
condition, however, everything is ina "down" state and its
impossible to tell what signal or condition was responsible
for the initial shutdown, as all operating parameters are now
"abnormal."
What to do: One technician | knew used a videocamera to
record the turbine control panel, so he could see what
happened (by indications on the gauges) first in an
automatic-shutdown event. Simply by looking at the panel
after the fact, there was no way to tell which signal shut the
turbine down, but the videotape playback would show what
happened in sequence, down to a frame-by-frame time
resolution.
Example #2: An alarm system Is falsely triggering, and you
suspect it may be due to a specific wire connection going
bad. Unfortunately, the problem never manifests itself while
you're watching it!
What to do: Many modern digital multimeters are equipped
with "record" settings, whereby they can monitor a voltage,
current, or resistance over time and note whether that
measurement deviates substantially from a regular value.
This is an invaluable tool for use in "intermittent" electronic
system failures.
Likely failures in proven systems
The following problems are arranged in order from most
likely to least likely, top to bottom. This order has been
determined largely from personal experience
troubleshooting electrical and electronic problems in
automotive, industry, and home applications. This order also
assumes a circuit or system that has been proven to function
as designed and has failed after substantial operation time.
Problems experienced in newly assembled circuits and
systems do not necessarily exhibit the same probabilities of
occurrence.
Operator error
A frequent cause of system failure is error on the part of
those human beings operating it. This cause of trouble is
placed at the top of the list, but of course the actual
likelihood depends largely on the particular individuals
responsible for operation. When operator error is the cause
of a failure, it is unlikely that it will be admitted prior to
investigation. | do not mean to suggest that operators are
incompetent and irresponsible -- quite the contrary: these
people are often your best teachers for learning system
function and obtaining a history of failure -- but the reality of
human error cannot be overlooked. A positive attitude
coupled with good interpersonal skills on the part of the
troubleshooter goes a long way in troubleshooting when
human error is the root cause of failure.
Bad wire connections
As incredible as this may sound to the new student of
electronics, a high percentage of electrical and electronic
system problems are caused by a very simple source of
trouble: poor (i.e. open or shorted) wire connections. This is
especially true when the environment is hostile, including
such factors as high vibration and/or a corrosive
atmosphere. Connection points found in any variety of plug-
and-socket connector, terminal strip, or splice are at the
greatest risk for failure. The category of "connections" also
includes mechanical switch contacts, which can be thought
of as a high-cycle connector. Improper wire termination lugs
(such as a compression-style connector crimped on the end
of a solid wire -- a definite faux pas) can cause high-
resistance connections after a period of trouble-free service.
It should be noted that connections in low-voltage systems
tend to be far more troublesome than connections in high-
voltage systems. The main reason for this is the effect of
arcing across a discontinuity (circuit break) in higher-voltage
systems tends to blast away insulating layers of dirt and
corrosion, and may even weld the two ends together if
sustained long enough. Low-voltage systems tend not to
generate such vigorous arcing across the gap of a circuit
break, and also tend to be more sensitive to additional
resistance in the circuit. Mechanical switch contacts used in
low-voltage systems benefit from having the recommended
minimum wetting current conducted through them to
promote a healthy amount of arcing upon opening, even if
this level of current is not necessary for the operation of
other circuit components.
Although open failures tend to more common than shorted
failures, "shorts" still constitute a substantial percentage of
wiring failure modes. Many shorts are caused by degradation
of wire insulation. This, again, is especially true when the
environment is hostile, including such factors as high
vibration, high heat, high humidity, or high voltage. It is rare
to find a mechanical switch contact that is failed shorted,
except in the case of high-current contacts where contact
"welding" may occur in overcurrent conditions. Shorts may
also be caused by conductive buildup across terminal strip
sections or the backs of printed circuit boards.
A common case of shorted wiring is the ground fault, where
a conductor accidently makes contact with either earth or
chassis ground. This may change the voltage(s) present
between other conductors in the circuit and ground, thereby
causing bizarre system malfunctions and/or personnel
hazard.
Power supply problems
These generally consist of tripped overcurrent protection
devices or damage due to overheating. Although power
supply circuitry is usually less complex than the circuitry
being powered, and therefore should figure to be less prone
to failure on that basis alone, it generally handles more
power than any other portion of the system and therefore
must deal with greater voltages and/or currents. Also,
because of its relative design simplicity, a system's power
supply may not receive the engineering attention it
deserves, most of the engineering focus devoted to more
glamorous parts of the system.
Active components
Active components (amplification devices) tend to fail with
greater regularity than passive (non-amplifying) devices,
due to their greater complexity and tendency to amplify
overvoltage/overcurrent conditions. Semiconductor devices
are notoriously prone to failure due to electrical transient
(voltage/current surge) overloading and thermal (heat)
overloading. Electron tube devices are far more resistant to
both of these failure modes, but are generally more prone to
mechanical failures due to their fragile construction.
Passive components
Non-amplifying components are the most rugged of all, their
relative simplicity granting them a statistical advantage
over active devices. The following list gives an approximate
relation of failure probabilities (again, top being the most
likely and bottom being the least likely):
e Capacitors (shorted), especially e/ectrolytic capacitors.
The paste electrolyte tends to lose moisture with age,
leading to failure. Thin dielectric layers may be
punctured by overvoltage transients.
e Diodes open (rectifying diodes) or shorted (Zener
diodes).
e Inductor and transformer windings open or shorted to
conductive core. Failures related to overheating
(insulation breakdown) are easily detected by smell.
Resistors open, almost never shorted. Usually this is due
to overcurrent heating, although it is less frequently
caused by overvoltage transient (arc-over) or physical
damage (vibration or impact). Resistors may also change
resistance value if overheated!
Likely failures in unproven systems
"All men are liable to error; "
John Locke
Whereas the last section deals with component failures in
systems that have been successfully operating for some
time, this section concentrates on the problems plaguing
brand-new systems. In this case, failure modes are generally
not of the aging kind, but are related to mistakes in design
and assembly caused by human beings.
Wiring problems
In this case, bad connections are usually due to assembly
error, such as connection to the wrong point or poor
connector fabrication. Shorted failures are also seen, but
usually involve misconnections (conductors inadvertently
attached to grounding points) or wires pinched under box
covers.
Another wiring-related problem seen in new systems is that
of electrostatic or electromagnetic interference between
different circuits by way of close wiring proximity. This kind
of problem is easily created by routing sets of wires too close
to each other (especially routing signal cables close to
power conductors), and tends to be very difficult to identify
and locate with test equipment.
Power supply problems
Blown fuses and tripped circuit breakers are likely sources of
trouble, especially if the project in question is an addition to
an already-functioning system. Loads may be larger than
expected, resulting in overloading and subsequent failure of
power supplies.
Defective components
In the case of a newly-assembled system, component fault
probabilities are not as predictable as in the case of an
operating system that fails with age. Any type of component
-- active or passive -- may be found defective or of imprecise
value "out of the box" with roughly equal probability,
barring any specific sensitivities in shipping (i.e fragile
vacuum tubes or electrostatically sensitive semiconductor
components). Moreover, these types of failures are not
always as easy to identify by sight or smell as an age- or
transient-induced failure.
Increasingly seen in large systems using microprocessor-
based components, "programming" issues can still plague
non-microprocessor systems in the form of incorrect time-
delay relay settings, limit switch calibrations, and drum
switch sequences. Complex components having
configuration "jumpers" or switches to control behavior may
not be "programmed" properly.
Components may be used in a new system outside of their
tolerable ranges. Resistors, for example, with too low of
power ratings, of too great of tolerance, may have been
installed. Sensors, instruments, and controlling mechanisms
may be uncalibrated, or calibrated to the wrong ranges.
Design error
Perhaps the most difficult to pinpoint and the slowest to be
recognized (especially by the chief designer) is the problem
of design error, where the system fails to function simply
because it cannot function as designed. This may be as
trivial as the designer specifying the wrong components in a
system, or as fundamental as a system not working due to
the designer's improper knowledge of physics.
| once saw a turbine control system installed that used a
low-pressure switch on the lubrication oil tubing to shut
down the turbine if oil pressure dropped to an insufficient
level. The oil pressure for lubrication was supplied by an oil
pump turned by the turbine. When installed, the turbine
refused to start. Why? Because when it was stopped, the oil
pump was not turning, thus there was no oil pressure to
lubricate the turbine. The low-oil-pressure switch detected
this condition and the control system maintained the turbine
in shutdown mode, preventing it from starting. This isa
classic example of a design flaw, and it could only be
corrected by a change in the system logic.
While most design flaws manifest themselves early in the
operational life of the system, some remain hidden until just
the right conditions exist to trigger the fault. These types of
flaws are the most difficult to uncover, as the troubleshooter
usually overlooks the possibility of design error due to the
fact that the system is assumed to be "proven." The example
of the turbine lubrication system was a design flaw
impossible to ignore on start-up. An example of a "hidden"
design flaw might be a faulty emergency coolant system for
a machine, designed to remain inactive until certain
abnormal conditions are reached -- conditions which might
never be experienced in the life of the system.
Potential pitfalls
Fallacious reasoning and poor interpersonal relations
account for more failed or belabored troubleshooting efforts
than any other impediments. With this in mind, the aspiring
troubleshooter needs to be familiar with a few common
troubleshooting mistakes.
Trusting that a brand-new component will always be
good. While it is generally true that a new component will
be in good condition, it is not a/ways true. It is also possible
that a component has been mis-labeled and may have the
wrong value (usually this mis-labeling is a mistake made at
the point of distribution or warehousing and not at the
manufacturer, but again, not always!).
Not periodically checking your test equipment. This is
especially true with battery-powered meters, as weak
batteries may give spurious readings. When using meters to
safety-check for dangerous voltage, remember to test the
meter on a known source of voltage both before and after
checking the circuit to be serviced, to make sure the meter
is in proper operating condition.
Assuming there is only one failure to account for the
problem. Single-failure system problems are ideal for
troubleshooting, but sometimes failures come in multiple
numbers. In some instances, the failure of one component
may lead to a system condition that damages other
components. Sometimes a component in marginal condition
goes undetected for a long time, then when another
component fails the system suffers from problems with both
components.
Mistaking coincidence for causality. Just because two
events occurred at nearly the same time does not
necessarily mean one event caused the other! They may be
both consequences of a common cause, or they may be
totally unrelated! If possible, try to duplicate the same
condition suspected to be the cause and see if the event
suspected to be the coincidence happens again. If not, then
there is either no causal relationship as assumed. This may
mean there is no causal relationship between the two events
whatsoever, or that there is a causal relationship, but just
not the one you expected.
Self-induced blindness. After a long effort at
troubleshooting a difficult problem, you may become tired
and begin to overlook crucial clues to the problem. Take a
break and let someone else look at it for a while. You will be
amazed at what a difference this can make. On the other
hand, it is generally a bad idea to solicit help at the start of
the troubleshooting process. Effective troubleshooting
involves complex, multi-level thinking, which is not easily
communicated with others. More often than not, "team
troubleshooting" takes more time and causes more
frustration than doing it yourself. An exception to this rule is
when the knowledge of the troubleshooters is
complementary: for example, a technician who knows
electronics but not machine operation, teamed with an
operator who knows machine function but not electronics.
Failing to question the troubleshooting work of
others on the same job. This may sound rather cynical
and misanthropic, but it is sound scientific practice. Because
it is easy to overlook important details, troubleshooting data
received from another troubleshooter should be personally
verified before proceeding. This is a common situation when
troubleshooters "change shifts" and a technician takes over
for another technician who is leaving before the job is done.
It is important to exchange information, but do not assume
the prior technician checked everything they said they did,
or checked it perfectly. I've been hindered in my
troubleshooting efforts on many occasions by failing to
verify what someone else told me they checked.
Being pressured to "hurry up." When an important
system fails, there will be pressure from other people to fix
the problem as quickly as possible. As they say in business,
"time is money." Having been on the receiving end of this
pressure many times, | can understand the need for
expedience. However, in many cases there is a higher
priority: caution. If the system in question harbors great
danger to life and limb, the pressure to "hurry up" may
result in injury or death. At the very least, hasty repairs may
result in further damage when the system is restarted. Most
failures can be recovered or at least temporarily repaired in
short time if approached intelligently. Improper "fixes"
resulting in haste often lead to damage that cannot be
recovered in short time, if ever. If the potential for greater
harm is present, the troubleshooter needs to politely address
the pressure received from others, and maintain their
perspective in the midst of chaos. Interpersonal skills are
just as important in this realm as technical ability!
Finger-pointing. It is all too easy to blame a problem on
someone else, for reasons of ignorance, pride, laziness, or
some other unfortunate facet of human nature. When the
responsibility for system maintenance is divided into
departments or work crews, troubleshooting efforts are often
hindered by blame cast between groups. "It's a mechanical
problem... its an electrical problem... its an instrument
problem..." ad infinitum, ad nauseum, is all too common in
the workplace. | have found that a positive attitude does
more to quench the fires of blame than anything else.
On one particular job, | was summoned to fix a problem ina
hydraulic system assumed to be related to the electronic
metering and controls. My troubleshooting isolated the
source of trouble to a faulty control valve, which was the
domain of the millwright (mechanical) crew. | knew that the
millwright on shift was a contentious person, so | expected
trouble if | simply passed the problem on to his department.
Instead, | politely explained to him and his supervisor the
nature of the problem as well as a brief synopsis of my
reasoning, then proceeded to help him replace the faulty
valve, even though it wasn't "my" responsibility to do so. As
a result, the problem was fixed very quickly, and | gained
the respect of the millwright.
Contributors
Contributors to this chapter are listed in chronological order
of their contributions, from most recent to first. See
Appendix 2 (Contributor List) for dates and contact
information.
Alejandro Gamero Divasto (January 2002): contributed
troubleshooting tips regarding potential hazards of
Swapping two similar components, avoiding pressure placed
on the troubleshooter, perils of "team" troubleshooting,
wisdom of recording system history, operator error as a
cause of failure, and the perils of finger-pointing.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
Next
—
nts
E¢
—_
—/ | 4]
Lessons In Electric Circuits
-- Volume V
Chapter 9
CIRCUIT SCHEMATIC
SYMBOLS
Wires and connections
Power sources
Resistors
Capacitors
Inductors
Mutual inductors
Switches, hand actuated
Switches, process actuated
Switches, electrically actuated (relays)
Connectors
Diodes
Transistors, bipolar
Transistors, insulated-gate field-effect (IGFET or
MOSFET)
Transistors, hybrid
Thyristors
Integrated circuits
Electron tubes
Wires and connections
Older convention
Connected Not connected
Newer convention
Connected Not connected
oy
Older electrical schematics showed connecting wires
crossing, while non-connecting wires "jumped" over each
other with little half-circle marks. Newer electrical
schematics show connecting wires joining with a dot, while
non-connecting wires cross with no dot. However, some
people still use the older convention of connecting wires
crossing with no dot, which may create confusion.
For this reason, | opt to use a hybrid convention, with
connecting wires unambiguously connected by a dot, and
non-connecting wires unambiguously "jumping" over one
another with a half-circle mark. While this may be frowned
upon by some, it leaves no room for interpretational error: in
each case, the intent is clear and unmistakable:
Convention used in this book
Connected Not connected
+
Power sources
DC voltage DC voltage AC voltage
rf
Variable
DC voltage DC current
A diagonal arrow _
represents variability |
Tr for any component! z
Generator AC current
oO ©
Resistors
Fixed-value Rheostat
> 0 #
Potentiometer Tapped Thermistor
~ | = @
Photoresistor
6)
Capacitors
Non-polarized Polarized (top positive)
ye cai i: aie oe
7 +R Ft
Variable
Ho
Inductors
Fixed-value lron core
330 Ol 4
Variable Variac Tapped
BF SF
Mutual tnductors
Step-up/step-down
Transformer transformer Variac
Saturable
Transformer Transformer Transformer reactor
ee ee ||
Synchro
4OF
Switches, hand actuated
Ree Ale ae ae
SPST toggle ol
normally open DPST toggle
_
—_e-s— ee. oe
SPST toggle |
normally closed
DPDT toggle
ee =
SPDT toggle SPST joystick
position of dot
» on circle indicates
—- — joystick direction
=5
Pushbutton
normally open
r
Pushbutton
normally closed
bah
4PDT toggle
Switches, process actuated
Normally open shown on top; normally closed on bottom
i: et nn Sn oo
Level Pressure Flow Temperature
~ Se -
A
Ns. , Y
Bo Electronic M
Limit Limit Speed
—_e—=>— E
nan ‘
—_+12—
> a.
1s
|
R
It is very important to keep in mind that the "normal"
contact status of a process-actuated switch refers to its
status when the process is absent and/or inactive, not
"normal" in the sense of process conditions as expected
during routine operation. For instance, a normally-closed
low-flow detection switch installed on a coolant pipe will be
maintained in the actuated state (open) when there is
regular coolant flow through the pipe. If the coolant flow
stops, the flow switch will go to its "normal" (unactuated)
status of closed.
A limit switch is one actuated by contact with a moving
machine part. An electronic limit switch senses mechanical
motion, but does so using light, magnetic fields, or other
non-contact means.
Switches, electrically actuated
(relays)
Relay components, "ladder logic” notation style
4+ VWeHy ©
Generic Electronic Relay coil, Relay coil,
| t electromechanical electronic
Relays, electronic schematic notation style
aa}
Connectors
—¥ —
Plug Jack
(male) — (female)
Receptacle Household
(female)
power
connectors
Plug
(male)
Diodes
Generic Schottky
A > K . —pf- i
Zener Light-emitting
A's
Tunnel Varactor
»—pl- K A IE K
A = Anode
K = Cathode
Plug
Shockley
—>»)>—
Plug & Jack
connected
Multi-conductor
plug/jack set
Jack
Constant current
DEX
Step recovery
sb «
Vacuum tube
P
Transistors, bipolar
... With case
Bipolar NPN Bipolar PNP
B B
ae - “3 * ay
Photo- ; ;
v Dual-emitter NPN Dual-emitter PNP
B
GY). eS ; o
E;
Darlington pair | E = Emitter Sziklai pair
; B = Base ;
C = Collector
c c
E E
Transistors, junction field-effect
(JFET)
N-channel P-channel ... with case
G G
S = Source
G = Gate
D = Drain
Transistors, insulated-gate field-
effect (IGFET or MOSFET)
N-channel P-channel N-channel P-channel
depletion depletion enhancement enhancement
Ee fee [ a
fb, yb. fl. HL.
ss ss ss ss
N-channel P-channel N-channel P-channel
depletion depletion enhancement enhancement
[ bs [ =
hs, gle Ges ete
S = Source ... with case
G = Gate
D = Drain
SS = Substrate
Transistors, hybrid
IGBT (NPN) IGBT (PNP)
ws ee cs
= s = 7
IGBT (N-channel) IGBT (P-channel)
— —
Be ie ie oe ie
E = Emitter
G = Gate
C = Collector
Thyristors
... With case
&
... with case
Shockley DIAC SCR LASCR
\
A aan K + A K A K
ay aay
TRIAC ai
MT a. MT, MT; 2S MT,
GTO UJT B, A = Anode
rs K = Cathode
Ms ~C G = Gate
MT = Main Terminal
E = Emitter
B = Base
Integrated circuits
Operational amplifier (alternative) Norton op-amp
1
Inverter AND gate OR gate XOR gate
Inverter NAND gate NOR gate XNOR gate
>
V
V
Negative-AND Negative-OR
Buffer gate gate
>
U
V
Gate with open- Gate with Schmitt
collector output trigger input
G
B
S-R Latch Enabled S-R Latch S-R Flip-flop
S Q S Q
E C
R Q R Q
D Latch D Flip-flop J-K Flip-flop
D Q D Q J Q
E C C
Q Q K Q
Electron tubes
Diode Glow tube Phototube
Pp Cc
eke
oO
=x
=
Triode Tetrode Beam tetrode
P P P
s
G G
C) ih &
c c c
H, H, H, H, H, H
Pentode Pentode Thyratron
P P P
ap
s G
G G
a es &
c HH, c Hy He c HH,
Ignitron Cathode Ray Tube
P = Plate S = Screen
G = Grid A = Anode
C = Cathode H = Heater
| = Ignitor Sup = Suppressor
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
— 4 —»
—|}|4/l—
Lessons In Electric Circuits
-- Volume V
Chapter 10
PERIODIC TABLE OF THE
ELEMENTS
e Table (landscape view)
e Table (portrait view)
e Data
Table (landscape view)
See Figure below.
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Periodic table of chemical elements.
Table (portrait view)
1 14
B wa
H 1 ee He 2
Hyon Periodic Table of the Elements Haun
1.co74 bie tonmaab 4.00080
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3/Be 4 a0 “iK 19 mn mer B a hi ed aioe oa eae: Hag v
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aa Qo12B2 20.0883 . mic mans bs ‘aon y= nl COST ‘aioe 1ae4 an rm
2' 2* 4s'
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No NiMg 12 contpun A SI 14 |P 16/cl 17 ate
Socum | Macnedui Mumntuim Silcon Prose Ssutur —
2207s | 3.20 Mots = 22.0ess 2208
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K pica 20 |Se aT 2\V 2\cr 24|Mn 25/Fe 23)Cco a|N @B\cu ain w\aa Ge oton As B/S 4 Br 35 |r %
Poassum| Cakum | Scandum) Ttartum | Vanacm) Chromium) Marganes@ = iron Cobalt Nickel Copper a satu Arseric | Sdertum | Bromine et
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eres 1BAOSS | 1011S | HORSES) 144.24 (145) 150.3 151.985 1S72S5 |158.00534) B25 | 164002) 16725 | BAG21|) 17.04 1.967
Sci'ea* 4t'Scl'ea* | 4fea* ret 4ec* 41*6a* 4fes* 4f Sci'8e* | 4%60* at" Ga* at" ea* at" * 4f*0* ae | 4tsal'ea*
20/Th @ | Pa o1ju 92 |Np 9 )Pu 4) Am Sicm /Bk or icr Es | Fm 100|/Md =101)/No 102}Lr 108
Acinpe = Adium | Thom am) Wrarium | Nepunum) Plutium|4mekium| Curum |Beretum (Calforium Eretemum) Fermium |ttendete Nebelum | Lawrencam
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cf Ts* ens Pet Ts* | Beef ts* | Stet | Stee rs*| sree rs* | Sf eefTs* | aPecPrs® | a! Gefrs* | a! eefro* | Sr eePTs* | Sr ecPTs* | efm* 6d'7s*
Periodic table of chemical elements.
Data
Atomic masses shown in parentheses indicate the most
stable isotope (longest half-life) known.
Electron configuration data was taken from Douglas C.
Giancoli's Physics, 3rd edition. Average atomic masses were
taken from Kenneth W. Whitten's, Kenneth D. Gailey's, and
Raymond E. Davis' General Chemistry, 3rd edition. In the
latter book, the masses were specified as 1985 IUPAC values.
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—|| +4]
— 4 —
Appendix 1
ABOUT THIS BOOK
Purpose
They say that necessity is the mother of invention. At least
in the case of this book, that adage is true. As an industrial
electronics instructor, | was forced to use a sub-standard
textbook during my first year of teaching. My students were
daily frustrated with the many typographical errors and
obscure explanations in this book, having spent much time
at home struggling to comprehend the material within.
Worse yet were the many incorrect answers in the back of
the book to selected problems. Adding insult to injury was
the $100+ price.
Contacting the publisher proved to be an exercise in futility.
Even though the particular text | was using had been in
print and in popular use for a couple of years, they claimed
my complaint was the first they'd ever heard. My request to
review the draft for the next edition of their book was met
with disinterest on their part, and | resolved to find an
alternative text.
Finding a Suitable alternative was more difficult than | had
imagined. Sure, there were plenty of texts in print, but the
really good books seemed a bit too heavy on the math and
the less intimidating books omitted a lot of information | felt
was important. Some of the best books were out of print, and
those that were still being printed were quite expensive.
It was out of frustration that | compiled Lessons in Electric
Circuits from notes and ideas | had been collecting for years.
My primary goal was to put readable, high-quality
information into the hands of my students, but a secondary
goal was to make the book as affordable as possible. Over
the years, | had experienced the benefit of receiving free
instruction and encouragement in my pursuit of learning
electronics from many people, including several teachers of
mine in elementary and high school. Their selfless
assistance played a key role in my own studies, paving the
way for a rewarding career and fascinating hobby. If only |
could extend the gift of their help by giving to other people
what they gavetome...
So, | decided to make the book freely available. More than
that, | decided to make it "open," following the same
development model used in the making of free software
(most notably the various UNIX utilities released by the Free
Software Foundation, and the Linux operating system,
whose fame Is growing even as | write). The goal was to
copyright the text -- so as to protect my authorship -- but
expressly allow anyone to distribute and/or modify the text
to suit their own needs with a minimum of legal
encumbrance. This willful and formal revoking of standard
distribution limitations under copyright is whimsically
termed copyleft. Anyone can "copyleft" their creative work
simply by appending a notice to that effect on their work,
but several Licenses already exist, covering the fine legal
points in great detail.
The first such License | applied to my work was the GPL --
General Public License -- of the Free Software Foundation
(GNU). The GPL, however, is intended to copyleft works of
computer software, and although its introductory language
is broad enough to cover works of text, its wording is not as
clear as it could be for that application. When other, less
specific copyleft Licenses began appearing within the free
software community, | chose one of them (the Design
Science License, or DSL) as the official notice for my project.
In "copylefting" this text, | guaranteed that no instructor
would be limited by a text insufficient for their needs, as |
had been with error-ridden textbooks from major publishers.
I'm sure this book in its initial form will not satisfy everyone,
but anyone has the freedom to change it, leveraging my
efforts to suit variant and individual requirements. For the
beginning student of electronics, learn what you can from
this book, editing it as you feel necessary if you come across
a useful piece of information. Then, if you pass it on to
someone else, you will be giving them something better
than what you received. For the instructor or electronics
professional, feel free to use this as a reference manual,
adding or editing to your heart's content. The only "catch" is
this: if you plan to distribute your modified version of this
text, you must give credit where credit is due (to me, the
Original author, and anyone else whose modifications are
contained in your version), and you must ensure that
whoever you give the text to is aware of their freedom to
similarly share and edit the text. The next chapter covers
this process in more detail.
It must be mentioned that although | strive to maintain
technical accuracy in all of this book's content, the subject
matter is broad and harbors many potential dangers.
Electricity maims and kills without provocation, and
deserves the utmost respect. | strongly encourage
experimentation on the part of the reader, but only with
circuits powered by small batteries where there is no risk of
electric shock, fire, explosion, etc. High-power electric
circuits should be left to the care of trained professionals!
The Design Science License clearly states that neither | nor
any contributors to this book bear any liability for what is
done with its contents.
The use of SPICE
One of the best ways to learn how things work is to follow
the inductive approach: to observe specific instances of
things working and derive general conclusions from those
observations. In science education, labwork is the
traditionally accepted venue for this type of learning,
although in many cases labs are designed by educators to
reinforce principles previously learned through lecture or
textbook reading, rather than to allow the student to learn
on their own through a truly exploratory process.
Having taught myself most of the electronics that | know, |
appreciate the sense of frustration students may have in
teaching themselves from books. Although electronic
components are typically inexpensive, not everyone has the
means or opportunity to set up a laboratory in their own
homes, and when things go wrong there's no one to ask for
help. Most textbooks seem to approach the task of education
from a deductive perspective: tell the student how things
are supposed to work, then apply those principles to specific
instances that the student may or may not be able to
explore by themselves. The inductive approach, as useful as
it is, is hard to find in the pages of a book.
However, textbooks don't have to be this way. | discovered
this when | started to learn a computer program called
SPICE. It is a text-based piece of software intended to model
circuits and provide analyses of voltage, current, frequency,
etc. Although nothing is quite as good as building real
circuits to gain knowledge in electronics, computer
simulation is an excellent alternative. In learning how to use
this powerful tool, | made a discovery: SPICE could be used
within a textbook to present circuit simulations to allow
students to "observe" the phenomena for themselves. This
way, the readers could learn the concepts inductively (by
interpreting SPICE's output) as well as deductively (by
interpreting my explanations). Furthermore, in seeing SPICE
used over and over again, they should be able to
understand how to use it themselves, providing a perfectly
safe means of experimentation on their own computers with
circuit simulations of their own design.
Another advantage to including computer analyses in a
textbook is the empirical verification it adds to the concepts
presented. Without demonstrations, the reader is left to take
the author's statements on faith, trusting that what has
been written is indeed accurate. The problem with faith, of
course, is that it is only as good as the authority in which it
is placed and the accuracy of interpretation through which it
is understood. Authors, like all human beings, are liable to
err and/or communicate poorly. With demonstrations,
however, the reader can immediately see for themselves
that what the author describes is indeed true.
Demonstrations also serve to clarify the meaning of the text
with concrete examples.
SPICE is introduced early in volume | (DC) of this book
series, and hopefully in a gentle enough way that it doesn't
create confusion. For those wishing to learn more, a chapter
in this volume (volume V) contains an overview of SPICE
with many example circuits. There may be more flashy
(graphic) circuit simulation programs in existence, but SPICE
is free, a virtue complementing the charitable philosophy of
this book very nicely.
Acknowledgements
First, | wish to thank my wife, whose patience during those
many and long evenings (and weekends!) of typing has
been extraordinary.
| also wish to thank those whose open-source software
development efforts have made this endeavor all the more
affordable and pleasurable. The following is a list of various
free computer software used to make this book, and the
respective programmers:
e GNU/Linux Operating System -- Linus Torvalds, Richard
Stallman, and a host of others too numerous to mention.
e Vim text editor -- Bram Moolenaar and others.
Xcircuit drafting program -- Tim Edwards.
SPICE circuit simulation program -- too many
contributors to mention.
e T-X text processing system -- Donald Knuth and others.
e Texinfo document formatting system -- Free Software
Foundation.
¢ LATEX document formatting system -- Leslie Lamport and
others.
e Gimp image manipulation program -- too many
contributors to mention.
Appreciation is also extended to Robert L. Boylestad, whose
first edition of Introductory Circuit Analysis taught me more
about electric circuits than any other book. Other important
texts in my electronics studies include the 1939 edition of
The "Radio" Handbook, Bernard Grob's second edition of
Introduction to Electronics I, and Forrest Mims' original
Engineer's Notebook.
Thanks to the staff of the Bellingham Antique Radio
Museum, who were generous enough to let me terrorize their
establishment with my camera and flash unit.
| wish to specifically thank Jeffrey Elkner and all those at
Yorktown High School for being willing to host my book as
part of their Open Book Project, and to make the first effort
in contributing to its form and content. Thanks also to David
Sweet (website: [*]) and Ben Crowell (website: [*]) for
providing encouragement, constructive criticism, and a
wider audience for the online version of this book.
Thanks to Michael Stutz for drafting his Design Science
License, and to Richard Stallman for pioneering the concept
of copyleft.
Last but certainly not least, many thanks to my parents and
those teachers of mine who saw in me a desire to learn
about electricity, and who kindled that flame into a passion
for discovery and intellectual adventure. | honor you by
helping others as you have helped me.
Tony Kuphaldt, July 2001
"A candle loses nothing of its light when lighting
another"
Kahlil Gibran
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R.
Kuphaldt, under the terms and conditions of the Design
Science License.
=—||4]l_—
—| | +]
Appendix 2
CONTRIBUTOR LIST
How to contribute to this book
As a copylefted work, this book is open to revision and expansion by
any interested parties. The only "catch" is that credit must be given
where credit is due. This /s a copyrighted work: it is notin the public
domain!
If you wish to cite portions of this book in a work of your own, you
must follow the same guidelines as for any other copyrighted work.
Here is a Sample from the Design Science License:
The Work is copyright the Author. All rights to the Work are reserved
by the Author, except as specifically described below. This License
describes the terms and conditions under which the Author permits you
to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright Law.
Your right to operate, perform, read or otherwise interpret and/or
execute the Work is unrestricted; however, you do so at your own risk,
because the Work comes WITHOUT ANY WARRANTY -- see Section 7 ("NO
WARRANTY") below.
If you wish to modify this book in any way, you must document the
nature of those modifications in the "Credits" section along with your
name, and ideally, information concerning how you may be
contacted. Again, the Design Science License:
Permission is granted to modify or sample from a copy of the Work,
producing a derivative work, and to distribute the derivative work
under the terms described in the section for distribution above,
provided that the following terms are met:
(a) The new, derivative work is published under the terms of this
License.
(b) The derivative work is given a new name, so that its name or
title can not be confused with the Work, or with a version of
the Work, in any way.
(c) Appropriate authorship credit is given: for the differences
between the Work and the new derivative work, authorship is
attributed to you, while the material sampled or used from
the Work remains attributed to the original Author; appropriate
notice must be included with the new work indicating the nature
and the dates of any modifications of the Work made by you.
Given the complexities and security issues surrounding the
maintenance of files comprising this book, it is recommended that
you submit any revisions or expansions to the original author (Tony R.
Kuphaldt). You are, of course, welcome to modify this book directly by
editing your own personal copy, but we would all stand to benefit
from your contributions if your ideas were incorporated into the
online “master copy” where all the world can see it.
Credits
All entries arranged in alphabetical order of surname. Major
contributions are listed by individual name with some detail on the
nature of the contribution(s), date, contact info, etc. Minor
contributions (typo corrections, etc.) are listed by name only for
reasons of brevity. Please understand that when | classify a
contribution as “minor,” it is in no way inferior to the effort or value of
a “major” contribution, just smaller in the sense of less text changed.
Any and all contributions are gratefully accepted. | am indebted to all
those who have given freely of their own knowledge, time, and
resources to make this a better book!
Dennis Crunkilton
« Date(s) of contribution(s):October 2005 to present
e Nature of contribution:Ch 1, added permitivity, capacitor and
inductor formulas, wire table; 10/2005.
e Nature of contribution:Ch 1, expanded dielectric table,
10232.eps, copied data from Volume 1, Chapter 13; 10/2005.
¢« Nature of contribution: Mini table of contents, all chapters
except appedicies; html, latex, ps, pdf; See Devel/tutorial.hAtmI;
01/2006.
¢ Nature of contribution: Changed CH2 from “Resistor color
codes” to “Color codes”; Added wiring color codes; 10/2007.
¢ Contact at: dcrunkilton(at)att(dot)net
Alejandro Gamero Divasto
« Date(s) of contribution(s): January 2002
e Nature of contribution: Suggestions related to
troubleshooting: caveat regarding swapping two similar
components as a troubleshooting tool; avoiding pressure placed
on the troubleshooter; perils of "team" troubleshooting; wisdom of
recording system history; operator error as a cause of failure; and
the perils of finger-pointing.
Tony R. Kuphaldt
Date(s) of contribution(s): 1996 to present
Nature of contribution: Original author.
Contact at: liec0@lycos.com
Your name here
Date(s) of contribution(s): Month and year of contribution
Nature of contribution: Insert text here, describing how you
contributed to the book.
Contact at: my email@provider.net
Typo corrections and other “minor” contributions
The students of Bellingham Technical College's Instrumentation
program.
Bernard Sheehan (January 2005), Typographical error correction
in "Right triangle trigonometry" section Chapter 5:
TRIGONOMETRY REFERENCE (two formulas for tan x the second
one reads tan x = cos x/sin x it Should be cot x = cos x/sin x-
changes to 01001.eps previously made)
Michiel van Bolhuis (April 2007) Typo Ch 1,
s/picofards/picofarads.
Chirvasuta Constantin (April 2003) Identified error in quadratic
equation formula.
Colin Creitz (May 2007) Chapters: several, s/it's/its.
Jeff DeFreitas (March 2006)Improve appearance: replace “/" and
"/" Chapters: Al, A2.
Gerald Gardner (January 2003) Suggested adding Imperial
gallons conversion to table.
Geoff Hosking (July 2006) Typo correction in Conductors and
Insulators chapter, Critical Temperatures of Superconductors:
s/degrees Kelvin/Kelvins.
Harvey Lew (??? 2003) Typo correction in Trig chapter:
"tangent" should have been "cotangent".
Len Nunn (May 2008) Typo correction in Calculus chapter:
"dx/d(a*x)" in error, 11042.png .
Don Stalkowski (June 2002) Technical help with PostScript-to-
PDF file format conversion.
¢ Joseph Teichman (June 2002) Suggestion and technical help
regarding use of PNG images instead of JPEG.
¢ Mark44@allaboutcircuits.com (March 2008) Ch 4, Clarification
of division by zero.
¢ Timothy Unregistered@allaboutcircuits.com (Feb 2008)
Changed default roman font to newcent.
Imranullah Syed (Feb 2008) Suggested centering of
uncaptioned schematics.
e Unregistered@allaboutcircuits.com (Aug 2008) formatting of
PDF off pps 130-136.
e D Crunkilton (Dec 2009) added missing images 10232.eps
10233.eps 10238.eps 10239.eps 10241.eps
« webbie@allaboutcircuits.com (Aug 2010) Chl,
S/usefull/useful/.
e D. Crunkilton (June 2011) hi.latex, header file; updated link to
openbookproject.net .
« NRG@allaboutcircuits.com (May 2012) Ch 2, s/are coded
are/are coded/ .
¢ RobinGriffiths@allaboutcircuits.com (May 2012) Ch1,
images renumbered 1023[12389].png 10241.png and source text
to not conflict with volume 2 image numbers.
e DC (Feb 2020) Ch 3, Reformatted text tables to html/latex .
Lessons In Electric Circuits copyright (C) 2000-2020 Tony R. Kuphaldt,
under the terms and conditions of the Design Science License.
—|/]|+4|l\—
—/ | 4]
Appendix 3
DESIGN SCIENCE LICENSE
Copyright © 1999-2000 Michael Stutz stutz@dsl.org
Verbatim copying of this document is permitted, in any
medium.
0. Preamble
Copyright law gives certain exclusive rights to the author of
a work, including the rights to copy, modify and distribute
the work (the "reproductive," "adaptative," and
"distribution" rights).
The idea of "copyleft" is to willfully revoke the exclusivity of
those rights under certain terms and conditions, so that
anyone can copy and distribute the work or properly
attributed derivative works, while all copies remain under
the same terms and conditions as the original.
The intent of this license is to be a general "copyleft" that
can be applied to any kind of work that has protection under
copyright. This license states those certain conditions under
which a work published under its terms may be copied,
distributed, and modified.
Whereas "design science" is a strategy for the development
of artifacts as a way to reform the environment (not people)
and subsequently improve the universal standard of living,
this Design Science License was written and deployed as a
strategy for promoting the progress of science and art
through reform of the environment.
1. Definitions
"License" shall mean this Design Science License. The
License applies to any work which contains a notice placed
by the work's copyright holder stating that it is published
under the terms of this Design Science License.
"Work" shall mean such an aforementioned work. The
License also applies to the output of the Work, only if said
output constitutes a "derivative work" of the licensed Work
as defined by copyright law.
“Object Form" shall mean an executable or performable form
of the Work, being an embodiment of the Work in some
tangible medium.
"Source Data" shall mean the origin of the Object Form,
being the entire, machine-readable, preferred form of the
Work for copying and for human modification (usually the
language, encoding or format in which composed or
recorded by the Author); plus any accompanying files,
scripts or other data necessary for installation, configuration
or compilation of the Work.
(Examples of "Source Data" include, but are not limited to,
the following: if the Work is an image file composed and
edited in 'PNG' format, then the original PNG source file is
the Source Data; if the Work is an MPEG 1.0 layer 3 digital
audio recording made from a 'WAV' format audio file
recording of an analog source, then the original WAV file is
the Source Data; if the Work was composed as an
unformatted plaintext file, then that file is the the Source
Data; if the Work was composed in LaTex, the LaTeX file(s)
and any image files and/or custom macros necessary for
compilation constitute the Source Data.)
"Author" shall mean the copyright holder(s) of the Work.
The individual licensees are referred to as "you."
2. Rights and copyright
The Work is copyright the Author. All rights to the Work are
reserved by the Author, except as specifically described
below. This License describes the terms and conditions
under which the Author permits you to copy, distribute and
modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright law.
Your right to operate, perform, read or otherwise interpret
and/or execute the Work is unrestricted; however, you do so
at your own risk, because the Work comes WITHOUT ANY
WARRANTY -- see Section 7 ("NO WARRANTY") below.
3. Copying and distribution
Permission is granted to distribute, publish or otherwise
present verbatim copies of the entire Source Data of the
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and disclaimer of warranty, where applicable, is
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Permission is granted to distribute, publish or otherwise
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medium, under the terms for distribution of Source Data
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(a) The Source Data is included in the same distribution,
distributed under the terms of this License; or
(bo) A written offer is included with the distribution, valid for
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print (whichever is longer), with a publicly-accessible
address (such as a URL on the Internet) where, for a charge
not greater than transportation and media costs, anyone
may receive a copy of the Source Data of the Work
distributed according to the section above; or
(c) A third party's written offer for obtaining the Source Data
at no cost, as described in paragraph (b) above, is included
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non-commercial party, and only if you received the Object
Form of the Work along with such an offer.
You may copy and distribute the Work either gratis or for a
fee, and if desired, you may offer warranty protection for the
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The aggregation of the Work with other works which are not
based on the Work -- such as but not limited to inclusion ina
publication, broadcast, compilation, or other media -- does
not bring the other works in the scope of the License; nor
does such aggregation void the terms of the License for the
Work.
4. Modification
Permission is granted to modify or sample from a copy of the
Work, producing a derivative work, and to distribute the
derivative work under the terms described in the section for
distribution above, provided that the following terms are
met:
(a) The new, derivative work is published under the terms of
this License.
(ob) The derivative work is given a new name, so that its
name or title can not be confused with the Work, or with a
version of the Work, in any way.
(c) Appropriate authorship credit is given: for the differences
between the Work and the new derivative work, authorship
is attributed to you, while the material sampled or used from
the Work remains attributed to the original Author;
appropriate notice must be included with the new work
indicating the nature and the dates of any modifications of
the Work made by you.
5. No restrictions
You may not impose any further restrictions on the Work or
any of its derivative works beyond those restrictions
described in this License.
6. Acceptance
Copying, distributing or modifying the Work (including but
not limited to sampling from the Work in a new work)
indicates acceptance of these terms. If you do not follow the
terms of this License, any rights granted to you by the
License are null and void. The copying, distribution or
modification of the Work outside of the terms described in
this License is expressly prohibited by law.
If for any reason, conditions are imposed on you that forbid
you to fulfill the conditions of this License, you may not
copy, distribute or modify the Work at all.
If any part of this License is found to be in conflict with the
law, that part shall be interpreted in its broadest meaning
consistent with the law, and no other parts of the License
Shall be affected.
7. No warranty
THE WORK IS PROVIDED "AS IS," AND COMES WITH
ABSOLUTELY NO WARRANTY, EXPRESS OR IMPLIED, TO THE
EXTENT PERMITTED BY APPLICABLE LAW, INCLUDING BUT
NOT LIMITED TO THE IMPLIED WARRANTIES OF
MERCHANTABILITY OR FITNESS FOR A PARTICULAR
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8. Disclaimer of liability
IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE
LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE
GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR
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THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR
OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
WORK, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH
DAMAGE.
END OF TERMS AND CONDITIONS
[ $Id: dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]
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